Charged Particles in Electromagnetic Fields

Electromagnetic fields expose a feature that is easy to miss in scalar-potential problems: the wave equation depends on potentials, while physical predictions must be independent of the gauge used to represent them. The resulting distinction between canonical and kinetic momentum organizes magnetic quantization, gauge-covariant probability flow, and interference around magnetic flux.

This chapter develops that structure for a spinless nonrelativistic particle in prescribed classical fields. Its central model is the uniform magnetic field, where transverse motion becomes an oscillator, each oscillator rung acquires a guiding-center degeneracy, and the number of states is fixed by magnetic flux.

Helpful background. The Electromagnetism Checklist supplies potentials and fields; the Time-Dependent Schrödinger Equation supplies Hamiltonian evolution; the Quantum Harmonic Oscillator supplies ladder energies and wavefunctions.

Current reviewed route

The reviewed public sequence is deliberately narrow: Minimal Coupling in Wave Mechanics establishes the gauge-covariant Hamiltonian, and Landau Levels derives and checks the ideal magnetic ladder. The chapter map then extends these calculations to gauge comparisons, flux counting, interference, and confinement.

Scope and conventions

The baseline Hamiltonian is

$$H = \frac{1}{2m} \left( -i\hbar\nabla-q\mathbf A \right)^2 +q\Phi,$$

where $q$ is the signed particle charge. The potentials produce

$$\mathbf E = -\nabla\Phi -\frac{\partial\mathbf A}{\partial t}, \qquad \mathbf B = \nabla\times\mathbf A.$$

Unless stated otherwise, uniform-field pages take

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

and use the positive cyclotron frequency

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

The sign of $qB$ still controls the orientation of cyclotron motion, the direction in which guiding-center labels are ordered, and the sign of orbital angular-momentum splittings.

The chapter treats electromagnetic fields as backgrounds. It does not quantize radiation, include pair creation, or derive gauge theory as a fundamental field theory. It also omits spin unless explicitly discussing how an extra spin degeneracy would alter counting. Spin and relativistic continuations cover the Pauli Equation and Dirac Equation. Zeeman structure, many-particle quantum Hall physics, disorder, edges, and radiative transitions require additional physics beyond the ideal orbital models developed here.

Minimal Coupling in Wave Mechanics owns the coordinate-space coupling and current. The structural interpretation of gauge redundancy belongs to Gauge Transformations in Quantum Mechanics, while this chapter keeps the first encounter computational.

Minimal coupling

Define the canonical momentum and kinetic momentum by

$$\mathbf p = -i\hbar\nabla, \qquad \boldsymbol\pi = \mathbf p-q\mathbf A.$$

Then

$$H = \frac{\boldsymbol\pi^2}{2m} +q\Phi, \qquad \mathbf v = \frac{\boldsymbol\pi}{m}.$$

The two momenta answer different questions. Canonical momentum is the differential generator used in a chosen representation. Kinetic momentum is proportional to mechanical velocity. In a magnetic field its components need not commute:

$$[\pi_i,\pi_j] = i\hbar q \sum_k\epsilon_{ijk}B_k.$$

For $\mathbf B=B\hat{\mathbf z}$,

$$[\pi_x,\pi_y] = i\hbar qB.$$

This noncommuting pair is the algebraic origin of the transverse oscillator. It also shows why replacing $\mathbf p$ by a numerical momentum too early can erase essential magnetic physics.

The square in the Hamiltonian is an operator product. For position-dependent $\mathbf A$,

$$\begin{aligned} (\mathbf p-q\mathbf A)^2\psi &= -\hbar^2\nabla^2\psi +iq\hbar(\nabla\cdot\mathbf A)\psi \\ &\quad +2iq\hbar\mathbf A\cdot\nabla\psi +q^2\mathbf A^2\psi. \end{aligned}$$

Keeping the covariant square intact until a gauge is chosen prevents ordering and sign errors.

