A quantum particle moves in 2 dimensions with Hamiltonian H:

$ H = \frac1{2m} ((P_1 + \frac12 eBX_2)^2 + (P_2 - \frac12 eBX_1)^2) $

For constants $e,B,m$ with $e$ and $B$ nonzero.

Show that the energy levels are of the form $ (n + \frac12)\bar h |eB|\frac{1}{m}$

The hint given is to define $\bar P$ and $\bar X$ as proportional to $P_1 + \frac12 eBX_2$ and $P_2 - \frac12 eBX_1$ and show that the original Hamiltonian has the form

$\frac1{2m} P^2 + \frac12m\omega^2X^2$ for some $\omega$, where

$P_j = -i\bar h \frac{\partial}{\partial x_j}$ and $X_j = x_j$

We are given that this has energy levels $(n+\frac12)\bar h \omega$.

  • 2
    $\begingroup$ This has all the looks of a question about physics. $\endgroup$ – Mariano Suárez-Álvarez Nov 27 '16 at 18:05
  • $\begingroup$ When you actually get into it, it's about operators a lot more than any physical situation. The question came from a mathematics course so I figured it was best asked here. $\endgroup$ – Fahrenheit997 Nov 27 '16 at 18:28
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    $\begingroup$ The problem is that getting from what you wrote to the math is physics. $\endgroup$ – Mariano Suárez-Álvarez Nov 27 '16 at 18:37
  • $\begingroup$ I see what you mean; I think it's an issue with my phrasing rather than the actual problem, so I'll edit it now for clarity $\endgroup$ – Fahrenheit997 Nov 27 '16 at 19:47
  • $\begingroup$ Hint: choose an $a$ in $P = P_1 + \frac12 eB X_2, X = a(P_2 -\frac12 eB X_1)$ so that $[ P, X ] = P X - X P = -i\hbar$ $\endgroup$ – achille hui Nov 27 '16 at 20:10

Depending on how you want to approach this problem I've seen the following done: \begin{eqnarray} \hat{H} &=& \frac{1}{2m}[\hat{p}_x ^2+\hat{p}_x \hat{y}eB+\frac{1}{4}\hat{y}^2e^2B^2-\hat{p}_y ^2+\hat{p}_y \hat{x}eB+\frac{1}{4}\hat{x}^2e^2B^2]\\ &=&\frac{1}{2m}[\hat{p}_x^2+\hat{p}_y^2]+\hat{L}_z \frac{eB}{2m}+\frac{e^2 B^2}{8m}(\hat{x}^2+\hat{y}^2) \end{eqnarray} It's provable that $\hat{L}_z$ commutes with $\hat{p}_x ^2 + \hat{p}_y ^2$ and $\hat{x}^2 + \hat{y}^2$. You can thus form a complete set of commuting operators for $L_z$ and what appears to be a spring Hamiltonian.

You could probably calculate the levels of $L_z$ and the levels of a spring hamiltonian, so I'll leave the rest to you.

  • $\begingroup$ Don't know if you got the point you were at through the book telling you to go that route/telling you the answer, or if you derived it from the definition of canonical momentum and appreciation of azimuthal symmetry, but if it were the latter then I really admire your concise setup! $\endgroup$ – Kitter Catter Nov 27 '16 at 20:37

Not an answer, but some advice: you may have more luck if you help us translate the physics problem into a question about mathematics. For example, my bumbling attempt, knowing no quantum mechanics:

$H$ is an operator on complex-valued functions $\Psi$ over the real plane given by $$H\Psi = \frac{1}{m}\left[\left(-i \hbar\frac{\partial}{\partial x} + \frac{1}{2}eBy \right)^2+\left(-i \hbar\frac{\partial }{\partial y} - \frac{1}{2}eBx \right)^2\right]\Psi$$ where $\hbar, e, B$ and $m$ are (real?) constants.

How do I show that the eigenvalues of $H$ are of the form $\left(n+\frac{1}{2}\right)\hbar |eB|\frac{1}{m}$ for $n\in\mathbb{Z}$?


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