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Quantum Mechanics

Charged Particle in a Magnetic Field Hamiltonian

Hamiltonian

Section titled “Hamiltonian”

For charge qq in scalar and vector potentials Φ\Phi and A\mathbf A,

H=12m(p^−qA(r^,t))2+qΦ(r^,t).H = \frac{1}{2m} \left( \hat{\mathbf p}-q\mathbf A(\hat{\mathbf r},t) \right)^2 + q\Phi(\hat{\mathbf r},t).

The kinetic momentum is

π=p^−qA.\boldsymbol\pi = \hat{\mathbf p}-q\mathbf A.

For a uniform magnetic field and no scalar potential, this Hamiltonian produces Landau quantization in the transverse motion.

Assumptions

Section titled “Assumptions”
  • Nonrelativistic particle in prescribed classical electromagnetic potentials.
  • The charge sign qq is explicit.
  • Gauge choice and boundary conditions are specified.
  • Spin coupling is omitted unless a Pauli term is added.

Common Mistakes

Section titled “Common Mistakes”
  • Confusing canonical momentum p^\hat{\mathbf p} with kinetic momentum π\boldsymbol\pi.
  • Treating a gauge-dependent vector potential as directly observable.
  • Forgetting the scalar-potential term qΦq\Phi when electric fields are present.
  • Adding spin degeneracy or Zeeman splitting without stating it.
Section titled “Canonical Links”

References

Section titled “References”
  • 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.