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.
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}
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.
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}
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.
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}
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.
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}
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:
