The concentration of potassium
\(\text{K}^+\) ions in the internal sap of a plant cell (for example,
a fresh water alga) may exceed by a factor of \(10^4\) the
concentration of \(\text{K}^+\) ions in the pond water in which the
cell is growing. The chemical potential of the \(\text{K}^+\) ions is
higher in the sap because their concentration \(n\) is higher there.
Estimate the difference in chemical potential at \(300\text{K}\) and
show that it is equivalent to a voltage of \(0.24\text{V}\) across the
cell wall. Take \(\mu\) as for an ideal gas. Because the values of the
chemical potential are different, the ions in the cell and in the pond
are not in diffusive equilibrium. The plant cell membrane is highly
impermeable to the passive leakage of ions through it. Important
questions in cell physics include these: How is the high concentration
of ions built up within the cell? How is metabolic energy applied to
energize the active ion transport?
David adds
You might wonder why it is even remotely plausible to consider the
ions in solution as an ideal gas. The key idea here is that the ideal
gas entropy incorporates the entropy due to position dependence, and
thus due to concentration. Since concentration is what differs between
the cell and the pond, the ideal gas entropy describes this pretty
effectively. In contrast to the concentration dependence, the
temperature-dependence of the ideal gas chemical potential will not be
so great.
A diesel
engine requires no spark plug. Rather, the air in the cylinder is
compressed so highly that the fuel ignites spontaneously when
sprayed into the cylinder.
In this problem, you may treat air as an ideal gas, which satisfies
the equation \(pV = Nk_BT\). You may also use the property of an
ideal gas that the internal energy depends only on the temperature
\(T\), i.e. the internal energy does not change for an isothermal
process. For
air at the relevant range of temperatures the heat capacity at
fixed volume is given by \(C_V=\frac52Nk_B\), which means the internal energy
is given by \(U=\frac52Nk_BT\).
Note: in this problem you are expected to use only the equations
given and fundamental physics laws. Looking up the formula in a textbook
is not considered a solution at this level.
If the air is initially at room temperature
(taken as \(20^{o}C\)) and is then compressed adiabatically to \(\frac1{15}\)
of the original volume, what final temperature is attained (before fuel
injection)?
(Messy algebra) Purpose: Convince yourself that the expressions for kinetic energy in original and center of mass coordinates are equivalent. The same for angular momentum.
Consider a system of two particles of mass \(m_1\) and \(m_2\).
Show that the total kinetic energy of the system is the same as that of two
“fictitious” particles: one of mass \(M=m_1+m_2\) moving with the velocity of the
center of mass and one of mass \(\mu\) (the reduced mass) moving with the
velocity of the relative position.
Show that the total angular momentum of the system can similarly be decomposed
into the angular momenta of these two fictitious particles.
Find a set of functions that approximate the delta function \(\delta(x-a)\) with a sequence of isosceles triangles \(\delta_{\epsilon}(x-a)\), centered at \(a\), that get narrower and taller as the parameter \(\epsilon\) approaches zero.
Using the test function \(f(x)=3x^2\), find the value of
\[\int_{-\infty}^{\infty} f(x)\delta_{\epsilon}(x-a)\, dx\]
Then, show that the integral approaches \(f(a)\) in the limit that \(\epsilon \rightarrow 0\).
Static Fields 2023 (4 years)
Determine the following derivatives and evaluate the following integrals, all by hand. You should also learn how to check these answers on Wolfram Alpha.
EnergyEntropyBoltzmann probabilitiesThermal and Statistical Physics 2020 (3 years)
Consider a three-state system with
energies \((-\epsilon,0,\epsilon)\).
At infinite temperature, what are the probabilities of the three
states being occupied? What is the internal energy \(U\)? What is
the entropy \(S\)?
At very low temperature, what are the three probabilities?
What are the three probabilities at zero temperature? What is
the internal energy \(U\)? What is the entropy
\(S\)?
What happens to the probabilities if you allow the temperature
to be negative?
The internal energy of helium gas at temperature \(T\) is
to a very good approximation given by
\begin{align}
U &= \frac32 Nk_BT
\end{align}
Consider a very irreversible process in which a small bottle of
helium is placed inside a large bottle, which otherwise contains
vacuum. The inner bottle contains a slow leak, so that the helium
leaks into the outer bottle. The inner bottle contains one tenth
the volume of the outer bottle, which is insulated. What is the
change in temperature when this process is complete? How much of the
helium will remain in the small bottle?
Consider the bottle in a bottle problem in a previous problem set, summarized here.
A small bottle of
helium is placed inside a large bottle, which otherwise contains
vacuum. The inner bottle contains a slow leak, so that the helium
leaks into the outer bottle. The inner bottle contains one tenth
the volume of the outer bottle, which is insulated.
The volume of the small bottle is 0.001 m23 and the volume of the big bottle is 0.01 m3. The initial state of the gas in the small bottle was \(p=106\) Pa and its temperature \(T=300\) K. Approximate the helium gas as an ideal gas of equations of state \(pV=Nk_BT\) and \(U=\frac32 Nk_BT\).
How many molecules of gas does the large bottle contain? What is the final temperature of the gas?
Compute the integral \(\int \frac{{\mathit{\unicode{273}}} Q}{T}\) and the change of entropy \(\Delta S\) between the initial state (gas in the small bottle) and the final state (gas leaked in the big bottle).
PDMEnergy and Entropy 2021 (2 years)
In your kits for the Portable Partial Derivative Machine should be the following:
A 1ft by 1ft board with 5 holes and measuring tapes (the measuring tapes will be on the top side)
2 S-hooks
A spring with 3 strings attached
2 small cloth bags
4 large ball bearings
8 small ball bearings
2 vertical clamp pulleys
A ziploc bag containing
5 screws
5 hex nuts
5 washers
5 wing nuts
2 horizontal pulleys
To assemble the Portable PDM, start by placing the PDM on a table surface with the measuring tapes perpendicular to the table's edge and the board edge with 3 holes closest to you.
one screw should be put through each hole so that the threads stick out through the top side of the board. Next use a hex nut to secure each screw in place. It is not critical that they be screwed on any more than you can comfortably manage by hand.
After securing all 5 screws in place with a hex nut, put a washer on each screw.
Slide a horizontal pulley onto screws 1 and 2 (as labeled above).
On all 5 screws, add a wing nut to secure the other pieces. Again, it does not need to be tightened all the way as long as it is secure enough that nothing will fall off.
Using the middle wingnut/washer/screw (Screw 4), clamp the shortest of the strings tied to the spring.
Loop the remaining 2 looped-ends of string around the horizontal pulleys and along the measuring tape.
Using the string as a guide, clamp the vertical pulleys into place on the edge of the board.
Through the looped-end of each string, place 1 S-hook.
Put the other end of each s-hook through the hole in the small cloth bag.
Here is a poor photo of the final result, which doesn't show the two vertical pulleys. If you would like, you could view a video of the building process.
Clausius-ClapeyronThermal and Statistical Physics 2020
Calculate based on the Clausius-Clapeyron equation the value of
\(\frac{dT}{dp}\) near \(p=1\text{atm}\) for the liquid-vapor
equilibrium of water. The heat of vaporization at
\(100^\circ\text{C}\) is \(2260\text{ J g}^{-1}\). Express the result
in kelvin/atm.
In carbon monoxide
poisoning the CO replaces the \(\textsf{O}_{2}\) adsorbed on
hemoglobin (\(\text{Hb}\)) molecules in the blood. To show the effect,
consider a model for which each adsorption site on a heme may be
vacant or may be occupied either with energy \(\varepsilon_A\) by one
molecule \(\textsf{O}_{2}\) or with energy \(\varepsilon_B\) by one
molecule CO. Let \(N\) fixed heme sites be in equilibrium with
\(\textsf{O}_{2}\) and CO in the gas phases at concentrations such
that the activities are \(\lambda(\text{O}_2) = 1\times 10^{-5}\) and
\(\lambda(\text{CO}) = 1\times 10^{-7}\), all at body temperature
\(37^\circ\text{C}\). Neglect any spin multiplicity factors.
First consider the system in the absence of CO. Evaluate
\(\varepsilon_A\) such that 90 percent of the \(\text{Hb}\) sites
are occupied by \(\textsf{O}_{2}\). Express the answer in eV per
\(\textsf{O}_{2}\).
Now admit the CO under the specified conditions. Fine
\(\varepsilon_B\) such that only 10% of the Hb sites are occupied
by \(\textsf{O}_{2}\).
(Straightforward) Purpose: Discover that a system of two masses can be a central force system even when they are not interacting at all. Practice with center-of-mass coordinates.
Consider two particles of equal mass \(m\). The forces on the
particles are \(\vec F_1=0\) and \(\vec F_2=F_0\hat{x}\). If the
particles are initially at rest at the origin, find the position,
velocity, and acceleration of the center of mass as functions of
time. Solve this problem in two ways,
with theorems about the center of mass motion,
without theorems
about the center of mass motion.
Write a short description
comparing the two solutions.
(Quick) Purpose: Recognize the definition of a central force. Build experience about which common physical situations represent central forces and which don't.
Which of the following forces can be central forces? which cannot? If the force CAN be a central force, explain the circumstances that would allow it to be a central force.
The force on a test mass \(m\) in a gravitational field \(\vec{g~}\),
i.e. \(m\vec g\)
The force on a test charge \(q\) in an electric field \(\vec E\), i.e.
\(q\vec E\)
The force on a test charge \(q\) moving at velocity \(\vec{v~}\) in a
magnetic field \(\vec B\), i.e. \(q\vec v \times \vec B\)
Centrifugal potentialThermal and Statistical Physics 2020
A circular cylinder of radius \(R\)
rotates about the long axis with angular velocity \(\omega\). The
cylinder contains an ideal gas of atoms of mass \(M\) at temperature
\(T\). Find an expression for the dependence of the concentration
\(n(r)\) on the radial distance \(r\) from the axis, in terms of
\(n(0)\) on the axis. Take \(\mu\) as for an ideal gas.
Static Fields 2023 (3 years)
A charged spiral in the \(x,y\)-plane has 6 turns from the origin out to a maximum radius \(R\) , with \(\phi\) increasing proportionally to the distance from the center of the spiral. Charge is distributed on the spiral so that the charge density increases linearly as the radial distance from the center increases. At the center of the spiral the linear charge density is
\(0~\frac{\textrm{C}}{\textrm{m}}\). At the end of the spiral, the linear charge
density is \(13~\frac{\textrm{C}}{\textrm{m}}\). What is the total charge on the
spiral?
Find the rectangular coordinates of the point where the angle \(\frac{5\pi}{3}\) meets the unit circle. If this were a point in the complex plane, what would be the rectangular and exponential forms of the complex number? (See figure.)
On the following diagrams, mark both \(\theta\) and \(\sin\theta\) for \(\theta_1=\frac{5\pi}{6}\) and \(\theta_2=\frac{7\pi}{6}\). Write one to three sentences about how these two representations are related to each other. (For example, see: this PHET)
In economics, the term utility is roughly related to overall
happiness. Many things affect your happiness, including the amount of
money you have and the amount of coffee you drink. We cannot directly
measure your happiness, but we can measure how much money you
are willing to give up in order to obtain coffee or bagels. If we
assume you choose wisely, we can thus determine that your happiness
increases when you decrease your amount of money by that amount in
exchange for increasing your coffee consumption. Thus money is a
(poor) measure of happiness or utility.
Money is also a nice quantity because it is conserved---just like
energy! You may gain or lose money, but you always do so by a
transaction. (There are some exceptions to the conservation of money,
but they involve either the Fed, counterfeiters, or destruction of
cash money, and we will ignore those issues.)
In this problem, we will assume that you have bought all the coffee
and bagels you want (and no more), so that your happiness has been
maximized. Thus you are in equilibrium with the coffee shop. We will
assume further that you remain in equilibrium with the coffee shop at
all times, and that you can sell coffee and bagels back to the coffee
shop at cost.*
Thus your savings \(S\) can be considered to be a function of your
bagels \(B\) and coffee \(C\). In this problem we will also discuss the
prices \(P_B\) and \(P_C\), which you may not assume are
independent of \(B\) and \(C\).
It may help to imagine that you could possibly buy out the local supply of coffee,
and have to import it at higher costs.
The prices of bagels and coffee \(P_B\) and \(P_C\) have derivative
relationships between your savings and the quantity of coffee and
bagels that you have. What are the units of these prices? What is
the mathematical definition of \(P_C\) and \(P_B\)?
Write down the total differential of your savings, in terms of
\(B\), \(C\), \(P_B\) and \(P_C\).
Solve for the total differential of your net worth. Your net worth
\(W\) is the sum of your total savings plus the value of the coffee and
bagels that you own. From the total differential, relate your amount of
coffee and bagels to partial derivatives of your net worth.
