This activity reinforces the strategies students have been practicing on each system by letting them create their own matrix operators and columns on the hydrogen atom and do some calculations with them.
This lab gives students a chance to take data on the first day of class (or later, but I prefer to do it the first day of class). It provides an immediate context for thermodynamics, and also gives them a chance to experimentally measure a change in entropy. Students are required to measure the energy required to melt ice and raise the temperature of water, and measure the change in entropy by integrating the heat capacity.
heatentropywaterice Found in: Energy and Entropy course(s)Found in: Ice Calorimetry Sequence sequence(s)
These notes from week 6 of https://paradigms.oregonstate.edu/courses/ph441 cover the ideal gas from a grand canonical standpoint starting with the solutions to a particle in a three-dimensional box. They include a number of small group activities.
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 \(y(x)\).
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.
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.
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.
Sketch the function \(f(x)g(x)\) for the default functions.
Make a list of properties of the functions \(f(x)\) and \(g(x)\) that you used to make your sketch. Pay attention to special cases and symmetries.
Check your sketch by clicking on the green check box, i.e. refer to authority.
Plug other functions into the applet and check that your list of properties is accurate and complete, i.e. check many cases.
Part II
Use the applet to find the values of the following integrals for integer \(m\) and \(m'\):
\begin{align*}
&\int_0^{2\pi} \sin mx\;\sin m'x \;dx\\[12pt]
& \int_0^{2\pi} \sin mx\;\cos m'x \;dx\\[12pt]
&\int_0^{2\pi} \cos mx\;\cos m'x \;dx
\end{align*}
Part III
Do a simple change of variables in your integrals to convince yourself of the following:
For integer \(m\) and \(m'\)
\begin{align*}
&\int_0^L \sin\tfrac{2\pi mx}{L}\;\sin\tfrac{2\pi m'x}{L} \;dx= \begin{cases}\frac{L}{2} \mbox{ if } m = m'\\ 0 \mbox{ if } m \neq m'\end{cases}\\[12pt]
& \int_0^L \sin\tfrac{2\pi mx}{L}\;\cos\tfrac{2\pi m'x}{L} \;dx= \begin{cases}0 \mbox{ if } m= m'\\ 0 \mbox{ if } m \neq m'\end{cases}\\[12pt]
&\int_0^L \cos\tfrac{2\pi mx}{L}\;\cos\tfrac{2\pi m'x}{L} \;dx= \begin{cases}\frac{L}{2} \mbox{ if } m= m'\neq 0\\ L \mbox{ if } m=m'=0\\0 \mbox{ if } m \neq m'\end{cases}\\[12pt]
\end{align*}
Hint: Recall that the function transformation \(f(x)\rightarrow f(\alpha x)\) shrinks or expands the function \(f(x)\) along the \(x\)-axis. See GMM: Function Transformations
Part IV
Compare your results for sines and cosines integrated over a whole period (this example) to what you know about the energy eigenstates of a quantum infinite square well, i.e. compare to a known example. How are these examples the same or different? Can you use the applet to explore the infinite square well case?
Students become acquainted with the Spins Simulations of Stern-Gerlach Experiments and record measurement probabilities of spin components for a spin-1/2 system. Students start developing intuitions for the results of quantum measurements for this system.
This lecture introduces the idea of entropy, including the relationship between entropy and multiplicity as well as the relationship between changes in entropy and heat.
Explain the consequences of energy and angular momentum conservation in a system of two particles interacting via a central force, in both classical and quantum systems.
Use effective potential diagrams to determine properties of classical orbits.
Solve for the quantum properties of a particle confined to a ring, rigid rotor, and the hydrogen atom in several different representations.
Relate the state of a quantum system (ring, rigid rotor, hydrogen atom) to graphs of a wave function.
Apply Schrödinger time dependence to central force systems (ring, rigid rotor, H atom).
Mathmatics Content Learning Objectives
Solve ordinary differential equations using power series methods.
Use eigen expansions as an orthonormal basis.
Solve the initial value problem for partial differential equations with more than one spatial variable.
Professional Learning Objectives
Communicate scientific ideas in writing and with other representations (e.g. graphs, code), using good scientific language and practices, concisely and without ambiguity.
Learn to work with and communicate productively and respectfully with peers and collaborators of different backgrounds.
Cite the information and ideas obtained from or with others in a clear and professional manner.
Communicate in a timely and professional manner with others in the work environment when things don't go to plan.
For this project, you will develop your own homework question, and write the model answer. You will be practicing an important skill: how to pose a tractable quantitative question that leads to interesting insights. You have freedom to choose a topic that you find especially interesting/intriguing. Within this topic, look for questions where a quantitative result can give insight. Think broadly about topics: many topics that you encounter outside of the physics classroom are full of
interesting physics. Please talk with the instructor about your topic ideas.
This is a multi-step, multi-week project with many opportunities to get feedback from the instructor. The main steps are:
Propose some possible topics and questions - receive feedback.
Write a draft of the question and answer - receive feedback.
Write a revised version of the question and model answer and submit a final portfolio. The final portfolio will show how your project developed. Steps 1, 2 and 3 will be included in the final portfolio.
A good question will lead you into “the unknown” (an area of knowledge
that is new to you). Your model answer will show how coarse-grained modeling of the system, followed by quantitative reasoning/calculations, can shed light on the unknown. For guidance on style and difficulty, look at examples of rich-context questions that are set by the instructor during the course.
The final version of your question should be posed in a way that is accessible to a well prepared PH315 student. The final version of your model answer should be written in a style similar to a physics textbook. Explain each step so that a PH315 student can clearly understand how you constructed your solution.