Gauge covariance

A real function $\chi(\mathbf r,t)$ generates the transformation

$$\mathbf A' = \mathbf A+\nabla\chi, \qquad \Phi' = \Phi-\frac{\partial\chi}{\partial t},$$

together with

$$\psi' = U_\chi\psi, \qquad U_\chi = \exp\left( \frac{iq\chi}{\hbar} \right).$$

The field strengths are unchanged. More importantly, the covariant derivative transforms with the state:

$$\left( -i\hbar\nabla-q\mathbf A' \right)\psi' = U_\chi \left( -i\hbar\nabla-q\mathbf A \right)\psi.$$

The time derivative and scalar potential transform as a matching pair, so the Schrödinger equation has the same physical content. For a time-dependent gauge transformation the Hamiltonian transformation includes the derivative of $U_\chi$; it is not merely $H'=U_\chi H U_\chi^\dagger$. This is why one should compare dynamics and observables, not bare canonical energies under unrelated time-dependent phase conventions.

The distinction among invariant, covariant, and gauge-dependent objects is useful:

Object	Gauge behavior	Physical use
$\mathbf E$, $\mathbf B$	Invariant	Local electromagnetic fields
$\lvert\psi\rvert^2$	Invariant	Probability density
$\psi$	Changes by $U_\chi$	Gauge-dependent state representative
$\mathbf A$, $\Phi$	Change with $\chi$	Potentials used in the wave equation
$\boldsymbol\pi\psi$	Covariant	Mechanical momentum amplitudes
Canonical labels such as a Landau-gauge $k_y$	Representation dependent	Convenient basis bookkeeping
Closed-loop phase $e^{iq\oint\mathbf A\cdot d\mathbf r/\hbar}$	Invariant	Interference and holonomy

Gauge Transformations: First Encounter derives these rules and separates choosing a gauge from changing the physical state.

Gauge-covariant probability flow

The probability density remains

$$\rho=\lvert\psi\rvert^2.$$

The conserved current for minimal coupling is

$$\mathbf j = \frac{1}{m} \operatorname{Re} \left[ \psi^* \left( -i\hbar\nabla-q\mathbf A \right) \psi \right].$$

Equivalently,

$$\mathbf j = \frac{\hbar}{2mi} \left( \psi^*\nabla\psi -\psi\nabla\psi^* \right) -\frac{q}{m}\mathbf A\lvert\psi\rvert^2.$$

It satisfies

$$\frac{\partial\rho}{\partial t} +\nabla\cdot\mathbf j = 0.$$

If $\psi=R e^{iS/\hbar}$, then

$$\mathbf j = \frac{\rho}{m} \left( \nabla S-q\mathbf A \right).$$

The phase gradient and vector potential are separately gauge dependent, but their combination is invariant. Using the free-particle current after introducing $\mathbf A$ breaks this cancellation.

Uniform magnetic field

Classically, the Lorentz force rotates transverse velocity at frequency $\omega_c$ while leaving motion parallel to $\mathbf B$ free. Quantum mechanically, the transverse Hamiltonian is

$$H_\perp = \frac{\pi_x^2+\pi_y^2}{2m}.$$

Introduce the magnetic length

$$\ell_B = \sqrt{ \frac{\hbar}{\lvert q\rvert B} }$$

and $s=\operatorname{sgn}(qB)$. The operator

$$a = \frac{\ell_B}{\sqrt2\,\hbar} \left( \pi_x+i s\pi_y \right)$$

satisfies $[a,a^\dagger]=1$, giving

$$H_\perp = \hbar\omega_c \left( a^\dagger a+\frac12 \right).$$

The field creates oscillator algebra without an ordinary scalar potential well. In three dimensions,

$$H = H_\perp +\frac{p_z^2}{2m},$$

so quantization is transverse and the longitudinal spectrum remains continuous in the ideal infinite system.

Particle in a Uniform Magnetic Field develops the classical-to-quantum setup, sign conventions, and the split between cyclotron and guiding-center motion.

Landau levels

Choose Landau gauge,

$$\mathbf A_L=Bx\,\hat{\mathbf y}.$$

Then

$$H_\perp = \frac{1}{2m} \left[ p_x^2 +(p_y-qBx)^2 \right].$$

Because $p_y$ commutes with this Hamiltonian, write

$$\psi(x,y) = \frac{e^{ik_y y}}{\sqrt{L_y}} \varphi(x).$$

The remaining equation is a harmonic oscillator centered at

$$x_0 = \frac{\hbar k_y}{qB},$$

with frequency $\omega_c$. Its energy does not depend on $k_y$:

$$E_n = \hbar\omega_c \left( n+\frac12 \right), \qquad n=0,1,2,\ldots .$$

The integer $n$ labels cyclotron excitation. The continuous or discretized $k_y$ label locates the guiding center and spans the degeneracy inside one Landau level. Treating both labels as independent energy excitations double-counts the physical structure.