Given the polar basis kets written as a superposition of Cartesian kets
\begin{eqnarray*}
\left|{\hat{s}}\right\rangle &=& \cos\phi \left|{\hat{x}}\right\rangle + \sin\phi \left|{\hat{y}}\right\rangle \\
\left|{\hat{\phi}}\right\rangle &=& -\sin\phi \left|{\hat{x}}\right\rangle + \cos\phi \left|{\hat{y}}\right\rangle
\end{eqnarray*}
Find the following quantities:
\[\left\langle {\hat{x}}\middle|{\hat{s}}\right\rangle ,\quad \left\langle {\hat{y}}\middle|{{\hat{s}}}\right\rangle ,\quad \left\langle {\hat{x}}\middle|{\hat{\phi}}\right\rangle ,\quad \left\langle {\hat{y}}\middle|{\hat{\phi}}\right\rangle \]
Given a vector written in the polar basis
\[\left|{\vec{v}}\right\rangle = a\left|{\hat{s}}\right\rangle + b\left|{\hat{\phi}}\right\rangle \]
where \(a\) and \(b\) are known. Find coefficients \(c\) and \(d\) such that
\[\left|{\vec{v}}\right\rangle = c\left|{\hat{x}}\right\rangle + d\left|{\hat{y}}\right\rangle \]
Do this by using the completeness relation:
\[\left|{\hat{x}}\right\rangle \left\langle {\hat{x}}\right| + \left|{\hat{y}}\right\rangle \left\langle {\hat{y}}\right| = 1\]
Using a completeness relation, change the basis of the spin-1/2 state
\[\left|{\Psi}\right\rangle = g\left|{+}\right\rangle + h\left|{-}\right\rangle \]
into the \(S_y\) basis. In otherwords, find \(j\) and \(k\) such that
\[\left|{\Psi}\right\rangle = j\left|{+}\right\rangle _y + k\left|{-}\right\rangle _y\]
Find \(dA\) on the surface of an (open) cone in both cylindrical and spherical coordinates. Hint: Be smart about how you coordinatize the cone.
Using integration, find the surface area of an (open) cone with height \(H\) and radius \(R\). Do this problem in both cylindrical and spherical coordinates.
Shown below is a contour plot of a scalar field, \(\mu(x,y)\). Assume that \(x\)
and \(y\) are measured in meters and that \(\mu\) is measured in kilograms.
Four points are indicated on the plot.
Determine \(\frac{\partial\mu}{\partial x}\) and \(\frac{\partial\mu}{\partial
y}\) at each of the four points.
On a printout of the figure, draw a qualitatively accurate vector at each point corresponding to the
gradient of \(\mu(x,y)\) using your answers to part a above. How did you choose
a scale for your vectors? Describe how the direction of the gradient vector is
related to the contours on the plot and what property of the contour map is
related to the magnitude of the gradient vector.
Evaluate the gradient of
\(h(x,y)=(x+1)^2\left(\frac{x}{2}-\frac{y}{3}\right)^3\)
at the
point \((x,y)=(3,-2)\).
Use the cross product to find the components of the unit vector
\(\mathbf{\boldsymbol{\hat n}}\) perpendicular to the plane shown in the figure below, i.e.
the plane joining the points \(\{(1,0,0),(0,1,0),(0,0,1)\}\).
Charge is distributed throughout the volume of a dielectric cube with
charge density \(\rho=\beta z^2\), where \(z\) is the height from the
bottom of the cube, and where each side of the cube has length \(L\).
What is the total charge inside the cube? Do this problem in two ways as both
a single integral and as a triple integral.
On a different cube: Charge is distributed on the surface of a cube with charge density \(\sigma=\alpha z\) where \(z\) is the height from the bottom of the cube, and where each side of the cube has length \(L\). What is the total charge on the cube? Don't forget about the top
and bottom of the cube.
Shown above is a two-dimensional cross-section of a vector field. All the parallel cross-sections of this field look exactly the same.
Determine the direction of the curl at points A, B, and C.
A solid cylinder with radius \(R\) and height \(H\) has its
base on the \(x,y\)-plane and is
symmetric around the \(z\)-axis. There is a fixed volume charge density
on the cylinder \(\rho=\alpha z\). If the cylinder is spinning with period \(T\):
Static Fields 2023 (4 years)
The current density in a cylindrical wire of radius \(R\) is given by
\(\vec{J}(\vec{r})=\alpha s^3\cos^2\phi\,\hat{z}\). Find the total current in the wire.
Fermi-Dirac functionThermal and Statistical Physics 2020Derivative of Fermi-Dirac function Show that the magnitude of the slope of the Fermi-Direc function \(f\) evaluated at the Fermi
level \(\varepsilon =\mu\) is inversely proportional to its temperature. This means that at lower temperatures the Fermi-Dirac
function becomes dramatically steeper.
Energy and Entropy 2021 (2 years)
Use the NIST web site “Thermophysical Properties of Fluid Systems” to answer the following questions. This site is an excellent resource for finding experimentally measured properties of fluids.
Find the partial derivatives
\[\left(\frac{\partial {S}}{\partial {T}}\right)_{p}\]
\[\left(\frac{\partial {S}}{\partial {T}}\right)_{V}\]
where \(p\) is the pressure, \(V\) is the volume, \(S\) is the entropy, and \(T\) is the temperature. Please find these derivatives for one gram of methanol at one atmosphere of pressure and at room temperature.
Why does it take only two variables to define the state?
Why are the derivatives above different?
What do the words isobaric, isothermal, and isochoric mean?
At low temperatures, a diatomic molecule can be well described as a
rigid rotor. The Hamiltonian of such a system is simply
proportional to the square of the angular momentum
\begin{align}
H &= \frac{1}{2I}L^2
\end{align}
and the energy eigenvalues are
\begin{align}
E_{\ell m} &= \hbar^2 \frac{\ell(\ell+1)}{2I}
\end{align}
What is the energy of the ground state and the first and
second excited states of the \(H_2\) molecule? i.e. the lowest three distinct energy eigenvalues.
At room temperature, what is the relative probability of
finding a hydrogen molecule in the \(\ell=0\) state versus finding it
in any one of the \(\ell=1\) states? i.e. what is
\(P_{\ell=0,m=0}/\left(P_{\ell=1,m=-1} + P_{\ell=1,m=0} + P_{\ell=1,m=1}\right)\)
At what temperature is the value of this ratio 1?
At room temperature, what is the probability of
finding a hydrogen molecule in any one of the \(\ell=2\) states versus
that of finding it in the ground state? i.e. what is
\(P_{\ell=0,m=0}/\left(P_{\ell=2,m=-2} + P_{\ell=2,m=-1} + \cdots + P_{\ell=2,m=2}\right)\)
For an infinitesimally thin cylindrical shell of radius \(b\) with uniform surface
charge density \(\sigma\), the electric field is zero for \(s<b\) and \(\vec{E}=
\frac{\sigma b}{\epsilon_0 s}\, \hat s\) for \(s > b\). Use the differential form
of Gauss' Law to find the charge density everywhere in space.
\(\left\langle {\Psi}\middle|{\Psi}\right\rangle =1\) Identify and discuss the dimensions of \(\left|{\Psi}\right\rangle \).
For a spin \(\frac{1}{2}\) system, \(\left\langle {\Psi}\middle|{+}\right\rangle \left\langle {+}\middle|{\Psi}\right\rangle + \left\langle {\Psi}\middle|{-}\right\rangle \left\langle {-}\middle|{\Psi}\right\rangle =1\). Identify and discuss the dimensions of \(\left|{+}\right\rangle \) and \(\left|{-}\right\rangle \).
In the position basis \(\int \left\langle {\Psi}\middle|{x}\right\rangle \left\langle {x}\middle|{\Psi}\right\rangle dx = 1\). Identify and discuss the dimesions of \(\left|{x}\right\rangle \).
You are on a hike. The altitude nearby is
described by the function \(f(x, y)= k x^{2}y\), where \(k=20 \mathrm{\frac{m}{km^3}}\)
is a constant, \(x\) and \(y\) are east and north coordinates,
respectively, with units of kilometers. You're standing at the spot
\((3~\mathrm{km},2~\mathrm{km})\) and there is a cottage located at \((1~\mathrm{km}, 2~\mathrm{km})\). You drop your water bottle and the water spills out.
Plot the function \(f(x, y)\) and also its level curves in your favorite plotting software. Include images of these graphs.
Special note: If you use a computer program written by someone else, you must reference that appropriately.
In which direction in space does the water flow?
At the spot you're standing, what is the slope of the ground in the direction of the cottage?
Does your result to part (c) make sense from the graph?
The distance \(\left\vert\vec r -\vec r\,{}'\right\vert\) between
the point \(\vec r\) and the point
\(\vec r'\) is a coordinate-independent, physical and geometric quantity. But,
in practice, you will need to know how to express this quantity in different
coordinate systems.
Find the distance \(\left\vert\vec r -\vec r\,{}'\right\vert\) between
the point \(\vec r\) and the point \(\vec
r'\) in rectangular coordinates.
Show that this same distance written in cylindrical coordinates is:
\begin{equation}
\left|\vec r -\vec r\,{}'\right| =\sqrt{s^2+s\,{}'^2-2ss\,{}'\cos(\phi-\phi\,{}') +(z-z\,{}')^2}
\end{equation}
Show that this same distance written in spherical coordinates is:
\begin{equation}
\left\vert\vec r -\vec r\,{}'\right\vert
=\sqrt{r'^2+r\,{}^2-2rr\,{}'
\left[\sin\theta\sin\theta\,{}'\cos(\phi-\phi\,{}')
+\cos\theta\cos\theta\,{}'\right]}
\end{equation}
Now assume that \(\vec r\,{}'\) and \(\vec r\) are in the \(x\)-\(y\) plane. Simplify
the previous two formulas.
Let us imagine a new mechanics in which the allowed occupancies
of an orbital are 0, 1, and 2. The values of the energy associated
with these occupancies are assumed to be \(0\), \(\varepsilon\), and
\(2\varepsilon\), respectively.
Derive an expression for the ensemble average occupancy
\(\langle N\rangle\), when the system composed of this orbital is in
thermal and diffusive contact with a resevoir at temperature \(T\)
and chemical potential \(\mu\).
Return now to the usual quantum mechanics, and derive an expression
for the ensemble average occupancy of an energy level which is
doubly degenerate; that is, two orbitals have the identical energy
\(\varepsilon\). If both orbitals are occupied the toal energy is
\(2\varepsilon\). How does this differ from part (a)?
Consider the vector field \(\vec F=(x+2)\hat{x} +(z+2)\hat{z}\).
Calculate the divergence of \(\vec F\).
In which direction does the vector field \(\vec F\) point on the plane
\(z=x\)? What is the value of \(\vec F\cdot \hat n\) on this plane
where \(\hat n\) is the unit normal to the plane?
Verify the divergence theorem for this vector field where the volume
involved is drawn below. (“Verify” means calculate both sides of the divergence theorem, separately, for this example and show that they are the same.)
Consider a mass \(\mu\)
in the potential shown in the graph below.
You give the
mass a push so that its initial
angular momentum is \(\ell\ne 0\) for a given fixed value of \(\ell\).
Give the definition of a central force system and briefly explain why
this situation qualifies.
Make a sketch of the graph of the effective potential for this situation.
How
should you push the puck to establish a circular orbit?
(i.e. Characterize the initial position, direction of push, and strength of the
push. You do NOT need to solve any equations.)
BRIEFLY discuss the possible orbit shapes that can arise from this effective
potential. Include a discussion of whether the orbits are open or closed,
bound or unbound, etc.
Make sure that you refer to your sketch of the effective potential
in your discussions, mark any points of physical significance on the sketch,
and describe the range of parameters relevant to each type of orbit.
Include a discussion of the role of the total energy of the orbit.
Einstein condensation temperature Starting
from the density of free particle orbitals per unit energy range
\begin{align}
\mathcal{D}(\varepsilon) =
\frac{V}{4\pi^2}\left(\frac{2M}{\hbar^2}\right)^{\frac32}\varepsilon^{\frac12}
\end{align} show that the lowest temperature at which the total number
of atoms in excited states is equal to the total number of atoms is
\begin{align}
T_E &=
\frac1{k_B}
\frac{\hbar^2}{2M}
\left(
\frac{N}{V}
\frac{4\pi^2}{\int_0^\infty\frac{\sqrt{\xi}}{e^\xi-1}d\xi}
\right)^{\frac23}
T_E &=
\end{align} The infinite sum may be numerically evaluated to be 2.612.
Note that the number derived by integrating over the density of
states, since the density of states includes all the states
except the ground state.
Note: This problem is solved in the text itself. I intend to
discuss Bose-Einstein condensation in class, but will not derive this
result.
Static Fields 2023 (5 years)
Consider a thin charged rod of length \(L\) standing along the \(z\)-axis
with the bottom end on the \(xy\)-plane.
The charge density \(\lambda\) is constant.
Find the electric field at the point \((0,0,2L)\).
Consider the finite line with a uniform charge density from class.
Write an integral expression for the electric field at any point in space due
to the finite line. In addition to your usual physics sense-making, you must
include a clearly labeled figure and discuss what happens to the direction of
the unit vectors as you integrate.Consider the finite line with a uniform
charge density from class.