The final portfolio will be graded based on the following components. Percentages add up to 110% because of bonus points. However, you can't earn more than 100%.
The Question (35%)(start on a new page)
Give context to motivate the question.
Include a visual aid, such as an image from the internet and/or a hand-drawn schematic.
A well prepared PH315 student should understand what the question is asking them to do. The question might need scaffolding into separate steps, part (a), part (b), etc.
The question should invite the application of physics concept(s) and physical reasoning. Ideally, the student will need to set up (or utilize) a physical model to answer the question.
Include any quantities that are needed to solve the question. Alternatively, you may decide that some quantities can be easily estimated or found on Google.
List any references you used when developing the question (for example a web address).
The Solution (45%)(start on a new page)
Explain your solution to the question (all steps) using clear communication and following the Mathematical Communication Guidelines (Writing answers to HW questions). A PH315 student should be able to understand how you constructed your solution. Your reasoning should be based on valid physical principles and consistent with the physics involved.
Include at least one diagram that helps explain some aspect of the physics in your solution.
Remember that calculations and physical reasoning are at the heart of your answer. If your calculations and physical reasoning are getting obscured by other details, you should consider a more concise way to explain things.
Interpret your answer: make sense of what your answer means.
If you gained insight/inspiration from an outside source, list any helpful references.
Bonus points (10%)
Bonus points may be earned if you worked on a particularly difficult/challenging problem. Bonus points will not enable you to score more than 100%, but may make up for other deficiencies.
Appendix (15%)(start on a new page, not required for the draft)
The appendix serves as evidence that you engaged in the process of drafting and revising your work.
The appendix includes
Your first list of three project ideas, and the instructor comments.
Your first draft of the question/solutions, the peer reviewer comments and the instructor comments.
Other suggestions/comments your received from the instructor.
Process Memo (5%)(start on a new page, not required for the draft)
Write a few sentences addressing each bullet point:
What went well? What do you like about your question and solution?
What technical knowledge did you gain while doing this term project?
In what ways did the project help strengthen your ability to make quantitative estimates about physical systems?
Do you give permission for the instructor to share your question (only the question, not the model answer) with the class? The question would only be accessible on this Canvas website. Authors of questions will remain anonymous.
Moving the sun to a different part of the galaxy with
a stellar
engine.
Non-rocket
space launch,
including space
tethers. (Word of caution: I have struggled to design a
coarse-grain model for the space tether concept. I haven't found a
good resource yet.)
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.
Students implement a finite-difference approximation for the kinetic energy operator as a matrix, and then use numpy to solve for eigenvalues and eigenstates, which they visualize.
Review, as much as necessary, how to do matrix addition, multiplication of a matrix by a scalar, matrix multiplication, determinant of a matrix (\(2\times 2\) and \(3\times 3\)), and inverse of a matrix (\(2\times 2\) only). You might find the information at the following links useful.
Students explore what linear transformation matrices do to vectors. The whole class discussion compares & contrasts several different types of transformations (rotation, flip, projections, “scrinch”, scale) and how the properties of the matrices (the determinant, symmetries, which vectors are unchanged) are related to these transformations.
(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.
Students work in small groups to use the Biot-Savart law
\[\vec{B}(\vec{r}) =\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r}^{\,\prime})\times \left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}\]
to find an integral expression for the magnetic field, \(\vec{B}(\vec{r})\), due to a spinning ring of charge.
In an optional extension, students find a series expansion for \(\vec{B}(\vec{r})\) either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable.
Add an extra half hour or more to the time estimate for the optional extension.
Students consider the relation (1) between the angular momentum and magnetic moment for a current loop and (2) the force on a magnetic moment in an inhomogeneous magnetic field. Students make a (classical) prediction of the outcome of a Stern-Gerlach experiment.
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 \(mB\ll kT\).
to perform a magnetic vector potential calculation using the superposition principle;
to decide which form of the superposition principle to use, depending on the dimensions of the current density;
how to find current from total charge \(Q\), period \(T\), and the geometry of the problem, radius \(R\);
to write the distance formula \(\vec{r}-\vec{r'}\) in both the numerator and denominator of the superposition principle in an appropriate mix of cylindrical coordinates and rectangular basis vectors;
Groups are asked to analyze the following standard problem:
Two identical lumps of clay of (rest) mass m collide head on, with each
moving at 3/5 the speed of light. What is the mass of the resulting lump of
clay?
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 \(\text{km}\).
This activity allows students to puzzle through indexing, the from of operators in quantum mechanics, and working with the new quantum numbers on the sphere in an applied context.
This small group activity introduces students to constrained optimization problems.
Students work in small groups to optimize a simple function on a given region.
The whole class wrap-up discussion emphasizes the importance of the boundary.
Students compute probabilities and averages given a probability density in one dimension. This activity serves as a soft introduction to the particle in a box, introducing all the concepts that are needed.
A beam of spin-\(\frac{1}{2}\) particles is prepared in the initial state \[ \left\vert \psi\right\rangle = \sqrt{\frac{2}{5}}\; |+\rangle_x - \sqrt{\frac{3}{5}}\; |-\rangle_x \](Note: this state is written in the \(S_x\) basis!)
What are the possible results of a measurement of \(S_x\), with what probabilities?
Repeat part a for measurements of \(S_z\).
Suppose you start with a particle in the state given above, measure \(S_x\), and happen to get \(+\hbar /2\). You then take that same particle and measure \(S_z\). What are the possible results and with what probability would you measure each possible result?
In this remote-friendly activity, students use a microwave oven (and optionally a thermometer) to measure the latent heat of melting for water (and optionally the heat capacity). From these they compute changes in entropy. See also Ice Calorimetry Lab.
