Charged Particles in Electromagnetic Fields

Electromagnetic fields couple a particle’s motion to scalar and vector potentials. This chapter explains how to use those potentials without confusing a convenient gauge choice with a physical prediction. The examples treat a spinless nonrelativistic particle of signed charge $q$ in prescribed classical fields.

Helpful background. The Electromagnetism Checklist relates potentials to fields; the Time-Dependent Schrödinger Equation supplies Hamiltonian evolution; the Quantum Harmonic Oscillator supplies the ladder used for magnetic quantization.

Mechanical momentum and magnetic quantization

The Hamiltonian and mechanical momentum are

$$H=\frac{(\mathbf p-q\mathbf A)^2}{2m}+q\Phi, \qquad \mathbf p=-i\hbar\nabla, \qquad \boldsymbol\pi=\mathbf p-q\mathbf A.$$

Mechanical velocity is $\boldsymbol\pi/m$. The square in $H$ is an operator product: derivatives act on the vector potential as well as on the wavefunction.

For a uniform field $\mathbf B=B\hat{\mathbf z}$ with $B>0$, the transverse mechanical momenta obey $[\pi_x,\pi_y]=i\hbar qB$. Their oscillator algebra gives

$$E_n=\hbar\omega_c\left(n+\frac12\right), \qquad \omega_c=\frac{|q|B}{m},\qquad n=0,1,\ldots .$$

These are the spinless Landau levels for transverse motion. In three dimensions there is also free motion parallel to the field. Different guiding-center locations give the degeneracy within a level; they are not additional oscillator excitations.

Gauge transformations change both the potentials and the state representative:

$$\mathbf A'=\mathbf A+\nabla\chi,\qquad \Phi'=\Phi-\partial_t\chi,\qquad \psi'=e^{iq\chi/\hbar}\psi.$$

Density and the consistently calculated current remain unchanged. A time-dependent gauge transformation also changes the Hamiltonian by a time-derivative term, so its instantaneous eigenvalues need not remain unchanged. Compare physical observables and evolution using the transformed operators and states together.

Reading sequence

Article	What you will learn
Minimal Coupling	Hamiltonian, mechanical momentum, operator ordering, and current
Gauge Transformations	Why potentials and wavefunction phases must transform together
Uniform Magnetic Field	Cyclotron motion and the free longitudinal direction
Landau Levels	The spectrum and wavefunctions in a convenient gauge
Landau and Symmetric Gauges	Gauge transformations and basis changes within degenerate levels
Landau-Level Degeneracy	One bulk orbital state per flux quantum
Aharonov–Bohm Effect	Interference around inaccessible magnetic flux
Charged Oscillator	How a trap modifies magnetic motion and lifts degeneracy

Spin interactions require the Pauli Equation. The orbital models here also omit radiation, pair creation, and many-particle quantum Hall effects.

Exercises

In a large two-dimensional sample of area $A$, the number of bulk orbital states per Landau level is approximately $N=A|q|B/h$. What happens to both the level spacing and this state count when the magnetic field doubles? Why is $N$ not automatically the number of electrons?

Solution

The spacing $\hbar|q|B/m$ and the bulk orbital count both double. The count describes available one-particle states. Their occupancy depends on the number of particles, temperature, chemical potential, and statistics. Spin can introduce additional states or split energies; it is excluded here. Physical edges also modify the ideal bulk count and energies.

Two coherent paths enclose flux $\Phi_B$ while remaining in a field-free region. Their magnetic relative phase is $q\Phi_B/\hbar$. Find a flux increment that changes this phase by an integer multiple of $2\pi$.

Solution

An increment $h/|q|$ changes the phase by $2\pi\operatorname{sgn}(q)$. The interference phase factor is therefore periodic in this flux increment, with the other path phases held fixed. The effect depends on the enclosed flux even though the ideal paths experience no local magnetic force.

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.