Charged Harmonic Oscillator in a Magnetic Field

A charged harmonic oscillator in a magnetic field is the simplest exactly solvable model that combines ordinary confinement with Landau quantization. It is often called the Fock–Darwin problem in two dimensions.

The model interpolates between two canonical limits:

• when the magnetic field vanishes, it is the isotropic two-dimensional harmonic oscillator;
• when the harmonic confinement vanishes, it becomes the Landau-level problem, with a zero-frequency guiding-center mode.

This page treats the spinless single-particle model. Real quantum dots, trapped charged particles, and semiconductor devices can add spin, Zeeman splitting, anisotropic confinement, disorder, finite thickness, and interactions.

Helpful background. Quantum Harmonic Oscillator to recognize oscillator energies, lengths, and eigenfunctions.

Hamiltonian

Consider a particle of mass $m$ and charge $q$ moving in the $xy$ plane with harmonic confinement

$$V(r) = \frac12m\omega_0^2r^2, \qquad r^2=x^2+y^2,$$

and a uniform magnetic field

$$\mathbf B = B\hat{\mathbf z}, \qquad B\gt0.$$

Minimal coupling gives

$$\hat H = \frac{1}{2m} \left( \hat{\mathbf p}-q\mathbf A \right)^2 + \frac12m\omega_0^2r^2.$$

Use symmetric gauge,

$$\mathbf A = \frac{B}{2} \left( -y\,\hat{\mathbf x} + x\,\hat{\mathbf y} \right).$$

Define the signed cyclotron frequency

$$\omega_q = \frac{qB}{m}$$

and the positive cyclotron frequency

$$\omega_c = \frac{\lvert q\rvert B}{m}.$$

Expanding the Hamiltonian gives

$$\hat H = \frac{\hat p_x^2+\hat p_y^2}{2m} + \frac12m\Omega^2r^2 - \frac{\omega_q}{2}\hat L_z,$$

where

$$\Omega = \sqrt{ \omega_0^2+\frac{\omega_c^2}{4} }$$

and

$$\hat L_z = x\hat p_y-y\hat p_x.$$

The magnetic field has done two things. It added a quadratic diamagnetic term, increasing the effective radial oscillator frequency from $\omega_0$ to $\Omega$, and it coupled to angular momentum through $-\omega_q\hat L_z/2$.

The magnetic field splits the two circular oscillator modes. At $B=0$ they are degenerate at $\omega_0$; as confinement is removed relative to the field, one mode approaches the cyclotron frequency and the other becomes the guiding-center zero mode.

Spectrum

The eigenstates can be labeled by a radial quantum number

$$n_r=0,1,2,\ldots$$

and an angular quantum number

$$\mu\in\mathbb Z, \qquad \hat L_z\psi=\hbar\mu\,\psi.$$

The Fock–Darwin energies are

$$E_{n_r,\mu} = \hbar\Omega \left( 2n_r+\lvert\mu\rvert+1 \right) - \frac{\hbar\omega_q}{2}\mu .$$

The first term is the isotropic two-dimensional oscillator spectrum with frequency $\Omega$. The second term splits states with opposite angular momentum. The sign of the splitting depends on the sign of $qB$.

Equivalently, the system can be written in terms of two circular oscillator modes with frequencies

$$\omega_- = \Omega-\frac{\omega_c}{2}, \qquad \omega_+ = \Omega+\frac{\omega_c}{2}.$$

The corresponding energies can be written as

$$E_{n_-,n_+} = \hbar\omega_- \left( n_-+\frac12 \right) + \hbar\omega_+ \left( n_++\frac12 \right),$$

with $n_-=n_r+(\lvert\mu\rvert+s\mu)/2$, $n_+=n_r+(\lvert\mu\rvert-s\mu)/2$, and $s=\operatorname{sgn}(qB)$. Thus the charge sign sets which circular mode carries positive angular momentum. This circular-mode form makes the Landau-level limit especially transparent.

Wavefunctions

Because symmetric gauge preserves rotations about the $z$ axis, the eigenfunctions have angular form

$$\psi_{n_r,\mu}(r,\phi) = e^{i\mu\phi}R_{n_r,\mu}(r).$$

The radial functions are two-dimensional oscillator functions with frequency $\Omega$. With the angular factor $e^{i\mu\phi}$ above and measure $r\,dr\,d\phi$, their normalized form is

$$R_{n_r,\mu}(r) =\sqrt{\frac{n_r!}{\pi\ell_\Omega^2(n_r+\lvert\mu\rvert)!}} \left( \frac{r}{\ell_\Omega} \right)^{\lvert\mu\rvert} L_{n_r}^{\lvert\mu\rvert} \left( \frac{r^2}{\ell_\Omega^2} \right) \exp\left( - \frac{r^2}{2\ell_\Omega^2} \right),$$

where

$$\ell_\Omega = \sqrt{ \frac{\hbar}{m\Omega} }.$$

The magnetic field changes the radial length scale through $\Omega$ and changes the energies through the angular-momentum splitting. It does not remove the usefulness of angular-momentum labels; in symmetric gauge, they are the natural labels.

Limits

Zero Magnetic Field

When $B\to0$,

$$\omega_c\to0, \qquad \Omega\to\omega_0.$$

The spectrum becomes

$$E_{n_r,\mu} \to \hbar\omega_0 \left( 2n_r+\lvert\mu\rvert+1 \right),$$

the ordinary isotropic two-dimensional oscillator spectrum. States with the same

$$N=2n_r+\lvert\mu\rvert$$

belong to the same oscillator shell.

