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### Disclaimer

Peeter’s lecture notes from class. These may be incoherent and rough.

These are notes for the UofT course PHY1520, Graduate Quantum Mechanics, taught by Prof. Paramekanti, covering [1] chap. 2 content.

### problem set note.

In the problem set we’ll look at interference patterns for two slit electron interference like that of fig. 1, where a magnetic whisker that introduces flux is added to the configuration.

### Aharonov-Bohm effect (cont.)

Why do we have the zeros at integral multiples of \( h/q \)? Consider a particle in a circular trajectory as sketched in fig. 3

FIXME: Prof mentioned:

\begin{equation}\label{eqn:qmLecture7:20}

\phi_{\textrm{loop}} = q \frac{ h p/ q }{\Hbar} = 2 \pi p

\end{equation}

… I’m not sure what that was about now.

In classical mechanics we have

\begin{equation}\label{eqn:qmLecture7:40}

\oint p dq

\end{equation}

The integral zero points are related to such a loop, but the \( q \BA \) portion of the momentum \( \Bp – q \BA \) needs to be considered.

### Superconductors

After cooling some materials sufficiently, superconductivity, a complete lack of resistance to electrical flow can be observed. A resistivity vs temperature plot of such a material is sketched in fig. 4.

Just like \ce{He^4} can undergo Bose condensation, superconductivity can be explained by a hybrid Bosonic state where electrons are paired into one state containing integral spin.

The Little-Parks experiment puts a superconducting ring around a magnetic whisker as sketched in fig. 6.

This experiment shows that the effective charge of the circulating charge was \( 2 e \), validating the concept of Cooper-pairing, the Bosonic combination (integral spin) of electrons in superconduction.

### Motion around magnetic field

\begin{equation}\label{eqn:qmLecture7:140}

\omega_{\textrm{c}} = \frac{e B}{m}

\end{equation}

We work with what is now called the Landau gauge

\begin{equation}\label{eqn:qmLecture7:60}

\BA = \lr{ 0, B x, 0 }

\end{equation}

This gives

\begin{equation}\label{eqn:qmLecture7:80}

\begin{aligned}

\BB

&= \spacegrad \cross \BA \\

&= \lr{ \partial_x A_y – \partial_y A_x } \zcap \\

&= B \zcap.

\end{aligned}

\end{equation}

An alternate gauge choice, the symmetric gauge, is

\begin{equation}\label{eqn:qmLecture7:100}

\BA = \lr{ -\frac{B y}{2}, \frac{B x}{2}, 0 },

\end{equation}

that also has the same magnetic field

\begin{equation}\label{eqn:qmLecture7:120}

\begin{aligned}

\BB

&= \spacegrad \BA \\

&= \lr{ \partial_x A_y – \partial_y A_x } \zcap \\

&= \lr{ \frac{B}{2} – \lr{ – \frac{B}{2} } } \zcap \\

&= B \zcap.

\end{aligned}

\end{equation}

We expect the physics for each to have the same results, although the wave functions in one gauge may be more complicated than in the other.

Our Hamiltonian is

\begin{equation}\label{eqn:qmLecture7:160}

\begin{aligned}

H

&= \inv{2 m} \lr{ \Bp – e \BA }^2 \\

&= \inv{2 m} \hat{p}_x^2 + \inv{2 m} \lr{ \hat{p}_y – e B \xhat }^2

\end{aligned}

\end{equation}

We can solve after noting that

\begin{equation}\label{eqn:qmLecture7:180}

\antisymmetric{\hat{p}_y}{H} = 0

\end{equation}

means that

\begin{equation}\label{eqn:qmLecture7:200}

\Psi(x,y) = e^{i k_y y} \phi(x)

\end{equation}

The eigensystem

\begin{equation}\label{eqn:qmLecture7:220}

H \psi(x, y) = E \phi(x, y) ,

\end{equation}

becomes

\begin{equation}\label{eqn:qmLecture7:240}

\lr{ \inv{2 m} \hat{p}_x^2 + \inv{2 m} \lr{ \Hbar k_y – e B \xhat}^2 } \phi(x)

= E \phi(x).

\end{equation}

This reduced Hamiltonian can be rewritten as

\begin{equation}\label{eqn:qmLecture7:320}

H_x

= \inv{2 m} p_x^2 + \inv{2 m} e^2 B^2 \lr{ \xhat – \frac{\Hbar k_y}{e B} }^2

\equiv \inv{2 m} p_x^2 + \inv{2} m \omega^2 \lr{ \xhat – x_0 }^2

\end{equation}

where

\begin{equation}\label{eqn:qmLecture7:260}

\inv{2 m} e^2 B^2 = \inv{2} m \omega^2,

\end{equation}

or

\begin{equation}\label{eqn:qmLecture7:280}

\omega = \frac{ e B}{m} \equiv \omega_{\textrm{c}}.

\end{equation}

and

\begin{equation}\label{eqn:qmLecture7:300}

x_0 = \frac{\Hbar}{k_y}{e B}.

\end{equation}

But what is this \( x_0 \)? Because \( k_y \) is not really specified in this problem, we can consider that we have a zero point energy for every \( k_y \), but the oscillator position is shifted for every such value of \( k_y \). For each set of energy levels fig. 8 we can consider that there is a different zero point energy for each possible \( k_y \).

This is an infinitely degenerate system with an infinite number of states for any given energy level.

This tells us that there is a problem, and have to reconsider the assumption that any \( k_y \) is acceptable.

To resolve this we can introduce periodic boundary conditions, imagining that a square is rotated in space forming a cylinder as sketched in fig. 9.

Requiring quantized momentum

\begin{equation}\label{eqn:qmLecture7:340}

k_y L_y = 2 \pi n,

\end{equation}

or

\begin{equation}\label{eqn:qmLecture7:360}

k_y = \frac{2 \pi n}{L_y}, \qquad n \in \mathbb{Z},

\end{equation}

gives

\begin{equation}\label{eqn:qmLecture7:380}

x_0(n) = \frac{\Hbar}{e B} \frac{ 2 \pi n}{L_y},

\end{equation}

with \( x_0 \le L_x \). The range is thus restricted to

\begin{equation}\label{eqn:qmLecture7:400}

\frac{\Hbar}{e B} \frac{ 2 \pi n_{\textrm{max}}}{L_y} = L_x,

\end{equation}

or

\begin{equation}\label{eqn:qmLecture7:420}

n_{\textrm{max}} = \underbrace{L_x L_y}_{\text{area}} \frac{ e B }{2 \pi \Hbar }

\end{equation}

That is

\begin{equation}\label{eqn:qmLecture7:440}

\begin{aligned}

n_{\textrm{max}}

&= \frac{\Phi_{\textrm{total}}}{h/e} \\

&= \frac{\Phi_{\textrm{total}}}{\Phi_0}.

\end{aligned}

\end{equation}

Attempting to measure Hall-effect systems, it was found that the Hall conductivity was quantized like

\begin{equation}\label{eqn:qmLecture7:460}

\sigma_{x y} = p \frac{e^2}{h}.

\end{equation}

This quantization is explained by these Landau levels, and this experimental apparatus provides one of the more accurate ways to measure the fine structure constant.

# References

[1] Jun John Sakurai and Jim J Napolitano. *Modern quantum mechanics*. Pearson Higher Ed, 2014.