Perform the integral to find the \(z\)-component of the electric field. In
addition to your usual physics sense-making, you must compare your result to
the gradient of the electric potential we found in class. (If you want to
challenge yourself, do the \(s\)-component as well!)
energyBoltzmann factorstatistical mechanicsheat capacityThermal and Statistical Physics 2020
Consider a system of
fixed volume in thermal contact with a resevoir. Show that the mean
square fluctuations in the energy of the system is \begin{equation}
\left<\left(\varepsilon-\langle\varepsilon\rangle\right)^2\right>
= k_BT^2\left(\frac{\partial U}{\partial T}\right)_{V}
\end{equation} Here \(U\) is the conventional symbol for
\(\langle\varepsilon\rangle\). Hint: Use the partition function
\(Z\) to relate \(\left(\frac{\partial U}{\partial T}\right)_V\) to
the mean square fluctuation. Also, multiply out the term
\((\cdots)^2\).
For electrons with an energy \(\varepsilon\gg mc^2\), where
\(m\) is the mass of the electron, the energy is given by
\(\varepsilon\approx pc\) where \(p\) is the momentum. For electrons
in a cube of volume \(V=L^3\) the momentum takes the same values as
for a non-relativistic particle in a box.
Show that in this extreme relativistic limit the Fermi energy of a
gas of \(N\) electrons is given by \begin{align}
\varepsilon_F &= \hbar\pi c\left(\frac{3n}{\pi}\right)^{\frac13}
\end{align} where \(n\equiv \frac{N}{V}\) is the number density.
Show that the total energy of the ground state of the gas is
\begin{align}
U_0 &= \frac34 N\varepsilon_F
\end{align}
The goal of this problem is
to show that once we have maximized the entropy and found the
microstate probabilities in terms of a Lagrange multiplier \(\beta\),
we can prove that \(\beta=\frac1{kT}\) based on the statistical
definitions of energy and entropy and the thermodynamic definition
of temperature embodied in the thermodynamic identity.
The internal energy and
entropy are each defined as a weighted average over microstates:
\begin{align}
U &= \sum_i E_i P_i & S &= -k_B\sum_i P_i \ln P_i
\end{align}:
We saw in clase that the probability of each microstate can be given
in terms of a Lagrange multiplier \(\beta\) as
\begin{align}
P_i &= \frac{e^{-\beta E_i}}{Z}
&
Z &= \sum_i e^{-\beta E_i}
\end{align}
Put these probabilities into the above weighted averages in
order to relate \(U\) and \(S\) to \(\beta\). Then make use of the
thermodynamic identity
\begin{align}
dU = TdS - pdV
\end{align}
to show that \(\beta = \frac1{kT}\).
The goal of this problem
is to show that once we have maximized the entropy and found the
microstate probabilities in terms of a Lagrange multiplier \(\beta\),
we can prove that \(\beta=\frac1{kT}\) based on the statistical
definitions of energy and entropy and the thermodynamic definition of
temperature embodied in the thermodynamic identity.
The internal energy and entropy are each defined as a weighted average
over microstates: \begin{align}
U &= \sum_i E_i P_i & S &= -k_B\sum_i P_i \ln P_i
\end{align} We saw in clase that the probability of each microstate
can be given in terms of a Lagrange multiplier \(\beta\) as
\begin{align}
P_i &= \frac{e^{-\beta E_i}}{Z}
&
Z &= \sum_i e^{-\beta E_i}
\end{align} Put these probabilities into the above weighted averages
in order to relate \(U\) and \(S\) to \(\beta\). Then make use of the
thermodynamic identity \begin{align}
dU = TdS - pdV
\end{align} to show that \(\beta = \frac1{kT}\).
Suppose that a
system of \(N\) atoms of type \(A\) is placed in diffusive contact
with a system of \(N\) atoms of type \(B\) at the same temperature and
volume.
Show that after diffusive equilibrium is reached the total entropy
is increased by \(2Nk\ln 2\). The entropy increase \(2Nk\ln 2\) is
known as the entropy of mixing.
If the atoms are identical (\(A=B\)), show that there is no increase
in entropy when diffusive contact is established. The difference has
been called the Gibbs paradox.
Since the Helmholtz free energy is lower for the mixed \(AB\) than
for the separated \(A\) and \(B\), it should be possible to extract
work from the mixing process. Construct a process that could extract
work as the two gasses are mixed at fixed temperature. You will
probably need to use walls that are permeable to one gas but not the
other.
Note
This course has not yet covered work, but it was covered in
Energy and Entropy, so you may need to stretch your memory to finish
part (c).
Make sure that you have memorized the following identities and can use them in simple algebra problems:
\begin{align}
e^{u+v}&=e^u \, e^v\\
\ln{uv}&=\ln{u}+\ln{v}\\
u^v&=e^{v\ln{u}}
\end{align}
Consider a system which has an internal energy \(U\) defined by:
\begin{align}
U &= \gamma V^\alpha S^\beta
\end{align}
where \(\alpha\), \(\beta\) and \(\gamma\) are constants. The internal
energy is an extensive quantity. What constraint does this place on
the values \(\alpha\) and \(\beta\) may have?
Start with \(d\vec{r}\) in rectangular, cylindrical, and spherical
coordinates. Use these expressions to write the scalar area elements \(dA\) (for different coordinate equals constant surfaces) and the volume element \(d\tau\). It might help you to think of the following surfaces: The various sides of a rectangular box, a finite cylinder with a top and a bottom, a half cylinder, and a hemisphere with both a curved and a flat side, and a cone.
(Use the equation for orbit shape.) Gain experience with unusual force laws.
In science fiction movies, characters often talk about a spaceship “spiralling in” right before it hits the planet. But all orbits in a \(1/r^2\) force are conic sections, not spirals. This spiralling in happens because the spaceship hits atmosphere and the drag from the atmosphere changes the shape of the orbit. But, in an alternate universe, we might have other force laws.
Find the force law for a mass \(\mu\), under the influence of a central-force field, that moves in a logarithmic spiral orbit given by \(r = ke^{\alpha \phi}\), where \(k\) and \(\alpha\) are constants.
Fermi gasgrand canonical ensemblestatistical mechanicsThermal and Statistical Physics 2020
(K&K 7.11) Show for a single
orbital of a fermion system that \begin{align}
\left<(\Delta N)^2\right> = \left<N\right>(1+\left<N\right>)
\end{align} if \(\left<N\right>\) is the average number of fermions in
that orbital. Notice that the fluctuation vanishes for orbitals with
energies far enough from the chemical potential \(\mu\) so that
\(\left<N\right>=1\) or \(\left<N\right>=0\).
Find the upward pointing flux of the electric field \(\vec E =E_0\,
z\, \hat z\) through the part of the surface \(z=-3 s^2 +12\)
(cylindrical coordinates) that sits above the \((x, y)\)--plane.
Static Fields 2023 (4 years)
Find the upward pointing flux of the vector field \(\boldsymbol{\vec{H}}=2z\,\boldsymbol{\hat{x}}
+\frac{1}{x^2+1}\boldsymbol{\hat{y}}+(3+2z)\boldsymbol{\hat{z}}\) through the rectangle \(R\) with one edge along the \(y\) axis and the other in the \(xz\)-plane along the line \(z=x\), with \(0\le y\le2\) and \(0\le x\le3\).
Find the Fourier transforms
of \(f(x)=\cos kx\) and \(g(x)=\sin kx\).
Find the Fourier transform of \(g(x)\) using the formula for the Fourier transform of a derivative and your result for the Fourier transform of \(f(x)\). Compare with your previous answer.
In quantum mechanics, the Fourier transform is the set of coefficients in the expansion of a quantum state in terms of plane waves, i.e. the function \(\tilde{f}(k)\) is a continuous histogram of how much each functions \(e^{ikx}\) contributes to the quantum state. What does the Fourier transform of the function \(\cos kx\) tell you about which plane waves make up this quantum state? Write a sentence or two about how this makes sense.
A one-dimensional
harmonic oscillator has an infinite series of equally spaced energy
states, with \(\varepsilon_n = n\hbar\omega\), where \(n\) is an
integer \(\ge 0\), and \(\omega\) is the classical frequency of the
oscillator. We have chosen the zero of energy at the state \(n=0\)
which we can get away with here, but is not actually the zero of
energy! To find the true energy we would have to add a
\(\frac12\hbar\omega\) for each oscillator.
Show that for a harmonic oscillator the free energy is
\begin{equation}
F = k_BT\log\left(1 - e^{-\frac{\hbar\omega}{k_BT}}\right)
\end{equation} Note that at high temperatures such that
\(k_BT\gg\hbar\omega\) we may expand the argument of the logarithm
to obtain \(F\approx k_BT\log\left(\frac{\hbar\omega}{kT}\right)\).
From the free energy above, show that the entropy is
\begin{equation}
\frac{S}{k_B} =
\frac{\frac{\hbar\omega}{kT}}{e^{\frac{\hbar\omega}{kT}}-1}
- \log\left(1-e^{-\frac{\hbar\omega}{kT}}\right)
\end{equation}
Entropy of a simple harmonic oscillatorHeat capacity of a simple harmonic oscillator
This entropy is shown in the nearby figure, as well
as the heat capacity.
The internal energy is of any ideal gas can be written as
\begin{align}
U &= U(T,N)
\end{align}
meaning that the internal energy depends only on the number of
particles and the temperature, but not the volume.* The ideal gas law
\begin{align}
pV &= Nk_BT
\end{align}
defines the relationship between \(p\), \(V\) and \(T\). You may take the
number of molecules \(N\) to be constant. Consider the free adiabatic
expansion of an ideal gas to twice its volume. “Free expansion”
means that no work is done, but also that the process is also
neither quasistatic nor reversible.
What is the change in entropy of the gas? How do you know
this?
Quantum MechanicsTime EvolutionSpin PrecessionExpectation ValueBohr FrequencyQuantum Fundamentals 2023 (3 years)
Consider a two-state quantum system with a Hamiltonian
\begin{equation}
\hat{H}\doteq
\begin{pmatrix}
E_1&0\\ 0&E_2
\end{pmatrix}
\end{equation}
Another physical observable \(M\) is described by the operator
\begin{equation}
\hat{M}\doteq
\begin{pmatrix}
0&c\\ c&0
\end{pmatrix}
\end{equation}
where \(c\) is real and positive. Let the initial state of the system be \(\left|{\psi(0)}\right\rangle
=\left|{m_1}\right\rangle \), where \(\left|{m_1}\right\rangle \) is the eigenstate corresponding to the larger of the two possible eigenvalues of \(\hat{M}\). What is the frequency of oscillation of the expectation value of \(M\)? This frequency is the Bohr frequency.
Static Fields 2023 (4 years)
Consider a thin charged rod of length \(L\) standing along the \(z\)-axis
with the bottom end on the \(x,y\)-plane.
The charge density \(\lambda_0\) is constant.
Find the total flux of the electric field through a closed cubical
surface with sides of length \(3L\) centered at the origin.
Give the general solution of the differential equation:
\[\frac{d^2 y}{dx^2}+Ay=0\]
Make sure that you can give the solution of this equation regardless of the geometric
names of the dependent and independent variables and for either
sign for the constant \(A\).
It is NOT necessary to show any work. You
may NOT, however, give a solution that has a negative number inside a square root.
I am testing whether you can recognize
this equation and remember its solution. This equation comes up over and over again
in physics, but disguised by different symbols. I am also testing whether you recognize that
the geometric character of the equation changes depending on the sign of \(A\).
Consider two noninteracting systems \(A\)
and \(B\). We can either treat these systems as separate, or as a
single combined system \(AB\). We can enumerate all states of the
combined by enumerating all states of each separate system. The
probability of the combined state \((i_A,j_B)\) is given by
\(P_{ij}^{AB} = P_i^AP_j^B\). In other words, the probabilities
combine in the same way as two dice rolls would, or the
probabilities of any other uncorrelated events.
Show that the entropy of the combined system \(S_{AB}\) is the
sum of entropies of the two separate systems considered
individually, i.e. \(S_{AB} = S_A+S_B\). This means that entropy is
extensive. Use the Gibbs entropy for this computation. You need
make no approximation in solving this problem.
Show that if you have \(N\) identical non-interacting systems,
their total entropy is \(NS_1\) where \(S_1\) is the entropy of a
single system.
Note
In real materials, we treat properties as being extensive even
when there are interactions in the system. In this case,
extensivity is a property of large systems, in which surface
effects may be neglected.
thermodynamicsMaxwell relationEnergy and Entropy 2020
The Gibbs free energy,
\(G\), is given by
\begin{align*}
G = U + pV - TS.
\end{align*}
Find the total differential of \(G\). As always, show your work.
Interpret the coefficients of the total differential \(dG\) in
order to find a derivative expression for the entropy \(S\).
From the total differential \(dG\), obtain a different
thermodynamic derivative that is equal to
\[ \left(\frac{\partial {S}}{\partial {p}}\right)_{T} \]
Consider a system that may be unoccupied with energy zero, or
occupied by one particle in either of two states, one of energy zero
and one of energy \(\varepsilon\). Find the Gibbs sum for this
system is in terms of the activity \(\lambda\equiv e^{\beta\mu}\).
Note that the system can hold a maximum of one particle.
Solve for the thermal average occupancy of the system in terms of
\(\lambda\).