Landau Levels is the canonical derivation of the spectrum and wavefunctions.

Landau gauge and symmetric gauge

The same field also follows from symmetric gauge,

$$\mathbf A_S = \frac12\mathbf B\times\mathbf r = \frac{B}{2} \left( -y\,\hat{\mathbf x} +x\,\hat{\mathbf y} \right).$$

The two potentials are related by

$$\mathbf A_L-\mathbf A_S = \nabla\left( \frac{Bxy}{2} \right),$$

so corresponding wavefunction representatives differ by

$$\psi_L = \exp\left( \frac{iqBxy}{2\hbar} \right) \psi_S.$$

Their practical strengths differ:

Gauge	Manifest organization	Natural geometry	Typical degeneracy label
Landau gauge	Translation along one direction	Strip or rectangle	$k_y$ or $x_0$
Symmetric gauge	Rotation about the field axis	Disk or circular trap	Angular label and guiding-center radius

Gauge transformation alone multiplies a state by a phase. Comparing the standard basis formulas also involves a change of basis inside a degenerate Landau-level subspace. Consequently, one Landau-gauge basis vector need not map to one symmetric-gauge angular-momentum basis vector.

Landau Gauge and Symmetric Gauge owns this basis comparison. The spectrum, total state count, and consistently transformed currents do not depend on which gauge makes the calculation convenient.

Flux-counting degeneracy

Place the particle in a large rectangle $L_x\times L_y$ and impose periodicity along $y$. Then

$$k_y = \frac{2\pi r}{L_y}, \qquad r\in\mathbb Z.$$

Adjacent guiding centers have separation

$$\Delta x_0 = \frac{2\pi\ell_B^2}{L_y}.$$

The number whose centers fit across width $L_x$ is therefore

$$N_\Phi \simeq \frac{L_xL_y}{2\pi\ell_B^2} = \frac{A}{2\pi\ell_B^2}.$$

Hence the bulk orbital density in each spinless Landau level is

$$\frac{N_\Phi}{A} = \frac{1}{2\pi\ell_B^2} = \frac{\lvert q\rvert B}{h}.$$

If $\Phi_B=BA$ and $\Phi_0=h/\lvert q\rvert$, the count is

$$N_\Phi = \frac{\lvert q\Phi_B\rvert}{h} = \frac{\lvert\Phi_B\rvert}{\Phi_0}.$$

This is one orbital state per flux quantum, not one electron per flux quantum: particle occupancy is an additional many-body question. On an infinite plane the degeneracy is infinite, so density per area is the meaningful statement. With physical edges the bulk formula is asymptotic; on a torus, consistent boundary conditions require integer total flux and make the ideal count exact.

For the fuller rectangular, symmetric-gauge, and guiding-center-algebra arguments, read Degeneracy of Landau Levels.

Aharonov–Bohm phase

Potentials carry global information even in a region where the magnetic field vanishes locally. For two coherent paths whose closed combination $C$ encloses inaccessible magnetic flux,

$$\Delta\varphi_{\mathrm{AB}} = \frac{q}{\hbar} \oint_C \mathbf A\cdot d\mathbf r = \frac{q\Phi_B}{\hbar}.$$

The observable object is the phase factor

$$\exp\left( \frac{iq}{\hbar} \oint_C\mathbf A\cdot d\mathbf r \right),$$

not the value of $\mathbf A$ at a point. It is invariant under ordinary single-valued gauge transformations and periodic under

$$\Phi_B \longrightarrow \Phi_B+\frac{h}{\lvert q\rvert}.$$

The effect is not caused by a local Lorentz force along ideal field-free paths. The accessible region has a hole, so a locally removable vector potential need not be removable by one smooth, single-valued phase convention over the whole region.

On a flux-threaded ring,

$$E_n(\Phi_B) = \frac{\hbar^2}{2I} \left( n-\frac{q\Phi_B}{h} \right)^2, \qquad n\in\mathbb Z.$$

The same physics can be represented by a vector potential in the Hamiltonian or by a twisted boundary condition after a local gauge removal. Continue to the dedicated Aharonov–Bohm Effect: First Encounter derivation and its geometric interpretation.

