*assignment*Free energy of a harmonic oscillator*assignment*Homework##### Free energy of a harmonic oscillator

Helmholtz free energy harmonic oscillator Thermal and Statistical Physics 2020A 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}

This entropy is shown in the nearby figure, as well as the heat capacity.

*assignment*Normalization of Quantum States*assignment*Homework##### Normalization of Quantum States

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}*assignment*Visualization of Wave Functions on a Ring*assignment*Homework##### Visualization of Wave Functions on a Ring

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.

*group*Gaussian Parameters*group*Small Group Activity10 min.

##### Gaussian Parameters

Periodic Systems 2022 Students use an applet to explore the role of the parameters \(N\), \(x_o\), and \(\sigma\) in the shape of a Gaussian \begin{equation} f(x)=Ne^{-\frac{(x-x_0)^2}{2\sigma^2}} \end{equation}*group*Fourier Transform of a Shifted Function*assignment*Quantum Particle in a 2-D Box*assignment*Homework##### Quantum Particle in a 2-D Box

Central Forces 2023 (3 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.

*assignment*Radiation in an empty box*assignment*Homework##### Radiation in an empty box

Thermal physics Radiation Free energy Thermal and Statistical Physics 2020As 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}

*face*Entropy and Temperature*face*Lecture120 min.

##### Entropy and Temperature

Thermal and Statistical Physics 2020paramagnet entropy temperature statistical mechanics

These lecture notes for the second week of Thermal and Statistical Physics involve relating entropy and temperature in the microcanonical ensemble, using a paramagnet as an example. These notes include a few small group activities.*assignment*Theta Parameters*assignment*Homework##### Theta Parameters

Static Fields 2023 (6 years)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}

*face*Thermal radiation and Planck distribution*face*Lecture120 min.

##### Thermal radiation and Planck distribution

Thermal and Statistical Physics 2020Planck distribution blackbody radiation photon statistical mechanics

These notes from the fourth week of Thermal and Statistical Physics cover blackbody radiation and the Planck distribution. They include a number of small group activities.-
Central Forces 2023 (3 years)
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:

- \begin{equation} \sum_{n=0}^3 n \end{equation}
- \begin{equation} \sum_{n=1}^5 e^{in\phi} \end{equation}
- \begin{equation} \sum_{n=0}^{\infty} a_n n(n-1)z^{n-2} \end{equation}