Show that the thermal average occupancy of the state at energy
\(\varepsilon\) is \begin{align}
\langle N(\varepsilon)\rangle =
\frac{\lambda e^{-\frac{\varepsilon}{kT}}}{\mathcal{Z}}
\end{align}
Find an expression for the thermal average energy of the system.
Allow the possibility that the orbitals at \(0\) and at
\(\varepsilon\) may each be occupied each by one particle at the
same time; Show that \begin{align}
\mathcal{Z} &= 1 + \lambda + \lambda e^{-\frac{\varepsilon}{kT}} +
\lambda^2 e^{-\frac{\varepsilon}{kT}}
\\
&= (1+\lambda)\left(1+e^{-\frac{\varepsilon}{kT}}\right)
\end{align} Because \(\mathcal{Z}\) can be factored as shown, we
have in effect two independent systems.
Consider the fields at a point
\(\vec{r}\) due to a point charge located at \(\vec{r}'\).
Write down an expression for the electrostatic potential \(V(\vec{r})\) at a point
\(\vec{r}\) due to a point charge located at \(\vec{r}'\). (There is nothing to
calculate here.)
Write down an expression for the electric field \(\vec{E}(\vec{r})\) at a point
\(\vec{r}\) due to a point charge located at \(\vec{r}'\). (There is nothing to
calculate here.)
Working in rectangular coordinates, compute the gradient of
\(V\).
Write several sentences comparing your answers to the last two
questions.
Learn more about the geometry of \(\vert \vec{r}-\vec{r'}\vert\) in two dimensions.
Make sketches of the following functions, by hand, on the same
axes:
\begin{align}
y &= \sin x\\
y &= \sin(2+x)
\end{align}
Briefly describe the role that the number 2 plays in the
shape of the second graph
Make a sketch of the graph
\begin{equation}
\vert \vec{r} - \vec{a} \vert = 2
\end{equation}
for each of the following values of \(\vec a\):
\begin{align}
\vec a &= \vec 0\\
\vec a &= 2 \hat x- 3 \hat y\\
\vec a &= \text{points due east and is 2 units long}
\end{align}
Derive a more familiar equation equivalent to
\begin{equation}
\vert \vec r - \vec a \vert = 2
\end{equation}
for arbitrary \(\vec a\), by expanding \(\vec r\) and \(\vec a\) in
rectangular coordinates. Simplify as much as possible. (Ok, ok, I
know this is a terribly worded question. What do I mean by “more
familiar"? What do I mean by “simplify as much as possible"? Why am
I making you read my mind? Try it anyway. Real life is not full of
carefully worded problems. Bonus points to anyone who can figure
out a better way of wording the question that doesn't give the point
away.)
Write a brief description of the geometric meaning of the equation
\begin{equation}
\vert \vec r - \vec a \vert = 2
\end{equation}
The gravitational field due to a spherical shell of matter (or equivalently, the
electric field due to a spherical shell of charge) is given by:
\begin{equation}
\vec g =
\begin{cases}
0&\textrm{for } r<a\\
-G \,\frac{M}{b^3-a^3}\,
\left( r-\frac{a^3}{r^2}\right)\, \hat r & \textrm{for } a<r<b\\
-G\,\frac{M}{r^2}\, \hat r & \textrm{for } r>b \\
\end{cases}
\end{equation}
This problem explores the consequences of the divergence
theorem for this shell.
Using the given description of the gravitational field, find the divergence of the
gravitational field everywhere in space. You will need to divide this
question up into three parts: \(r<a\), \(a<r<b\), and \(r>b\).
Briefly discuss the physical meaning of the divergence in this particular
example.
For this gravitational field, verify the divergence theorem on a
sphere, concentric with the shell, with radius \(Q\), where \(a<Q<b\).
("Verify" the divergence theorem means calculate the integrals from both sides of the divergence theorem and show that they give the same answer.)
Briefly discuss how this example would change if you were discussing the
electric field of a uniformly charged spherical shell.
VaporizationHeatThermal and Statistical Physics 2020
The pressure
of water vapor over ice is 518 Pa at \(-2^\circ\text{C}\). The vapor
pressure of water at its triple point is 611 Pa, at
0.01\(^\circ\text{C}\) (see
Estimate in \(\text{J
mol}^{-1}\) the heat of vaporization of ice just under freezing. How
does this compare with the heat of vaporization of water?
Show that for a reversible heat pump the energy required per unit of
heat delivered inside the building is given by the Carnot
efficiency: \begin{align}
\frac{W}{Q_H} &= \eta_C = \frac{T_H-T_C}{T_H}
\end{align} What happens if the heat pump is not reversible?
Assume that the electricity consumed by a reversible heat pump must
itself be generated by a Carnot engine operating between the even
hotter temperature \(T_{HH}\) and the cold (outdoors) temperature
\(T_C\). What is the ratio \(\frac{Q_{HH}}{Q_H}\) of the heat
consumed at \(T_{HH}\) (i.e. fuel burned) to the heat delivered at
\(T_H\) (in the house we want to heat)? Give numerical values for
\(T_{HH}=600\text{K}\); \(T_{H}=300\text{K}\);
\(T_{C}=270\text{K}\).
Draw an energy-entropy flow diagram for the combination heat
engine-heat pump, similar to Figures 8.1, 8.2 and 8.4 in the text
(or the equivalent but sloppier) figures in the course notes.
However, in this case we will involve no external work at all, only
energy and entropy flows at three temperatures, since the work done
is all generated from heat.
Stefan-Boltzmannblackbody radiationThermal and Statistical Physics 2020
A black (nonreflective) sheet of metal at high
temperature \(T_h\) is parallel to a cold black sheet of metal at temperature
\(T_c\). Each sheet has an area \(A\) which is much greater than the distance between them. The sheets are in vacuum, so energy can only be transferred by radiation.
Solve for the net power transferred between the two sheets.
A third black metal sheet is
inserted between the other two and is allowed to come to a steady
state temperature \(T_m\). Find the temperature of the middle sheet,
and solve for the new net power transferred between the hot and cold sheets.
This is the principle of the heat shield, and is part of how the James Web telescope shield works.
Optional: Find the power through an \(N\)-layer sandwich.
A helix with 17 turns has height \(H\) and radius \(R\). Charge is distributed on the helix so that the charge density increases like (i.e. proportional to) the square of the distance up the helix.
At the bottom of the helix the linear charge density is
\(0~\frac{\textrm{C}}{\textrm{m}}\). At the top of the helix, the linear charge
density is \(13~\frac{\textrm{C}}{\textrm{m}}\). What is the total charge on the
helix?
(Synthesis Problem: Brings together several different concepts from this unit.) Use effective potential diagrams for other than \(1/r^2\) forces.
Consider the frictionless motion of a hockey puck of mass \(m\) on a perfectly
circular bowl-shaped ice rink with radius \(a\). The central region of the bowl
(\(r < 0.8a\)) is perfectly flat and the sides of the ice bowl smoothly rise to a
height \(h\) at \(r = a\).
Draw a sketch of the potential energy for this system. Set the zero of
potential energy at the top of the sides of the bowl.
Situation 1: the puck is initially moving radially outward from the
exact center of the rink. What minimum velocity does the puck need
to escape the rink?
Situation 2: a stationary puck, at a distance \(\frac{a}{2}\) from the
center of the rink, is hit in such a way that it's initial velocity
\(\vec v_0\) is perpendicular to its position vector as measured from
the center of the rink. What is the total energy of the puck
immediately after it is struck?
In situation 2, what is the angular momentum of the puck immediately after it is struck?
Draw a sketch of the effective potential for situation 2.
In situation 2, for what minimum value of \(\vec v_0\) does the puck
just escape the rink?
Homogeneous, linear ODEs with constant coefficients were likely covered in your Differential Equations course (MTH 256 or equiv.). If you need a review, please see:
Plot your data I
Plot the temperature versus total energy
added to the system (which you can call \(Q\)). To do this, you will
need to integrate the power. Discuss this curve and any interesting
features you notice on it.
Plot your data II
Plot the heat capacity versus
temperature. This will be a bit trickier. You can find the heat
capacity from the previous plot by looking at the slope.
\begin{align}
C_p &= \left(\frac{\partial Q}{\partial T}\right)_p
\end{align}
This is what is called the heat capacity, which is the amount
of energy needed to change the temperature by a given amount. The
\(p\) subscript means that your measurement was made at constant
pressure. This heat capacity is actually the total heat capacity of
everything you put in the calorimeter, which includes the resistor
and thermometer.
Specific heat From your plot of \(C_p(T)\), work out the
heat capacity per unit mass of water. You may assume the effect of
the resistor and thermometer are negligible. How does your answer
compare with the prediction of the Dulong-Petit law?
Latent heat of fusion
What did the temperature do while the ice was melting? How much
energy was required to melt the ice in your calorimeter? How much
energy was required per unit mass? per molecule?
Entropy of fusion The change in entropy is easy to measure for a
reversible isothermal process (such as the slow melting of ice),
it is just
\begin{align}
\Delta S &= \frac{Q}{T}
\end{align}
where \(Q\) is the energy thermally added to the system and \(T\) is
the temperature in Kelvin. What is was change in the entropy of
the ice you melted? What was the change in entropy per
molecule? What was the change in entropy per molecule divided
by Boltzmann's constant?
Entropy for a temperature change Choose two temperatures
that your water reached (after the ice melted), and find the change
in the entropy of your water. This change is given by
\begin{align}
\Delta S &= \int \frac{{\mathit{\unicode{273}}} Q}{T} \\
&= \int_{t_i}^{t_f} \frac{P(t)}{T(t)}dt
\end{align}
where \(P(t)\) is the heater power as a function of time and \(T(t)\) is
the temperature, also as a function of time.
Consider one mole of an
ideal monatomic gas at 300K and 1 atm. First, let the gas expand
isothermally and reversibly to twice the initial volume; second, let
this be followed by an isentropic expansion from twice to four times
the original volume.
How much heat (in joules) is added to the gas in each of these two
processes?
What is the temperature at the end of the second process?
Suppose the first process is replaced by an irreversible expansion
into a vacuum, to a total volume twice the initial volume. What is
the increase of entropy in the irreversible expansion, in J/K?
Inhomogeneous, linear ODEs with constant coefficients are among the most straigtforward to solve, although the algebra can get messy. This content should have been covered in your Differential Equations course (MTH 256 or equiv.). If you need a review, please see:
The Method for Inhomogeneous Equations
or your differential equations text.
For the following inhomogeneous linear equation with constant coefficients, find the general solution for \(y(x)\).
The general solution of the homogeneous differential equation
\[\ddot{x}-\dot{x}-6 x=0\]
is
\[x(t)=A\, e^{3t}+ B\, e^{-2t}\]
where \(A\) and \(B\) are arbitrary constants that would be determined by the initial conditions of the problem.
Find a particular solution of the inhomogeneous differential equation
\(\ddot{x}-\dot{x}-6 x=-25\sin(4 t)\).
Find the general solution of \(\ddot{x}-\dot{x}-6 x=-25\sin(4 t)\).
Some terms in your general solution have an undetermined coefficients, while some coefficients are fully determined. Explain what is different about these two cases.
Find a particular solution of \(\ddot{x}-\dot{x}-6 x=12 e^{-3 t}\)
Find the general solution of \(\ddot{x}-\dot{x}-6 x=12 e^{-3 t}-25\sin(4 t)\)
How is this general solution related to the particular solutions you found in the previous parts of this question?
Can you add these particular solutions together with arbitrary coefficients to get a new particular solution?
Sense-making: Check your answer; Explicitly plug in your final answer in part (e) and check that it satisfies the differential equation.
None 2023
The properties that an inner product on an abstract vector space must satisfy can be found in:
Definition and Properties of an Inner Product.
Definition: The inner product for any two vectors in the vector space of
periodic functions with a given period (let's pick \(2\pi\) for simplicity)
is given by:
\[\left\langle {f}\middle|{g}\right\rangle =\int_0^{2\pi} f^*(x)\, g(x)\, dx\]
Show that the first property of inner products
\[\left\langle {f}\middle|{g}\right\rangle =\left\langle {g}\middle|{f}\right\rangle ^*\]
is satisfied for this definition.
Show that the second property of inner products
\[\left\langle {f}\right|\Big(\lambda\left|{g}\right\rangle + \mu \left|{h}\right\rangle \Big) = \lambda\left\langle {f}\middle|{g}\right\rangle +\mu\left\langle {f}\middle|{h}\right\rangle \]
is satisfied for this definition.
In class, you measured the isolength
stretchability and the isoforce stretchability of your systems in the
PDM. We found that for some systems these were very
different, while for others they were identical.
