assignment Homework

Quantum harmonic oscillator

Entropy Quantum harmonic oscillator Frequency Energy Thermal and Statistical Physics 2020

1. 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 \(N-1 \approx N\).

2. 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.

assignment Homework

Entropy and Temperature

Energy Temperature Ideal gas Entropy Thermal and Statistical Physics 2020

Suppose \(g(U) = CU^{3N/2}\), where \(C\) is a constant and \(N\) is the number of particles.

1. Show that \(U=\frac32 N k_BT\).

2. Show that \(\left(\frac{\partial^2S}{\partial U^2}\right)_N\) is negative. This form of \(g(U)\) actually applies to a monatomic ideal gas.

assignment Homework

Using Gradescope (AIMS)

AIMS Maxwell 2021 (2 years)

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:

1. Navigate to https://paradigms.oregonstate.eduhttps://www.gradescope.com and login
2. Select the appropriate course, such as "AIMS F21". (There will likely be only one course listed.)
3. Select the assignment called "Sample Assignment"
4. Follow the instructions to upload your assignment. (The preferred format is PDF.)
5. 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.
6. When you are finished, click "Submit"
7. After the assignments have been marked, you can log back in to see instructor comments.

assignment Homework

Vectors

vector geometry Static Fields 2023 (4 years)

Let \begin{align} \boldsymbol{\vec a} &= \boldsymbol{\hat x}-3\boldsymbol{\hat y}-\boldsymbol{\hat z}\\ \boldsymbol{\vec b} &= \boldsymbol{\hat x}+\boldsymbol{\hat y}+2\boldsymbol{\hat z}\\ {\boldsymbol{\vec c}} &= -2\boldsymbol{\hat x}-\boldsymbol{\hat y}+\boldsymbol{\hat z}\\ \boldsymbol{\vec d} &= -\boldsymbol{\hat x}-\boldsymbol{\hat y}+\boldsymbol{\hat z} \end{align}

Which pairs (if any) of these vectors

1. Are perpendicular?
2. Are parallel?
3. Have an angle less than \(\pi/2\) between them?
4. Have an angle of more than \(\pi/2\) between them?

assignment Homework

Inhomogeneous Linear ODEs with Constant Coefficients

Oscillations and Waves 2023 (2 years)

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.

1. Find a particular solution of the inhomogeneous differential equation \(\ddot{x}-\dot{x}-6 x=-25\sin(4 t)\).

2. Find the general solution of \(\ddot{x}-\dot{x}-6 x=-25\sin(4 t)\).

3. Some terms in your general solution have an undetermined coefficients, while some coefficients are fully determined. Explain what is different about these two cases.

4. Find a particular solution of \(\ddot{x}-\dot{x}-6 x=12 e^{-3 t}\)

5. 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?

6. Sense-making: Check your answer; Explicitly plug in your final answer in part (e) and check that it satisfies the differential equation.

assignment Homework

Differentials of Two Variables

Static Fields 2023 (8 years) Find the total differential of the following functions:

1. \(y=3u^2 + 4\cos 3v\)
2. \(y=3uv\)
3. \(y=3u^2\cos wv\)
4. \(y=u\cos(3v^2-2)\)

assignment Homework

Paramagnetism

Energy Temperature Paramagnetism Thermal 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.

assignment Homework

Quantum Particle in a 2-D Box

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.)

1. 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.
2. Find the Hamiltonian for a 2-D box and show that your eigenstates are indeed eigenstates and find a formula for the possible energies

3. 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*}

4. 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.

assignment Homework

Carbon monoxide poisoning

Equilibrium Absorbtion Thermal and Statistical Physics 2020

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.

1. 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 \(\textsf{O}_{2}\).

2. Now admit the CO under the specified conditions. Fine \(\varepsilon_B\) such that only 10% of the Hb sites are occupied by \(\textsf{O}_{2}\).

assignment Homework

Angular Momentum and Kinetic Energy in the Center of Mass

Central Forces 2023 (3 years)

(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 \(m_2\).

1. 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.
2. Show that the total angular momentum of the system can similarly be decomposed into the angular momenta of these two fictitious particles.