# Constant magnetic field applied to a quantum harmonic oscillator

I have a spinless particle of mass $m$ and charge $q$ which is an isotropic harmonic oscillator of frequency $\omega_0$, then I apply a constant magnetic field in the $z$ direction. We can show the Hamiltonian operator in this case is:

$$\hat {\mathcal H} = \frac{\hat {\mathbf p} ^2}{2m}-\frac{qB}{2m} \hat L_z + \frac{q^2B^2}{8m} (\hat x^2 + \hat y^2) + \frac{1}{2}m\omega_0^2 (\hat x^2+\hat y^2+\hat z^2)$$

Then the exercise asks me to immediately give the exact energy levels without doing any calculation, but I don't see how that's possible.

• The Hamiltonian is not separable in rectangular coordinates because of the $\hat L_z = i\hbar (\hat x \hat p_y - \hat y \hat p_x)$ term. My friend told me to just use the fact that $\left \{\hat {\mathcal H},\hat L^2, \hat L_z \right \}$ form a CSCO (complete set of commuting observables) and thus have the same eigenvectors, but I don't think that's true; they form a CSCO only if the potential is central.
• Changing the Hamiltonian to cylindrical or spherical coordinates doesn't seem to help.
• I could try via a perturbation method:

$$\hat {\mathcal H} = \hat {\mathcal H}_{0x} + \hat {\mathcal H}_{0y} + \hat {\mathcal H}_{0z} + \hat {\mathcal H}_{V}$$

with:

\begin{align*} \hat {\mathcal H}_{0x} &= \frac{\hat p_x ^2}{2m} + \frac{1}{2}m \left [ \left ( \frac{qB}{2m} \right )^2 + \omega_0 ^2 \right ]\hat x^2\\ \hat {\mathcal H}_{0y} &= \frac{\hat p_y ^2}{2m} + \frac{1}{2}m \left [ \left ( \frac{qB}{2m} \right )^2 + \omega_0 ^2 \right ]\hat y^2\\ \hat {\mathcal H}_{0x} &= \frac{\hat p_z ^2}{2m} + \frac{1}{2}m \omega_0 ^2 \hat z^2\\ \hat {\mathcal H}_{V} &= -\frac{qB}{2m} \hat L_z \end{align*}

But this requires doing some calculation and doesn't give the exact energy levels.

Thanks for the help!

## 3 Answers

You should be using cylindrical coordinates because the symmetry of the problem demands it. Doing anything else is just playing hard-headed.

You should split the hamiltonian into a $z$ component and a cylindrical radial component: \begin{align*} \hat {\mathcal H}_{\rho} &= \frac{\hat p_x ^2+\hat p_y^2}{2m} + \frac{1}{2}m \left ( \omega_\text{c}^2 + \omega_0 ^2 \right)\left(\hat x^2+\hat y^2\right)-\omega_\text{c} \hat L_z,\\ \hat {\mathcal H}_{z} &= \frac{\hat p_z ^2}{2m} + \frac{1}{2}m \omega_0 ^2 \hat z^2, \end{align*} where $\omega_\text{c}=\frac{qB}{2m}$ is the relevant cyclotron frequency. To solve this without any calculation:

• The $z$ component is immediate.
• The radial component has a 2D central potential $\mathcal H_{\rho,0}=\frac{\hat p_x ^2+\hat p_y^2}{2m} + \frac{1}{2}m \left ( \omega_\text{c}^2 + \omega_0 ^2 \right)\left(\hat x^2+\hat y^2\right)$ which commutes with $\hat L_z$, so you can simply add the energies: that is, $\mathcal H_{\rho,0}$ has a common eigenbasis with $\omega_\text{c} \hat L_z$, and you can diagonalize them simultaneously; this 2D harmonic oscillator is relatively standard but you can still solve it explicitly.

To do that, you can find the eigenenergies of $\mathcal H_{\rho,0}$ by splitting it into two 1D harmonic oscillators, but the product basis of those oscillators does not give you an angular-momentum eigenbasis. Instead, the thing to do is to note that the set of eigenfunctions with a fixed total excitation number $n$, i.e. the subspace $$\mathcal L_n=\mathrm{span} \left\{|n_x\rangle \otimes |n_y\rangle : n_x+n_y=n\right\}$$ is invariant under rotations (because the hamiltonian is) so you can diagonalize $\hat L_z$ in this subspace.

How do you do that, then? Well, you notice that your basis functions, of the form $H_{n_x}(x)H_{n_y}(y)$ once you remove the gaussian, are polynomials of degree $n$ in $x,y$ and with definite parity, and you're looking to decompose them as linear combinations of $(x\pm iy)^m$ for $m=\ldots,n-4,n-2,n$, which are the eigenfunctions of $\hat L_z$.

That then tells you what eigenvalues $m$ of $\hat L_z$ are allowed for each eigenspace of $\mathcal H_{\rho,0}$ with excitation number $n$, i.e. $m\leq n$ with both of the same parity. That's sufficient to fix the spectrum.

I don't think there is any need to change coordinates, or anything like that. Just note that $\hat L_z$ commutes with the Hamiltonian, so you can simultaneously diagonalise them. Now you can replace $\hat L_z$ with its eigenvalue, so that term becomes a constant, and the rest of the Hamiltonian just represents three decoupled harmonic oscillators.

• The problem is that the $x$ and $y$ oscillators don't individually commute with $\hat{L}_z$, only their combination does. So you can't solve for them in terms of independent $n_x,\ n_y$ (there is some nontrivial constraint condition on the expansion coefficients). – Michael Feb 23 '13 at 0:31

Just to add a different perspective, let me comment the following, with the punch line being that $L_z$ is proportional to a $J_y$ momentum operator. If you write $L_z$ in terms of the ladder operators, you get $L_z=-i\hbar (a_x^\dagger a_y-a_x a_y^\dagger)$. Then, taking into account the Schwinger oscillator model of angular momentum (where for example $J_+=a_x^\dagger a_y$) $L_z$ reads $L_z=2\hbar J_y$. On the other hand, $J^2=\frac{N}{2}(\frac{N}{2}+1)$, where $N$ is the sum of the occupation numbers $N_x$ and $N_y$, so $j=N/2$. With this in mind, the eigeinstates are $|j m_y\rangle$, the states with definite (Schwinger) total angular momentum $j$ and $m_y$. These can be obtained from the standard ones $|j m\rangle=|n_x n_y\rangle$ by a rotation of $-\pi/2$ around $\hat{x}$.

• Actually, combining the oscillators in $x$ and $y$ one can have, schematically, $H=N_++N_-+1$ and $L_z=N_+-N_-$. See this link link – Alan Garbarz Oct 11 '15 at 21:36