Show with algebra (NOT experiment) that the ratio of isolength stretchability to isoforce
stretchability is the same for both the left-hand side of the system and the right-hand side of the system.
i.e.:
\begin{align}
\frac{\left(\frac{\partial {x_L}}{\partial {F_L}}\right)_{x_R}}{\left(\frac{\partial {x_L}}{\partial {F_L}}\right)_{F_R}} &=
\frac{\left(\frac{\partial {x_R}}{\partial {F_R}}\right)_{x_L}}{\left(\frac{\partial {x_R}}{\partial {F_R}}\right)_{F_L}}
\label{eq:ratios}
\end{align}
Hint
You will need to make use of the cyclic chain
rule:
\begin{align}
\left(\frac{\partial {A}}{\partial {B}}\right)_{C} = -\left(\frac{\partial {A}}{\partial {C}}\right)_{B}\left(\frac{\partial {C}}{\partial {B}}\right)_{A}
\end{align}
Hint
You will also need the ordinary chain
rule:
\begin{align}
\left(\frac{\partial {A}}{\partial {B}}\right)_{D} = \left(\frac{\partial {A}}{\partial {C}}\right)_{D}\left(\frac{\partial {C}}{\partial {B}}\right)_{D}
\end{align}
The isothermal
compressibility is defined as
\begin{equation}
K_{T}=-\frac{1}{V} \left(\frac{\partial V}{\partial p}\right)_{T}
\end{equation}
\(K_T\) is be found by measuring the fractional change in volume when
the the pressure is slightly changed with the temperature held
constant. In contrast, the adiabatic compressibility is defined as
\begin{equation}
K_{S}=-\frac{1}{V} \left(\frac{\partial V}{\partial p}\right)_{S}
\end{equation}
and is measured by making a slight change in pressure without
allowing for any heat transfer. This is the compressibility, for
instance, that would directly affect the speed of sound. Show that
\begin{equation}
\frac{K_{T}}{K_{S}} = \frac{C_{p}}{C_{V}}
\end{equation}
Where the heat capacities at constant pressure and volume are given
by
\begin{align}
C_{p} &= T \left(\frac{\partial S}{\partial T}\right)_{p} \\
C_{V} &= T \left(\frac{\partial S}{\partial T}\right)_{V}
\end{align}
A particle in an infinite square well potential has an initial state vector
\[\left|{\Psi(0)}\right\rangle = A\big(\left|{\phi_1}\right\rangle -\left|{\phi_2}\right\rangle +i\left|{\phi_3}\right\rangle \big)\]
where \(|\phi_n\rangle\) are the energy eigenstates. You have previously found \(\left|{\Psi(t)}\right\rangle \) for this state.
Use a computer to graph the wave function \(\Psi(x,t)\) and probability density \(\rho(x,t)\). Choose a few interesting values of \(t\) to include in your submission.
Use a computer to calculate the probability of measuring the particle to be near the middle of the well (within 1% on either side) as a function of time. Include both your symbolic result and a graph in your submission.
Choose another location in the well, different from the location above. Use a computer to calculate the probability of measuring the particle to be near your chosen location (within 1% on either side) as a function of time. Include both your symbolic result and a graph in your submission.
Are there any locations in the well where the probability is independent of time? Explain how you determined your answer.
The time dependence for a wave function like this is complicated. Write a lengthy description in words about the major features of this wave function and its probability density, how they change in time, and why they change the way they do. Comment on any interesting features you noticed that you have not already discussed in the questions above and describe any additional things you learned from the process of solving this problem.
Carnot refridgeratorWorkEntropyThermal and Statistical Physics 2020
A 100W light bulb is
left burning inside a Carnot refridgerator that draws 100W. Can the
refridgerator cool below room temperature?
Find the electric field around a finite, uniformly charged, straight
rod, at a point a distance \(s\) straight out from the midpoint,
starting from Coulomb's Law.
Find the electric field around an infinite, uniformly charged,
straight rod, starting from the result for a finite rod.
Find the electric field around an infinite, uniformly charged,
straight wire, starting from the following expression for the electrostatic
potential:
\begin{equation}
V(\vec r)=\frac{2\lambda}{4\pi\epsilon_0}\, \ln\left( \frac{ s_0}{s} \right)
\end{equation}
Consider a collection of three charges arranged in a line along the
\(z\)-axis: charges \(+Q\) at \(z=\pm D\) and charge \(-2Q\) at \(z=0\).
Find the electrostatic potential at a point \(\vec{r}\) in the \(xy\)-plane at a
distance \(s\) from the center of the quadrupole.
The formula for the electrostatic potential \(V\) at a point \(\vec{r}\) due to a charge \(Q\) at the point \(\vec{r'}\) is given by:
\[
V(\vec{r})=\frac{1}{4\pi\epsilon_0} \frac{Q}{\vert \vec{r}-\vec{r'}\vert}
\]
Electrostatic potentials satisfy the superposition principle.
Assume \(s\gg D\). Find the first two non-zero terms of a power
series expansion to the electrostatic potential you found in the
first part of this problem.
A series of charges arranged in this way is called a linear
quadrupole. Why?
Static Fields 2023 (4 years)
Consider a collection of three charges arranged in a line along the
\(z\)-axis: charges \(+Q\) at \(z=\pm D\) and charge \(-2Q\) at \(z=0\).
Find the electrostatic potential at a point \(\vec{r}\) on the \(x\)-axis at a
distance \(x\) from the center of the quadrupole.
A series of charges arranged in this way is called a linear
quadrupole. Why?
(Algebra involving trigonometric functions) Purpose: Practice with polar equations.
The general equation for a straight line in polar coordinates is
given by:
\begin{equation}
r(\phi)=\frac{r_0}{\cos(\phi-\delta)}
\end{equation}
where \(r_0\) and \(\delta\) are constant parameters. Find the polar equation for the straight lines below. You do NOT need to evaluate any complicated trig or inverse trig functions. You may want to try plotting the general polar equation to figure out the roles of the parameters.
Consider a paramagnet, which is a
material with \(n\) spins per unit volume each of which may each be
either “up” or “down”. The spins have energy \(\pm mB\) where
\(m\) is the magnetic dipole moment of a single spin, and there is no
interaction between spins. The magnetization \(M\) is defined as the
total magnetic moment divided by the total volume. Hint: each
individual spin may be treated as a two-state system, which you have
already worked with above.
Plot of magnetization vs. B field
Find the Helmholtz free energy of a paramagnetic system (assume
\(N\) total spins) and show that \(\frac{F}{NkT}\) is a function of
only the ratio \(x\equiv \frac{mB}{kT}\).
Use the canonical ensemble (i.e. partition function and
probabilities) to find an exact expression for the total
magentization \(M\) (which is the total dipole moment per unit
volume) and the susceptibility \begin{align}
\chi\equiv\left(\frac{\partial M}{\partial
B}\right)_T
\end{align} as a function of temperature and magnetic field for the
model system of magnetic moments in a magnetic field. The result for
the magnetization is \begin{align}
M=nm\tanh\left(\frac{mB}{kT}\right)
\end{align} where \(n\) is the number of spins per unit volume. The figure shows what this magnetization looks like.
Show that the susceptibility is \(\chi=\frac{nm^2}{kT}\) in the
limit \(mB\ll kT\).
Static Fields 2023 (4 years)
Consider a rod of length \(L\) lying on the \(z\)-axis. Find an algebraic expression for the mass density of the rod if the mass density at \(z=0\) is \(\lambda_0\) and at \(z=L\) is \(7\lambda_0\) and you know that the mass density increases
Determine the total mass of each of the slabs below.
A square slab of side length \(L\) with thickness \(h\), resting on a
table top at \(z=0\), whose mass density is given by
\begin{equation}
\rho=A\pi\sin(\pi z/h).
\end{equation}
A square slab of side length \(L\) with thickness \(h\), resting on a
table top at \(z=0\), whose mass density is given by
\begin{equation}
\rho = 2A \Big( \Theta(z)-\Theta(z-h) \Big)
\end{equation}
An infinitesimally thin square sheet of side length \(L\), resting on
a table top at \(z=0\), whose surface density is given by
\(\sigma=2Ah\).
An infinitesimally thin square sheet of side length \(L\), resting on
a table top at \(z=0\), whose mass density is given by
\(\rho=2Ah\,\delta(z)\).
What are the dimensions of \(A\)?
Write several sentences comparing your answers to the different cases above.
Consider a white dwarf of mass \(M\) and radius \(R\). The dwarf
consists of ionized hydrogen, thus a bunch of free electrons and
protons, each of which are fermions. Let the electrons be degenerate
but nonrelativistic; the protons are nondegenerate.
Show that the order of magnitude of the gravitational self-energy is
\(-\frac{GM^2}{R}\), where \(G\) is the gravitational constant. (If
the mass density is constant within the sphere of radius \(R\), the
exact potential energy is \(-\frac53\frac{GM^2}{R}\)).
Show that the order of magnitude of the kinetic energy of the
electrons in the ground state is \begin{align}
\frac{\hbar^2N^{\frac53}}{mR^2}
\approx \frac{\hbar^2M^{\frac53}}{mM_H^{\frac53}R^2}
\end{align} where \(m\) is the mass of an electron and \(M_H\) is
the mas of a proton.
Show that if the gravitational and kinetic energies are of the same
order of magnitude (as required by the virial theorem of mechanics),
\(M^{\frac13}R \approx 10^{20} \text{g}^{\frac13}\text{cm}\).
If the mass is equal to that of the Sun (\(2\times 10^{33}g\)), what
is the density of the white dwarf?
It is believed that pulsars are stars composed of a cold degenerate
gas of neutrons (i.e. neutron stars). Show that for a neutron star
\(M^{\frac13}R \approx 10^{17}\text{g}^{\frac13}\text{cm}\). What is
the value of the radius for a neutron star with a mass equal to that
of the Sun? Express the result in \(\text{km}\).
Writing an operator in matrix notation in its own basis is easy: it is diagonal with the eigenvalues on the diagonal.
What if I want to calculate the matrix elements using a different basis??
The eigenvalue equation tells me what happens when an operator acts on its own eigenstate. For example: \(\hat{S}_y\left|{\pm}\right\rangle _y=\pm\frac{\hbar}{2}\left|{\pm}\right\rangle _y\)
In Dirac bra-ket notation, to know what an operator does to a ket, I needs to write the ket in the basis that is the eigenstates of the operator (in order to use the eigenvalue equation.)
One way to do this to stick completeness relationships into the braket:
\begin{eqnarray*}
\left\langle {+}\right|\hat{S_y}\left|{+}\right\rangle = \left\langle {+}\right|(I)\hat{S_y}(I)\left|{+}\right\rangle
\end{eqnarray*}
where \(I\) is the identity operator: \(I=\color{blue}{\left|{+}\right\rangle _{yy}\left\langle {+}\right|}\;+\;\color{blue}{\left|{-}\right\rangle _{yy}\left\langle {-}\right|}\). This effectively rewrite the \(\left|{+}\right\rangle \) in the \(\left|{\pm}\right\rangle _y\) basis.
Find the top row matrix elements of the operator \(\hat{S}_y\) in the \(S_z\) basis by inserting completeness relations into the brakets. (The answer is already on the Spins Reference Sheet, but I want you do demonstrate the calculation.)
Look up and memorize the power series to fourth order for \(e^z\), \(\sin z\), \(\cos z\), \((1+z)^p\) and \(\ln(1+z)\). For what values of \(z\) do these series converge?
Central Forces 2023 (3 years)
Show that if a linear combination of ring energy eigenstates is normalized, then
the coefficients must satisfy
\begin{equation}
\sum_{m=-\infty}^{\infty} \vert c_m\vert^2=1
\end{equation}
Nuclei of a particular isotope species contained in a crystal have
spin \(I=1\), and thus, \(m = \{+1,0,-1\}\). The interaction between
the nuclear quadrupole moment and the gradient of the crystalline
electric field produces a situation where the nucleus has the same
energy, \(E=\varepsilon\), in the state \(m=+1\) and the state \(m=-1\),
compared with an energy \(E=0\) in the state \(m=0\), i.e. each nucleus
can be in one of 3 states, two of which have energy \(E=\varepsilon\)
and one has energy \(E=0\).
Find the Helmholtz free energy \(F = U-TS\) for a crystal
containing \(N\) nuclei which do not interact with each other.
Find an expression for the entropy as a function of
temperature for this system. (Hint: use results of part a.)
Indicate what your results predict for the entropy at the
extremes of very high temperature and very low temperature.
Ideal gasEntropyTempuratureThermal and Statistical Physics 2020
Consider an ideal gas of
\(N\) particles, each of mass \(M\), confined to a one-dimensional
line of length \(L\). The particles have spin zero (so you can ignore
spin) and do not interact with one another. Find the entropy at
temperature \(T\). You may assume that the temperature is high enough
that \(k_B T\) is much greater than the ground state energy of one
particle.
Energy and Entropy 2021 (2 years)
We have the following equations of state for the total magnetization \(M\), and the entropy \(S\) of a paramagnetic system:
\begin{align}
M&=N\mu\, \frac{e^{\frac{\mu B}{k_B T}} - e^{-\frac{\mu B}{k_B T}}}
{e^{\frac{\mu B}{k_B T}} + e^{-\frac{\mu B}{k_B T}}}\\
S&=Nk_B\left\{\ln 2 +
\ln \left(e^{\frac{\mu B}{k_B T}}+e^{-\frac{\mu B}{k_B T}}\right)
+\frac{\mu B}{k_B T}
\frac{e^{\frac{\mu B}{k_B T}} - e^{-\frac{\mu B}{k_B T}}}
{e^{\frac{\mu B}{k_B T}} + e^{-\frac{\mu B}{k_B T}}}
\right\}
\end{align}
List variables in their proper positions in the middle columns of the charts below.