Confinement and the Fock–Darwin model

Add an isotropic two-dimensional trap,

$$V(r)=\frac12m\omega_0^2r^2,$$

and use symmetric gauge. The Hamiltonian becomes

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

where

$$\omega_q=\frac{qB}{m}, \qquad \Omega = \sqrt{ \omega_0^2+\frac{\omega_c^2}{4} }.$$

The Fock–Darwin spectrum is

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

Two circular-mode frequencies expose the limiting structure:

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

When $B\to0$, both recover the isotropic oscillator organization. When $\omega_0\to0$, one mode approaches zero and becomes guiding-center freedom, while the other approaches $\omega_c$ and carries cyclotron energy. Finite confinement lifts the extensive Landau degeneracy.

The dedicated Charged Harmonic Oscillator in a Magnetic Field treatment develops this interpolation and its quantum-dot interpretation.

Model hierarchy

Model	Energy-carrying structure	Degeneracy or second mode	Main caution
General minimal coupling	Kinetic momentum $\boldsymbol\pi$	Depends on fields and geometry	Preserve operator ordering and gauge covariance
Uniform magnetic field in 2D	Cyclotron oscillator	Guiding-center Hilbert space	Gauge labels are not universal observables
Uniform magnetic field in 3D	Landau ladder plus free $k_z$	Guiding center and longitudinal continuum	Field does not confine parallel motion
Flux-threaded ring	Shifted angular spectrum	Flux-periodic relabeling	Local field-free does not imply globally trivial phase
Fock–Darwin oscillator	Two circular modes	Trap lifts guiding-center degeneracy	Track the sign of $qB$ in angular splitting

The durable workflow is:

1. State the charge sign, fields, scalar potential, vector-potential convention, dimension, and boundary conditions.
2. Distinguish canonical momentum from kinetic momentum.
3. Identify gauge-invariant observables before interpreting basis labels.
4. Separate cyclotron excitation from guiding-center degeneracy.
5. Check the zero-field, unconfined, and large-area limits appropriate to the model.

Reading route

For a first pass:

1. Begin with Minimal Coupling in Wave Mechanics.
2. Derive the ideal orbital spectrum in Landau Levels.
3. Return to this gateway for the degeneracy, Aharonov–Bohm, and Fock–Darwin summaries.

Use the specialist pages below for gauge transformations, uniform-field setup, gauge comparisons, degeneracy, interference, and confinement.

Core Page Map

Page	Canonical role
Minimal Coupling in Wave Mechanics	Potentials, kinetic momentum, operator ordering, current, and phase form
Landau Levels	Landau-gauge solution, energy ladder, wavefunctions, and guiding-center split

Specialist Page Map

Continue according to the calculation you need:

• Gauge Transformations: First Encounter
• Particle in a Uniform Magnetic Field
• Landau Gauge and Symmetric Gauge
• Degeneracy of Landau Levels
• Aharonov–Bohm Effect: First Encounter
• Charged Harmonic Oscillator in a Magnetic Field

Common mistakes

• Using $\mathbf p$ as mechanical momentum after introducing $\mathbf A$.
• Expanding $(\mathbf p-q\mathbf A)^2$ as if derivatives did not act on $\mathbf A$.
• Changing the potentials without transforming the wavefunction phase.
• Calling a gauge-dependent wavefunction or canonical label directly observable.
• Using the free-particle current in a vector potential.
• Treating $qB/m$ as a positive frequency instead of separating sign from $\omega_c$.
• Interpreting the Landau-gauge $k_y$ label as an ordinary physical momentum.
• Comparing one Landau-gauge basis state with one symmetric-gauge basis state without allowing mixing inside the degenerate subspace.
• Counting an infinite-plane degeneracy without specifying area or density.
• Multiplying by spin degeneracy after Zeeman splitting has removed it.
• Confusing the single-particle flux quantum $h/\lvert q\rvert$ with the superconducting value for charge $2e$.
• Claiming that the Aharonov–Bohm effect makes the gauge-dependent vector potential itself observable.
• Forgetting that ideal three-dimensional Landau motion remains free along the field.
• Calling confined Fock–Darwin levels ordinary Landau levels after the trap has lifted their guiding-center degeneracy.