Weak Magnetic Field

For small $B$, the leading splitting is linear:

$$\Delta E \approx - \frac{\hbar\omega_q}{2}\mu.$$

The quadratic diamagnetic shift enters through

$$\Omega = \omega_0 + O(B^2).$$

Thus weak fields first split angular-momentum partners; the radial confinement shift is second order in $B$.

Landau-Level Limit

When $\omega_0\to0$,

$$\Omega\to\frac{\omega_c}{2}.$$

The circular-mode frequencies become

$$\omega_- \to 0, \qquad \omega_+ \to \omega_c.$$

The high-frequency circular mode is the cyclotron motion. The zero-frequency mode is the guiding-center freedom responsible for Landau-level degeneracy. A weak harmonic trap lifts that degeneracy by giving the guiding center a slow restoring energy.

This is the clean conceptual bridge: Landau levels are what remains when one circular oscillator mode has a finite cyclotron frequency and the other loses its restoring force.

Quantum-Dot Preview

The Fock–Darwin spectrum is a standard first model for a lateral quantum dot with approximately parabolic confinement in a perpendicular magnetic field. In that setting:

• $\omega_0$ models the confinement strength;
• $B$ tunes the cyclotron scale;
• angular-momentum states shift relative to each other as the field changes;
• spin and interactions must be added for realistic many-electron dots.

The single-particle spectrum is useful because it isolates the orbital physics. It should not be mistaken for a complete device model.

Common Mistakes

• Using the positive $\omega_c$ in the angular term without tracking the sign of $qB$.
• Forgetting the diamagnetic contribution to $\Omega$.
• Treating the model as one-dimensional; the angular-momentum term is intrinsically two-dimensional.
• Expecting Landau-level degeneracy to survive unchanged when $\omega_0\ne0$.
• Including Zeeman splitting without saying that spin has been added to the spinless model.
• Calling every magnetic oscillator spectrum “Landau levels” even when confinement has lifted the degeneracy.

Exercises

1. Expand the symmetric-gauge Hamiltonian and identify $\Omega$.

Solution

In symmetric gauge,

$$\mathbf A = \frac{B}{2} \left( -y\,\hat{\mathbf x} + x\,\hat{\mathbf y} \right),$$

so

$$\frac{1}{2m} \left( \hat{\mathbf p}-q\mathbf A \right)^2 = \frac{\hat p_x^2+\hat p_y^2}{2m} + \frac{q^2B^2}{8m}r^2 - \frac{qB}{2m}\hat L_z.$$

Adding $\frac12m\omega_0^2r^2$ gives

$$\frac12m \left( \omega_0^2+\frac{q^2B^2}{4m^2} \right) r^2.$$

Since $\omega_c=\lvert q\rvert B/m$,

$$\Omega = \sqrt{ \omega_0^2+\frac{\omega_c^2}{4} }.$$

2. Starting from the two-dimensional oscillator spectrum with frequency $\Omega$, derive the Fock–Darwin energies.

Solution

The isotropic two-dimensional oscillator energies are

$$E^{(0)}_{n_r,\mu} = \hbar\Omega \left( 2n_r+\lvert\mu\rvert+1 \right).$$

The angular-momentum term is

$$- \frac{\omega_q}{2}\hat L_z.$$

Since $\hat L_z\psi=\hbar\mu\psi$, this contributes

$$- \frac{\hbar\omega_q}{2}\mu.$$

Therefore

$$E_{n_r,\mu} = \hbar\Omega \left( 2n_r+\lvert\mu\rvert+1 \right) - \frac{\hbar\omega_q}{2}\mu.$$

3. Check the $B\to0$ limit.

Solution

As $B\to0$,

$$\omega_c\to0, \qquad \omega_q\to0, \qquad \Omega\to\omega_0.$$

The energy becomes

$$E_{n_r,\mu} \to \hbar\omega_0 \left( 2n_r+\lvert\mu\rvert+1 \right),$$

which is the isotropic two-dimensional oscillator spectrum.

4. Check the $\omega_0\to0$ limit using circular-mode frequencies.

Solution

If $\omega_0\to0$, then

$$\Omega\to\frac{\omega_c}{2}.$$

Therefore

$$\omega_- = \Omega-\frac{\omega_c}{2} \to 0,$$

and

$$\omega_+ = \Omega+\frac{\omega_c}{2} \to \omega_c.$$

The finite mode is the cyclotron oscillator. The zero mode is the guiding-center degeneracy of the Landau problem.

5. For $qB\gt0$, which sign of angular momentum is lowered by the magnetic term?

Solution

If $qB\gt0$, then $\omega_q\gt0$, and the angular term is

$$- \frac{\hbar\omega_q}{2}\mu.$$

Positive $\mu$ lowers the energy and negative $\mu$ raises it, all else equal. If the sign of $qB$ is reversed, this conclusion reverses.

References

• C. G. Darwin, “The diamagnetism of the free electron,” Mathematical Proceedings of the Cambridge Philosophical Society 27, 86-90, 1931.
• V. Fock, “Bemerkung zur Quantelung des harmonischen Oszillators im Magnetfeld,” Zeitschrift für Physik 47, 446-448, 1928.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.