Solve for the magnetic susceptibility, which is defined as:
\[\chi_B=\left(\frac{\partial M}{\partial B}\right)_T
\]
Using both the differentials (zapping with d) and chain rule diagram methods, find a chain rule for:
\[\left(\frac{\partial M}{\partial B}\right)_S
\]
Evaluate your chain rule. Sense-making: Why does this come out to zero?
EnergyTemperatureParamagnetismThermal and Statistical Physics 2020
Find the equilibrium value at temperature \(T\)
of the fractional magnetization \begin{equation}
\frac{\mu_{tot}}{Nm} \equiv \frac{2\langle s\rangle}{N}
\end{equation} of a system of \(N\) spins each of magnetic moment
\(m\) in a magnetic field \(B\). The spin excess is \(2s\). The energy
of this system is given by \begin{align}
U &= -\mu_{tot}B
\end{align} where \(\mu_{tot}\) is the total magnetization. Take the
entropy as the logarithm of the multiplicity \(g(N,s)\) as given in
(1.35 in the text): \begin{equation}
S(s) \approx k_B\log g(N,0) - k_B\frac{2s^2}{N}
\end{equation} for \(|s|\ll N\), where \(s\) is the spin excess, which
is related to the magnetization by \(\mu_{tot} = 2sm\). Hint:
Show that in this approximation \begin{equation}
S(U) = S_0 - k_B\frac{U^2}{2m^2B^2N},
\end{equation} with \(S_0=k_B\log g(N,0)\). Further, show that
\(\frac1{kT} = -\frac{U}{m^2B^2N}\), where \(U\) denotes
\(\langle U\rangle\), the thermal average energy.
For each of the following complex numbers \(z\), find \(z^2\), \(\vert z\vert^2\), and rewrite \(z\) in exponential form, i.e. as a magnitude times a complex exponential phase:
\(z_1=i\),
\(z_2=2+2i\),
\(z_3=3-4i\).
In quantum mechanics, it turns out that the overall phase for a state does not have any physical significance. Therefore, you will need to become quick at rearranging the phase of various states. For each of the vectors listed below, rewrite the vector as an overall complex phase times a new vector whose first component is real and positive.
\[\left|D\right\rangle\doteq
\begin{pmatrix}
7e^{i\frac{\pi}{6}}\\ 3e^{i\frac{\pi}{2}}\\ -1\\
\end{pmatrix}\\
\left|E\right\rangle\doteq
\begin{pmatrix}
i\\ 4\\
\end{pmatrix}\\
\left|F\right\rangle\doteq
\begin{pmatrix}
2+2i\\ 3-4i\\
\end{pmatrix}
\]
For each of the \(\left|{\psi_i}\right\rangle \) above, calculate the probabilities of spin component measurements along the \(x\), \(y\), and \(z\)-axes.
Look For a Pattern (and Generalize): Use your results from \((a)\) to comment on the importance of the overall phase and of the relative phases of the quantum state vector.
In the "Quizzes" section of Canvas, please fill out the "Photo Permission Form" to indicate what information you'd like me to post about you on the Physics Department Website.
Faculty & Students use this site to learn who is taking Paradigms and to network.
In our week on radiation, we saw that the Helmholtz free energy of a
box of radiation at temperature \(T\) is \begin{align}
F &= -8\pi \frac{V(kT)^4}{h^3c^3}\frac{\pi^4}{45}
\end{align} From this we also found the internal energy and entropy
\begin{align}
U &= 24\pi \frac{(kT)^4}{h^3c^3}\frac{\pi^4}{45} V \\
S &= 32\pi kV\left(\frac{kT}{hc}\right)^3 \frac{\pi^4}{45}
\end{align} Given these results, let us consider a Carnot engine that
uses an empty metalic piston (i.e. a photon gas).
Given \(T_H\) and \(T_C\), as well as \(V_1\) and \(V_2\) (the two
volumes at \(T_H\)), determine \(V_3\) and \(V_4\) (the two volumes
at \(T_C\)).
What is the heat \(Q_H\) taken up and the work done by the gas
during the first isothermal expansion? Are they equal to each other,
as for the ideal gas?
Does the work done on the two isentropic stages cancel each other,
as for the ideal gas?
Calculate the total work done by the gas during one cycle. Compare
it with the heat taken up at \(T_H\) and show that the energy
conversion efficiency is the Carnot efficiency.
Central Forces 2023 (3 years)
Show that the plane polar coordinates are equivalent
to spherical coordinates if we make the choices:
The direction of \(\theta=0\) in spherical coordinates is the same as the
direction of out of the plane in plane polar coordinates.
Given the correspondance above, then if we choose the \(\theta\) of spherical coordinates is to be \(\pi/2\), we restrict to the equatorial plane of spherical coordinates.
Potential energyHeat capacityThermal and Statistical Physics 2020
Consider a column of atoms each of mass \(M\) at temperature \(T\) in
a uniform gravitational field \(g\). Find the thermal average
potential energy per atom. The thermal average kinetic energy is
independent of height. Find the total heat capacity per atom. The
total heat capacity is the sum of contributions from the kinetic
energy and from the potential energy. Take the zero of the
gravitational energy at the bottom \(h=0\) of the column. Integrate
from \(h=0\) to \(h=\infty\). You may assume the gas is ideal.
In this course, two of the primary examples we will be using are the potential due
to gravity and the potential due to an electric charge. Both of these forces vary
like \(\frac{1}{r}\), so they will have many, many similarities. Most of the
calculations we do for the one case will be true for the other. But there are
some extremely important differences:
Find the value of the electrostatic potential energy of a system consisting of a
hydrogen nucleus and an electron separated by the Bohr radius. Find the value
of the gravitational potential energy of the same two particles at the same
radius. Use the same system of units in both cases. Compare and the contrast
the two answers.
Find the value of the electrostatic potential due to the nucleus of a hydrogen atom
at the Bohr radius. Find the gravitational potential due to the nucleus at
the same radius. Use the same system of units in both cases. Compare and
contrast the two answers.
Briefly discuss at least one other fundamental difference between
electromagnetic and gravitational systems. Hint: Why are we bound to the
earth gravitationally, but not electromagnetically?
It has been proposed to use the thermal
gradient of the ocean to drive a heat engine. Suppose that at a
certain location the water temperature is \(22^\circ\)C at the ocean
surface and \(4^{o}\)C at the ocean floor.
What is the maximum possible efficiency of an engine operating
between these two temperatures?
If the engine is to produce 1 GW of electrical power, what
minimum volume of water must be processed every second? Note that
the specific heat capacity of water \(c_p = 4.2\) Jg\(^{-1}\)K\(^{-1}\) and the
density of water is 1 g cm\(^{-3}\), and both are roughly constant
over this temperature range.
At a power plant that produces 1 GW
(\(10^{9} \text{watts}\)) of electricity, the steam turbines take in steam at a
temperature of \(500^{o}C\), and the waste energy is expelled into the
environment at \(20^{o}C\).
What is the maximum possible efficiency of this plant?
Suppose you arrange the power plant to expel its waste energy
into a chilly mountain river at \(15^oC\). Roughly how much money can
you make in a year by installing your improved hardware, if you sell
the additional electricity for 10 cents per kilowatt-hour?
At what rate will the plant expel waste energy into this river?
Assume the river's flow rate is 100 m\(^{3}/\)s. By how much will
the temperature of the river increase?
To avoid this “thermal pollution” of the river the plant could
instead be cooled by evaporation of river water. This is more
expensive, but it is environmentally preferable. At what rate must
the water evaporate? What fraction of the river must be evaporated?
Static Fields 2023 (6 years)
Use the formula for a Taylor series:
\[f(z)=\sum_{n=0}^{\infty} \frac{1}{n!} \frac{d^n f(a)}{dz^n} (z-a)^n\]
to find the series expansion for \(f(z)=e^{-kz}\) to second order
around \(z=3\).
Static Fields 2023 (6 years)
Use the formula for a Taylor series:
\[f(z)=\sum_{n=0}^{\infty} \frac{1}{n!} \frac{d^n f(a)}{dz^n} (z-a)^n\]
to find the series expansion for \(f(z)=\cos(kz)\) to second order
around \(z=2\).
Show that a Fermi electron gas in the ground state exerts a pressure
\begin{align}
p = \frac{\left(3\pi^2\right)^{\frac23}}{5}
\frac{\hbar^2}{m}\left(\frac{N}{V}\right)^{\frac53}
\end{align} In a uniform decrease of the volume of a cube every
orbital has its energy raised: The energy of each orbital is
proportional to \(\frac1{L^2}\) or to \(\frac1{V^{\frac23}}\).
Find an expression for the entropy of a Fermi electron gas in the
region \(kT\ll \varepsilon_F\). Notice that \(S\rightarrow 0\) as
\(T\rightarrow 0\).
(modified from
K&K 4.6) We discussed in class that \begin{align}
p &= -\left(\frac{\partial F}{\partial V}\right)_T
\end{align} Use this relationship to show that
\begin{align}
p &= -\sum_j \langle n_j\rangle\hbar \left(\frac{d\omega_j}{dV}\right),
\end{align}
where \(\langle n_j\rangle\) is the number of photons
in the mode \(j\);
Solve for the relationship between pressure and internal energy.
bose-einstein gasstatistical mechanicsThermal and Statistical Physics 2020
Consider one particle
confined to a cube of side \(L\); the concentration in effect is
\(n=L^{-3}\). Find the kinetic energy of the particle when in the
ground state. There will be a value of the concentration for which
this zero-point quantum kinetic energy is equal to the temperature
\(kT\). (At this concentration the occupancy of the lowest orbital is
of the order of unity; the lowest orbital always has a higher
occupancy than any other orbital.) Show that the concentration \(n_0\)
thus defined is equal to the quantum concentration \(n_Q\) defined by
(63): \begin{equation}
n_Q \equiv \left(\frac{MkT}{2\pi\hbar^2}\right)^{\frac32}
\end{equation} within a factor of the order of unity.
Find the entropy of a set of \(N\) oscillators of frequency
\(\omega\) as a function of the total quantum number \(n\). Use the
multiplicity function: \begin{equation}
g(N,n) = \frac{(N+n-1)!}{n!(N-1)!}
\end{equation} and assume that \(N\gg 1\). This means you can
make the Sitrling approximation that
\(\log N! \approx N\log N - N\). It also means that
\(N-1 \approx N\).
Let \(U\) denote the total energy \(n\hbar\omega\) of the
oscillators. Express the entropy as \(S(U,N)\). Show that the total
energy at temperature \(T\) is \begin{equation}
U = \frac{N\hbar\omega}{e^{\frac{\hbar\omega}{kT}}-1}
\end{equation} This is the Planck result found the hard
way. We will get to the easy way soon, and you will never again need
to work with a multiplicity function like this.
Central Forces 2023 (4 years)
You know that the normalized spatial eigenfunctions for a particle in a 1-D box of length
\(L\) are \(\sqrt{\frac{2}{L}}\sin{\frac{n\pi x}{L}}\). If you want the eigenfunctions for a particle in a 2-D
box, then you just multiply together the eigenfunctions for a 1-D box in each
direction. (This is what the separation of variables procedure tells you to do.)
Find the normalized eigenfunctions for a particle in a 2-D box with sides of length \(L_x\)
in the \(x\)-direction and length \(L_y\) in the \(y\)-direction.
Find the Hamiltonian for a 2-D box and show that your eigenstates are indeed eigenstates and find a formula for the possible energies
Any sufficiently smooth spatial wave function inside a 2-D box can be expanded in a double sum of the product wave functions, i.e.
\begin{equation}
\psi(x,y)=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\; \hbox{eigenfunction}_n(x)\;\hbox{eigenfunction}_m(y)
\end{equation}
Using your expressions from part (a) above, write out all the terms in this sum out to \(n=3\), \(m=3\). Arrange the terms, conventionally, in terms of increasing
energy.
You may find it easier to work in bra/ket notation:
\begin{align*}
\left|{\psi}\right\rangle &=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\left|{n}\right\rangle \left|{m}\right\rangle \\
&=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\left|{nm}\right\rangle
\end{align*}
Find a formula for the \(c_{nm}\)s in part (b). Find the formula first in bra ket notation and then rewrite it in wave function notation.
As discussed in
class, we can consider a black body as a large box with a small hole
in it. If we treat the large box a metal cube with side length \(L\)
and metal walls, the frequency of each normal mode will be given by:
\begin{align}
\omega_{n_xn_yn_z} &= \frac{\pi c}{L}\sqrt{n_x^2 + n_y^2 + n_z^2}
\end{align} where each of \(n_x\), \(n_y\), and \(n_z\) will have
positive integer values. This simply comes from the fact that a half
wavelength must fit in the box. There is an additional quantum number
for polarization, which has two possible values, but does not affect
the frequency. Note that in this problem I'm using different
boundary conditions from what I use in class. It is worth learning to
work with either set of quantum numbers. Each normal mode is a
harmonic oscillator, with energy eigenstates \(E_n = n\hbar\omega\)
where we will not include the zero-point energy
\(\frac12\hbar\omega\), since that energy cannot be extracted from the
box. (See the
Casimir effect
for an example where the zero point energy of photon modes does have
an effect.)