References

• Y. Aharonov and D. Bohm, “Significance of Electromagnetic Potentials in the Quantum Theory,” Physical Review 115, 485–491, 1959.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.
• R. E. Prange and S. M. Girvin, eds., The Quantum Hall Effect, 2nd ed., Springer, 1990.
• 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.

Exercises

1. Verify the gauge covariance of kinetic momentum under $\mathbf A'=\mathbf A+\nabla\chi$ and $\psi'=e^{iq\chi/\hbar}\psi$.

Solution

Differentiate the transformed state:

$$-i\hbar\nabla\psi' = e^{iq\chi/\hbar} \left( -i\hbar\nabla\psi +q\nabla\chi\,\psi \right).$$

Then

$$\begin{aligned} (&-i\hbar\nabla-q\mathbf A')\psi' \\ &= e^{iq\chi/\hbar} \left[ -i\hbar\nabla\psi +q\nabla\chi\,\psi -q\mathbf A\psi -q\nabla\chi\,\psi \right] \\ &= e^{iq\chi/\hbar} (-i\hbar\nabla-q\mathbf A)\psi. \end{aligned}$$

The cancellation is why the wavefunction phase must transform together with the potential.

2. In Landau gauge, derive the oscillator center and the two-dimensional Landau spectrum.

Solution

For $\mathbf A=Bx\,\hat{\mathbf y}$,

$$H_\perp = \frac{1}{2m} \left[ p_x^2+(p_y-qBx)^2 \right].$$

With $\psi=e^{ik_y y}\varphi(x)/\sqrt{L_y}$,

$$(p_y-qBx)^2 \longrightarrow (\hbar k_y-qBx)^2 = (qB)^2(x-x_0)^2,$$

where

$$x_0=\frac{\hbar k_y}{qB}.$$

The remaining equation is a harmonic oscillator of frequency $\lvert q\rvert B/m$, so

$$E_n = \hbar\omega_c \left( n+\frac12 \right).$$

The center $x_0$ changes with $k_y$, but the energy does not.

3. Count the bulk spinless degeneracy of one Landau level in a rectangle of area $A=L_xL_y$.

Solution

Periodic boundary conditions along $y$ give spacing

$$\Delta k_y=\frac{2\pi}{L_y}.$$

The corresponding guiding-center spacing is

$$\lvert\Delta x_0\rvert = \frac{\hbar\Delta k_y}{\lvert q\rvert B} = \frac{2\pi\ell_B^2}{L_y}.$$

Approximately $L_x/\lvert\Delta x_0\rvert$ centers fit in the sample, giving

$$N_\Phi \simeq \frac{L_xL_y}{2\pi\ell_B^2} = \frac{A}{2\pi\ell_B^2} = \frac{\lvert q\rvert BA}{h}.$$

This is an orbital count. Internal spin, valley, or layer factors may multiply it only if the Hamiltonian leaves those states degenerate.

4. A particle on a ring encloses flux $\Phi_B$. Show that increasing the flux by $h/\lvert q\rvert$ leaves the spectrum as a set unchanged.

Solution

The levels are

$$E_n(\Phi_B) = \frac{\hbar^2}{2I} \left( n-\frac{q\Phi_B}{h} \right)^2.$$

Let $s_q=q/\lvert q\rvert$. Under

$$\Phi_B \longrightarrow \Phi_B+\frac{h}{\lvert q\rvert},$$

the bracket becomes

$$n-\frac{q\Phi_B}{h}-s_q.$$

Relabeling the integer by $n'=n-s_q$ restores the original set of values. Individual labels move along the flux-dependent parabolas, but the spectrum as an unlabeled set is flux periodic.

5. Use the circular-mode frequencies of the Fock–Darwin model to recover both the zero-field oscillator and unconfined Landau limits.

Solution

The frequencies are

$$\omega_\pm = \Omega\pm\frac{\omega_c}{2}, \qquad \Omega = \sqrt{ \omega_0^2+\frac{\omega_c^2}{4} }.$$

When $B\to0$,

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

so both circular modes have frequency $\omega_0$, as required for the isotropic two-dimensional oscillator.

When $\omega_0\to0$,

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

and therefore

$$\omega_-\to0, \qquad \omega_+\to\omega_c.$$

The finite mode is cyclotron excitation; the zero mode is the guiding-center freedom that produces Landau degeneracy.