Note
This is a slight approximation, as the boundary conditions for light
are a bit more complicated. However, for large \(n\) values this gives
the correct result.
Show that the free energy is given by \begin{align}
F &= 8\pi \frac{V(kT)^4}{h^3c^3}
\int_0^\infty \ln\left(1-e^{-\xi}\right)\xi^2d\xi
\\
&= -\frac{8\pi^5}{45} \frac{V(kT)^4}{h^3c^3}
\\
&= -\frac{\pi^2}{45} \frac{V(kT)^4}{\hbar^3c^3}
\end{align} provided the box is big enough that
\(\frac{\hbar c}{LkT}\ll 1\). Note that you may end up with a
slightly different dimensionless integral that numerically evaluates
to the same result, which would be fine. I also do not expect you to
solve this definite integral analytically, a numerical confirmation
is fine. However, you must manipulate your integral until it
is dimensionless and has all the dimensionful quantities removed
from it!
Show that the entropy of this box full of photons at temperature
\(T\) is \begin{align}
S &= \frac{32\pi^5}{45} k V \left(\frac{kT}{hc}\right)^3
\\
&= \frac{4\pi^2}{45} k V \left(\frac{kT}{\hbar c}\right)^3
\end{align}
Show that the internal energy of this box full of photons at
temperature \(T\) is \begin{align}
\frac{U}{V} &= \frac{8\pi^5}{15}\frac{(kT)^4}{h^3c^3}
\\
&= \frac{\pi^2}{15}\frac{(kT)^4}{\hbar^3c^3}
\end{align}
(Simple graphing) Purpose: Discover some of the properties of reduced mass by exploring the graph.
Using your favorite graphing package, make a plot of the reduced
mass
\begin{equation}
\mu=\frac{m_1\, m_2}{m_1+m_2}
\end{equation}
as a function of \(m_1\) and \(m_2\). What about the shape
of this graph tells you something about the physical world that you
would like to remember? You should be able to find at least three
things. Hint: Think limiting cases.
Central Forces 2023 (3 years)
Consider the normalized wavefunction \(\Phi\left(\phi\right)\) for a
quantum mechanical particle of mass \(\mu\) constrained to move on a
circle of radius \(r_0\), given by:
\begin{equation}
\Phi\left(\phi\right)= \frac{N}{2+\cos(3\phi)}
\end{equation}
where \(N\) is the normalization constant.
Find \(N\).
Plot this wave function.
Plot the probability density.
Find the probability that if you measured \(L_z\) you would get \(3\hbar\).
What is the expectation value of \(L_z\) in this state?
Attached, you will find a table showing different representations of physical
quantities associated with a quantum particle confined to a ring. Fill in all of the missing entries. Hint: You may look ahead. We filled out a number of the entries throughout the table to give you hints about what the forms of the other entries might be.
pdf link for the Table
or doc link for the Table
Consider a hanging rectangular rubber sheet.
We will consider there to be two ways to get energy into or out of
this sheet: you can either stretch it vertically or horizontally.
The distance of vertical stretch we will call \(y\), and the distance
of horizontal stretch we will call \(x\).
If I pull the bottom down by a small distance \(\Delta y\), with no
horizontal force, what is the resulting change in width \(\Delta x\)?
Express your answer in terms of partial derivatives of the potential
energy \(U(x,y)\).
Consider a very light particle of mass \(\mu\) scattering from a very
heavy, stationary particle of mass \(M\). The force between the two
particles is a repulsive Coulomb force \(\frac{k}{r^2}\). The
impact parameter \(b\) in a scattering problem is defined to be the
distance which would be the closest approach if there were no
interaction (See Figure). The initial velocity (far from the
scattering event) of the mass \(\mu\) is \(\vec v_0\). Answer the
following questions about this situation in terms of \(k\), \(M\),
\(\mu\), \(\vec v_0\), and \(b\). (It is not necessarily wise to answer
these questions in order.)
What is the initial angular momentum of the system?
What is the initial total energy of the system?
What is the distance of closest approach \(r_{\rm{min}}\)
with the interaction?
Sketch the effective potential.
What is the angular momentum at \(r_{\rm{min}}\)?
What is the total energy of the system at \(r_{\rm{min}}\)?
What is the radial component of the velocity at \(r_{\rm{min}}\)?
What is the tangential component of the velocity at \(r_{\rm{min}}\)?
What is the value of the effective potential at \(r_{\rm{min}}\)?
For what values of the initial total energy are there bound orbits?
Using your results above, write a short essay describing this type
of scattering problem, at a level appropriate to share with another
Paradigm student.
Recall that, if you take an infinite number of terms, the power series for
\(\sin z\) and the function itself \(f(z)=\sin z\) are equivalent
representations of the same thing for all real numbers \(z\), (in
fact, for all complex numbers \(z\)). This is what it means for the power
series to “converge” for all \(z\). Not all power series converge for all values
of the argument of the function. More
commonly, a power series is only a valid, equivalent representation of a
function for some more restricted values of \(z\), EVEN IF YOUR KEEP AN INFINITE NUMBER OF TERMS. The technical name
for this idea is convergence--the series only "converges" to the
value of the function on some restricted domain, called the “interval” or “region of convergence.”
Find the power series for the function \(f(z)=\frac{1}{1+z^2}\). Then,
using the Geogebra applet from class as a model,
or some other computer algebra system like Mathematica or Maple, explore
the convergence of this series. Where does your series for this new
function converge? Can you tell anything about the region of
convergence from the graphs of the various approximations? Print out a
plot and write a brief description (a sentence or two) of the region
of convergence. You may need to include a lot of terms to see the effect of the region of convergence. You may also need to play with the values of \(z\) that you plot. Keep adding terms until you see a really strong effect!
Note: As a matter of professional ettiquette (or in some cases, as a legal
copyright requirement), if you use or modify a computer program written by
someone else, you should always acknowledge that fact briefly in whatever you
write up. Say something like: “This calculation was based on a (name
of software package) program titled (title) originally written by
(author) copyright (copyright date).”
Write (a good guess for) the following series using sigma \(\left(\sum\right)\) notation. (If you only know a few terms of a series, you don't know for sure how the series continues.)
Static Fields 2023 (4 years)
Find the surface area of a sphere using cylindrical coordinates. Note: The fact that you can describe spheres nicely in cylindrical coordinates underlies the equal area cylindrical map project that allows you to draw maps of the earth where everything has the correct area, even if the shapes seem distorted. If you want to plot something like population density, you need an area preserving map projection.
Consider a spherical shell with charge density \(\rho (\vec{r})=\alpha3e^{(k r)^3}\) between the inner radius \(a\) and the outer radius \(b\). The charge density is zero everywhere else. Use step functions to write this charge density as a single function valid everywhere in space.
Quantum Fundamentals 2023 (2 years)
The following two problems ask you to make Fermi estimates. In a good Fermi estimate, you start from basic scientific facts you already know or quantities that you can reasonably estimate based on your life experiences and then reason your way to estimate a quantity that you would not be able guess. You may look up useful conversion factors or constants. Use words, pictures, and equations to explain your reasoning:
Imagine that you send a pea-sized bead of silver through a Stern-Gerlach device oriented to measure the z-component of intrinsic spin. Estimate the total z-component of the intrinsic spin of the ball you would measure in the HIGHLY improbable case that every atom is spin up.
Protons, neutrons, and electrons are all spin-1/2 particles. Give a (very crude) order of magnitude estimate of the number of these particles in your body.
Quantum Fundamentals 2023
A spin-3/2 particle initially is in the state \(|\psi(0)\rangle = |\frac{1}{2}\rangle\). This particle is placed in an external magnetic field so that the Hamiltonian is proportional to the \(\hat{S}_x\) operator, \(\hat{H} = \alpha \hat{S}_x \doteq \frac{\alpha\hbar}{2}\begin{pmatrix}
0 & \sqrt{3} & 0 & 0\\ \sqrt{3} & 0 & 2 & 0\\ 0 & 2 & 0 & \sqrt{3} \\ 0 & 0 & \sqrt{3} & 0
\end{pmatrix}\)
Find the energy eigenvalues and energy eigenstates for the system.
Find \(|\psi(t)\rangle\).
List the outcomes of all possible measurements of \(S_x\) and find their probabilities. Explicitly identify any probabilities that depend on time.
List the outcomes of all possible measurements of \(S_z\) and find their probabilities. Explicitly identify any probabilities that depend on time.
eigenvectorsQuantum Fundamentals 2023
The operator \(\hat{S}_x\) for spin-1 may be written as:
defined by:
\[\hat{S}_x=\frac{\hbar}{\sqrt{2}}
\begin{pmatrix}
0&1&0\\ 1&0&1 \\ 0&1&0 \\
\end{pmatrix}
\]
Find the eigenvalues and eigenvectors of this matrix. Write the eigenvectors as both matrices and kets.
Confirm that the eigenstates you found give probabilities that match your expectation from the Spins simulation for spin-1 particles.
Quantum Fundamentals 2023
Two electrons are placed in a magnetic field in the \(z\)-direction. The initial state of the first electron is \(\frac{1}{\sqrt{2}}\begin{pmatrix}
1\\ i\\
\end{pmatrix}\) and the initial state of the second electron is \(\frac{1}{2}\begin{pmatrix}
\sqrt{3}\\ 1\\
\end{pmatrix}\).
Find the probabilty of measuring each particle to have spin-up in the \(x\)-, \(y\)-, and \(z\)-directions at \(t = 0\).
Find the probabilty of measuring each particle to have spin-up in the \(x\)-, \(y\)-, and \(z\)-directions at some later time \(t\).
Calculate the expectation values for \(S_x\), \(S_y\), and \(S_z\) for each particle as functions of time.
Are there any times when all the probabilities you have calculated are the same as they were at \(t = 0\)?
Energy and Entropy 2021 (2 years)
The spring constant \(k\) for a one-dimensional spring is defined by:
\[F=k(x-x_0).\]
Discuss briefly whether each of the variables in this equation is extensive or intensive.
In each of the following sums, shift the index \(n\rightarrow n+2\).
Don't forget to shift the limits of the sum as well. Then write out
all of the terms in the sum (if the sum has a finite number of
terms) or the first five terms in the sum (if the sum has an
infinite number of terms) and convince yourself that the two
different expressions for each sum are the same:
(Numbers and Units) Purpose: Gain experience with the relative sizes of objects and distances in the Solar System. Gain experience with realistic reduced masses.
Calculate the following quantities:
Find \({\vec r}_{\rm sun}-{\vec r}_{\rm cm}\) and \(\mu\) for the
Sun-Earth system. Compare \({\vec r}_{\rm sun}-{\vec r}_{\rm cm}\) to
the radius of the Sun and to the distance from the Sun to the Earth.
Compare \(\mu\) to the mass of the Sun and the mass of the Earth.
Repeat the calculation for the Sun-Jupiter system.
TemperatureRadiationThermal and Statistical Physics 2020
Calculate the
temperature of the surface of the Earth on the assumption that as a
black body in thermal equilibrium it reradiates as much thermal
radiation as it receives from the Sun. Assume also that the surface of
the Earth is a constant temperature over the day-night cycle. Use the
sun's surface temperature \(T_{\odot}=5800\text{K}\); and the sun's
radius \(R_{\odot}=7\times 10^{10}\text{cm}\); and the Earth-Sun
distance of \(1.5\times 10^{13}\text{cm}\).
Instructions for 2022: You will need to complete this assignment in a 15 minute appointment on Zoom or in person with one of the members of the teaching team between 1/21 and 10 pm on 1/26. Here is a link to a sign-up page.
You are required to watch a sample video for how to make symmetry arguments here. As demonstrated in the video you should bring with you to the meeting a cylinder, an observer, and a vector.
Use good symmetry arguments to find the possible direction for
the electric field due to a charged wire. Also, use good symmetry
arguments to find the possible functional dependence of the electric field
due to a charged wire. Rather than writing this up to turn in, you
should find a member of the teaching team
and make the arguments to them verbally.
SymmetryOrbitalsThermal and Statistical Physics 2020
Show that
\begin{align}
f(\mu+\delta) &= 1 - f(\mu-\delta)
\end{align}
This means that the probability that an orbital above the
Fermi level is occupied is equal to the probability an orbital
the same distance below the Fermi level being empty. These unoccupied orbitals are called holes.
Using a dot product, find the angle between any two line segments that join the
center of a regular tetrahedron to its vertices. Hint: Think of the
vertices of the tetrahedron as sitting at the vertices of a cube (at coordinates
(0,0,0), (1,1,0), (1,0,1) and (0,1,1)---you may need to build a model and play
with it to see how this works!)
Vector Calculus I 2022
You are climbing a hill along the steepest path, whose slope at your current location is \(1\over5\). There is another path branching off at an angle of \(30^\circ\) (\(\pi\over6\)). How steep is it?
differentialsStatic Fields 2023 (6 years)
The depth of a puddle in millimeters is given by
\[h=\frac{1}{10} \bigl(1+\sin(\pi xy)\bigr)\]
Your path through the puddle is given by
\[x=3t \qquad y=4t\]
and your current position is \(x=3\), \(y=4\), with \(x\) and \(y\) also in millimeters, and \(t\) in seconds.
At your current position, how fast is the depth of water through which you are walking changing per unit time?
At your current position, how fast is the depth of water through which you are walking changing per unit distance?
FOOD FOR THOUGHT (optional)
There is a walkway over the puddle at \(x=10\). At your current position, how fast is the depth of water through which you are walking changing per unit distance towards the walkway.
The function \(\theta(x)\) (the Heaviside or unit
step function) is a defined as:
\begin{equation}
\theta(x) =\begin{cases}
1 & \textrm{for}\; x>0 \\
0 & \textrm{for}\; x<0
\end{cases}
\end{equation}
This function is discontinuous at \(x=0\) and is generally taken to
have a value of \(\theta(0)=1/2\).
Make sketches of the following functions, by hand, on axes with the
same scale and domain. Briefly describe, using good scientific
writing that includes both words and equations, the role that the
number two plays in the shape of each graph:
\begin{align}
y &= \theta (x)\\
y &= 2+\theta (x)\\
y &= \theta(2+x)\\
y &= 2\theta (x)\\
y &= \theta (2x)
\end{align}
A positively charged (dielectric) spherical shell of inner radius
\(a\) and outer radius \(b\) with a spherically symmetric internal
charge density
\begin{equation}
\rho(\vec{r})=3\alpha\, e^{(kr)^3}
\end{equation}
A positively charged (dielectric) cylindrical shell of inner radius
\(a\) and outer radius \(b\) with a cylindrically symmetric internal
charge density
\begin{equation}
\rho(\vec{r})=\alpha\, \frac{1}{s}\, e^{ks}
\end{equation}
A current \(I\) flows down a cylindrical wire of radius \(R\).
If it is uniformly distributed over the surface, give a formula for
the surface current density \(\vec K\).
If it is distributed in such a way that the volume current density,
\(|\vec J|\), is inversely proportional to the distance from the axis,
give a formula for \(\vec J\).
Current \(I\) flows down a wire with square cross-section. The length of the square side is \(L\). If the current is uniformly distributed over the entire area, find the
current density
.
If the current is uniformly distributed over the outer surface only, find the
current density
.
Consider the diagram of \(T\) vs \(V\) for several different constant values of \(p\).
Translate this diagram to a \(p\) vs \(V\) w/ constant \(T\) graph, including the point \(A\). Complete your graph by hand and make a fairly accurate sketch by printing out the attached grid or in some other way making nice square axes with appropriate tick marks.
Are the lines that you drew straight or curved? What feature of the \(TV\) graph would have to change to change this result?
Sketch the line of constant temperature that passes through the point \(A\).
What are the values of all the thermodynamic variables associated with the point A?
(Straightforward algebra) Purpose: Discover the change of variables that allows you to go from the solution to the reduced mass system back to the original system. Practice solving systems of two linear equations.
For systems of particles, we used the formulas
\begin{align}
\vec{R}_{cm}&=\frac{1}{M}\left(m_1\vec{r}_1+m_2\vec{r}_2\right) \nonumber\\
\vec{r}&=\vec{r}_2-\vec{r}_1
\label{cm}
\end{align}
to switch from a rectangular coordinate system that is unrelated to the system to coordinates adapted to the center-of-mass. After you have solved the equations of motion in the center-of-mass coordinates, you may want to transform back to the original coordinate system. Find the inverse transformation, i.e. solve for:
\begin{align}
\vec{r}_1&=\\
\vec{r}_2&=
\end{align}
Hint: The system of equations (\ref{cm}) is linear, i.e. each variable is to the first power, even though the variables are vectors. In this case, you can use all of the methods you learned for solving systems of equations while keeping the variables vector valued, i.e. you can safely ignore the fact that the \(\vec{r}\)s are vectors while you are doing the algebra as long as you don't divide by a vector.
(Sketch limiting cases) Purpose: For two central force systems that share the same reduced mass system, discover how the motions of the original systems are the same and different.
The figure below shows the position vector \(\vec r\) and the orbit of
a “fictitious” reduced mass \(\mu\).
Suppose \(m_1=m_2\), Sketch the position vectors and orbits for \(m_1\) and \(m_2\) corresponding to \(\vec{r}\). Describe a common physics example of central force motion for which \(m_1=m_2\).
Quantum Fundamentals 2023 (3 years)
With the Spins simulation set for a spin 1/2 system, measure the probabilities of all the possible spin components for each of the unknown initial states \(\left|{\psi_3}\right\rangle \) and \(\left|{\psi_4}\right\rangle \).
Use your measured probabilities to find each of the unknown states as a linear superposition of the \(S_z\)-basis states \(\left|{+}\right\rangle \) and \(\left|{-}\right\rangle \).
Articulate a Process: Write a set of general instructions that would allow another student in next year's class to find an unknown state from measured probabilities.
Compare Theory with Experiment: Design an experiment that will allow you to test whether your prediction for each of the unknown states is correct. Describe your experiment here, clearly but succinctly, as if you were writing it up for a paper. Do the experiment and discuss your results.
Make a Conceptual Connection: In general, can you determine a quantum state with spin-component probability measurements in only two spin-component-directions? Why or why not?
Static Fields 2023
The question is meant to get you used to using the Canvas Discussion Board. Please go to the course Canvas page and find the Discussions tab on the left hand side. Find the Discussion titled Random and add one of the following:
You are given the following Gibbs free energy:
\begin{equation*}
G=-k T N \ln \left(\frac{a T^{5 / 2}}{p}\right)
\end{equation*}
where \(a\) is a constant (whose dimensions make the argument of the logarithm dimensionless).
Compute the entropy.
Work out the heat capacity at constant pressure \(C_p\).
Find the connection among \(V\), \(p\), \(N\), and \(T\), which is called the equation of state (Hint: find the volume as a partial derivative of the Gibbs free energy).
Task:Draw a right triangle. Put a circle around the right angle, that is, the angle that is \(\frac\pi2\) radians.
Preparing your submission:
Complete the assignment using your choice of technology. You may write your answers on paper, write them electronically (for instance using xournal), or typeset them (for instance using LaTeX).
If using software, please export to PDF. If writing by hand, please scan your work using the AIMS scanner if possible. You can also use a scanning app; Gradescope offers advice and suggested apps at this URL. The preferred format is PDF; photos or JPEG scans are less easy to read (and much larger), and should be used only if no alternative is available.)
Please make sure that your file name includes your own name and the number of the assignment, such as "Tevian2.pdf."
Using Gradescope:
We will arrange for you to have a Gradescope account, after which you should receive access instructions directly from them. To submit an assignment:
Select the appropriate course, such as "AIMS F21".
(There will likely be only one course listed.)
Select the assignment called "Sample Assignment"
Follow the instructions to upload your assignment.
(The preferred format is PDF.)
You will then be prompted to associate submitted pages with problem numbers by selecting pages on the right and questions on the left. (In this assignment, there is only one of each.) You may associate multiple problems with the same page if appropriate.
When you are finished, click "Submit"
After the assignments have been marked, you can log back in to see instructor comments.
phase transformationClausius-ClapeyronThermal and Statistical Physics 2020
Consider a phase
transformation between either solid or liquid and gas. Assume that the
volume of the gas is way bigger than that of the liquid or
solid, such that \(\Delta V \approx V_g\). Furthermore, assume that
the ideal gas law applies to the gas phase. Note: this problem
is solved in the textbook, in the section on the Clausius-Clapeyron
equation.
Solve for \(\frac{dp}{dT}\) in terms of the pressure of the vapor
and the latent heat \(L\) and the temperature.
Assume further that the latent heat is roughly independent of
temperature. Integrate to find the vapor pressure itself as a
function of temperature (and of course, the latent heat).
Central Forces 2023 (3 years)
Using either this
Geogebra applet
or this
Mathematica notebook, explore the wave functions on a ring. (Note: The Geogebra applet may be a little easier to use and understand and is accessible if you don't have access to Mathematica, but it is more limited in the wave functions that you can represent. Also, the animation is pretty jumpy in some browsers, especially Firefox. Imagine that the motion is smooth.)
Look at graphs of the following states
\begin{align}
\Phi_1(\phi)&=\frac{1}{\sqrt{2}}(\left|{2}\right\rangle +\left|{-2}\right\rangle )\\
\Phi_2(\phi)&=\frac{1}{\sqrt{2}}(\left|{2}\right\rangle -\left|{-2}\right\rangle )\\
\Phi_3(\phi)&=\frac{1}{\sqrt{2}}(\left|{2}\right\rangle +i\left|{-2}\right\rangle )
\end{align}
Write a short description of how these states differ from each other.
Find a state for which the probability density does not depend on time. Write the state in both ket and wave function notation. These are called stationary states. Generalize your result to give a characterization of the set of all possible states that are stationary states.
Find a state that is right-moving. Write the state in both ket and wave function notation. Generalize your result to give a characterization of the set of all possible states that are right-moving.
Find a state that is a standing wave. Write the state in both ket and wave function notation. Generalize your result to give a characterization of the set of all possible states that are standing waves.
You have a charge distribution on the \(x\)-axis composed of two point charges: one with charge
\(+3q\) located at \(x=-d\) and the other with charge \(-q\) located at \(x=+d\).
Sketch the charge distribution.
Write an expression for the volume charge density \(\rho (\vec{r})\)
everywhere in space.
Consider the following wave functions (over all space - not the infinite square well!):
\(\psi_a(x) = A e^{-x^2/3}\)
\(\psi_b(x) = B \frac{1}{x^2+2} \)
\(\psi_c(x) = C \;\mbox{sech}\left(\frac{x}{5}\right)\) (“sech” is the hyperbolic secant function.)
In each case:
normalize the wave function,
plot the wave function using Mathematica or other computer plotting tool (be sure to include the code you used and label your plots/axes appropriately),
find the probability that the particle is measured to be in the range \(0<x<1\).
The following are 3 different representations for the \(\textbf{same}\) state on a quantum ring for \(r_0=1\)
\begin{equation}
\left|{\Phi_a}\right\rangle
= i\sqrt{\frac{ 2}{12}}\left|{3}\right\rangle
- \sqrt{\frac{ 1}{12}}\left|{1}\right\rangle
+\sqrt{\frac{ 3}{12}}e^{i\frac{\pi}{4}}\left|{0}\right\rangle
-i\sqrt{\frac{ 2}{ 12}}\left|{-1}\right\rangle
+\sqrt{\frac{ 4}{12}}\left|{-3}\right\rangle
\end{equation}
\begin{equation}
\left| \Phi_b\right\rangle \doteq \left( \begin{matrix}
\vdots \\
i\sqrt{\frac{ 2}{12}}\\
0 \\
-\sqrt{\frac{ 1}{12}} \\
\sqrt{\frac{ 3}{12}}e^{i\frac{\pi}{4}} \\
-i\sqrt{\frac{ 2}{12}}\\
0 \\
\sqrt{\frac{4}{12} }\\
\vdots
\end{matrix}\right)
\begin{matrix}
\leftarrow m=0
\end{matrix}
\end{equation}
\begin{equation}
\Phi_c(\phi) \doteq \sqrt{\frac{1}{24 \pi}} \left( i\sqrt{2}e^{i 3 \phi} -e^{i\phi}
+\sqrt{3}e^{i\frac{\pi}{4}} -i \sqrt{2} e^{-i\phi} + \sqrt{4}e^{-i 3 \phi} \right)
\end{equation}
With each representation of the state given above, explicitly calculate the probability that \(L_z=-1\hbar\). Then, calculate all other non-zero probabilities for values of \(L_z\) with a method/representation of your choice.
Explain how you could be sure you calculated all of the non-zero probabilities.
If you measured the \(z\)-component of angular momentum to be \(3\hbar\), what would the
state of the particle be immediately after the measurement is made?
With each representation of the state given above, explicitly calculate the probability that \(E=\frac{9}{2}\frac{\hbar^2}{I}\). Then, calculate all other non-zero probabilities for values of \(E\) with a method of your choice.
If you measured the energy of the state to be \(\frac{9}{2}\frac{\hbar^2}{I}\), what would the state of the particle be immediately after the measurement is made?
In a solid, a free electron doesn't see” a bare nuclear charge
since the nucleus is surrounded by a cloud of other electrons. The
nucleus will look like the Coulomb potential close-up, but be
screened” from far away. A common model for such problems is
described by the Yukawa or screened potential:
\begin{equation}
U(r)= -\frac{k}{r} e^{-\frac{r}{\alpha}}
\end{equation}
Graph the potential, with and without the exponential term.
Describe how the Yukawa potential approximates the “real”
situation. In particular, describe the role of the parameter
\(\alpha\).
Draw the effective potential for the two choices \(\alpha=10\) and
\(\alpha=0.1\) with \(k=1\) and \(\ell=1\). For which value(s) of
\(\alpha\) is there the possibility of stable circular orbits?
