Skip to content
Quantum Mechanics

Minimal Coupling

Minimal coupling is the rule that replaces canonical momentum by gauge-covariant kinetic momentum:

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

For a spinless nonrelativistic particle in prescribed electromagnetic potentials, the Hamiltonian is

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 page Minimal Coupling in Wave Mechanics owns the detailed wave-equation treatment. This page explains why the replacement is the natural gauge-covariant structure and how to interpret it inside symmetry and geometry.

The Gauge-Covariant Momentum

Section titled “The Gauge-Covariant Momentum”

In position representation,

p^=−iℏ∇.\hat{\mathbf p} = -i\hbar\nabla.

A local phase transformation

ψ′=eiqχ/ℏψ\psi' = e^{iq\chi/\hbar}\psi

does not commute cleanly with ∇\nabla. The derivative produces an extra ∇χ\nabla\chi term. The page Local Phase Transformations derives this explicitly.

Minimal coupling introduces a vector potential so that

π^=−iℏ∇−qA\hat{\boldsymbol\pi} = -i\hbar\nabla-q\mathbf A

transforms covariantly when

A′=A+∇χ.\mathbf A' = \mathbf A+\nabla\chi.

The key identity is

π^′ψ′=eiqχ/ℏπ^ψ.\hat{\boldsymbol\pi}'\psi' = e^{iq\chi/\hbar} \hat{\boldsymbol\pi}\psi.

Thus π^ψ\hat{\boldsymbol\pi}\psi has the same local phase behavior as ψ\psi itself. This is the mathematical reason the kinetic momentum, not the canonical momentum alone, belongs in the Hamiltonian.

Covariant-Derivative Form

Section titled “Covariant-Derivative Form”

Define

Di=∂i−iqℏAi,Dt=∂t+iqℏΦ.D_i = \partial_i-\frac{iq}{\hbar}A_i, \qquad D_t = \partial_t+\frac{iq}{\hbar}\Phi.

Then

π^i=−iℏDi,\hat\pi_i = -i\hbar D_i,

and the minimally coupled Schrödinger equation is

iℏDtψ=−ℏ22m∑iDiDiψ.i\hbar D_t\psi = -\frac{\hbar^2}{2m} \sum_iD_iD_i\psi.

Under

ψ′=eiqχ/ℏψ,A′=A+∇χ,Φ′=Φ−∂tχ,\psi' = e^{iq\chi/\hbar}\psi, \qquad \mathbf A' = \mathbf A+\nabla\chi, \qquad \Phi' = \Phi-\partial_t\chi,

the covariant derivatives obey

Di′ψ′=eiqχ/ℏDiψ,Dt′ψ′=eiqχ/ℏDtψ.D_i'\psi' = e^{iq\chi/\hbar}D_i\psi, \qquad D_t'\psi' = e^{iq\chi/\hbar}D_t\psi.

The equation therefore has the same form in every gauge.

Classical Correspondence

Section titled “Classical Correspondence”

Minimal coupling also matches the classical charged-particle Lagrangian

L=12mr˙ 2+qr˙⋅A(r,t)−qΦ(r,t).L = \frac{1}{2}m\dot{\mathbf r}^{\,2} + q\dot{\mathbf r}\cdot\mathbf A(\mathbf r,t) - q\Phi(\mathbf r,t).

The canonical momentum is

p=∂L∂r˙=mr˙+qA,\mathbf p = \frac{\partial L}{\partial\dot{\mathbf r}} = m\dot{\mathbf r}+q\mathbf A,

so the mechanical, or kinetic, momentum is

mr˙=p−qA.m\dot{\mathbf r} = \mathbf p-q\mathbf A.

Quantization promotes this same combination to the operator π^\hat{\boldsymbol\pi}. This correspondence is useful, but the quantum reason is sharper: the combination is the one that transforms covariantly under local phase changes.

Canonical and Kinetic Momentum

Section titled “Canonical and Kinetic Momentum”

Canonical momentum and kinetic momentum answer different questions.

Canonical momentum:

p^=−iℏ∇\hat{\mathbf p} = -i\hbar\nabla

is tied to the chosen position-space phase convention and to ordinary translations. Kinetic momentum:

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

is tied to mechanical velocity:

v^=π^m.\hat{\mathbf v} = \frac{\hat{\boldsymbol\pi}}{m}.

The commutators show the field strength:

[r^i,π^j]=iℏδij,[\hat r_i,\hat\pi_j] = i\hbar\delta_{ij},

while

[π^i,π^j]=iℏq∑kϵijkBk.[\hat\pi_i,\hat\pi_j] = i\hbar q \sum_k \epsilon_{ijk}B_k.

The noncommuting kinetic momenta are the algebraic seed of Landau quantization and magnetic translations.

Potentials, Fields, and Curvature

Section titled “Potentials, Fields, and Curvature”

The electromagnetic fields are

B=∇×A,E=−∇Φ−∂tA.\mathbf B = \nabla\times\mathbf A, \qquad \mathbf E = -\nabla\Phi-\partial_t\mathbf A.

In geometric language, the potentials are connection data and the fields are curvature data. The potentials are not unique, but they organize how phases are compared at neighboring spacetime points.

This explains why the vector potential appears in the wave equation even though the Lorentz force is written in terms of E\mathbf E and B\mathbf B. Locally, fields determine forces. Quantum mechanically, potentials also determine phase transport. When the accessible region has nontrivial topology, phase holonomy can remain observable even where the local magnetic field vanishes along the path.

Pure Gauge and Holonomy

Section titled “Pure Gauge and Holonomy”

If

A=∇χ,Φ=−∂tχ,\mathbf A=\nabla\chi, \qquad \Phi=-\partial_t\chi,

then the fields vanish:

E=0,B=0.\mathbf E=\mathbf 0, \qquad \mathbf B=\mathbf 0.

On a simply connected region with a single-valued χ\chi, this potential can be removed by a gauge transformation. The wavefunction changes by the phase eiqχ/ℏe^{iq\chi/\hbar}, and no gauge-invariant local field remains.

The global caveat is important. In a multiply connected region, a potential can be locally pure gauge while still having nontrivial loop phase:

exp⁡(iqℏ∮CA⋅dr).\exp \left( \frac{iq}{\hbar} \oint_C \mathbf A\cdot d\mathbf r \right).

That is the geometric core of the Aharonov–Bohm Effect.

Spin and Nonminimal Terms

Section titled “Spin and Nonminimal Terms”

For a scalar nonrelativistic particle, minimal coupling gives the orbital electromagnetic coupling. A spin-1/21/2 particle also has magnetic moment coupling. In the ideal g=2g=2 Pauli Hamiltonian,

H=π^ 22m+qΦ−qℏ2mσ⋅B.H = \frac{\hat{\boldsymbol\pi}^{\,2}}{2m} + q\Phi - \frac{q\hbar}{2m} \boldsymbol\sigma\cdot\mathbf B.

The last term is not obtained by merely expanding the scalar Hamiltonian. It reflects spin structure. More general particles can also have anomalous magnetic moments, electric dipole couplings, polarizability terms, and other effective interactions. Those are called nonminimal couplings because they go beyond the replacement p^↦p^−qA\hat{\mathbf p}\mapsto\hat{\mathbf p}-q\mathbf A.

Scope and Boundaries

Section titled “Scope and Boundaries”

This page assumes prescribed classical electromagnetic potentials. It does not quantize the electromagnetic field, describe photon emission, include radiation reaction, or treat pair creation. Those questions require quantum electrodynamics or effective many-body light-matter theory.

Within nonrelativistic quantum mechanics, minimal coupling is enough for many standard systems: charged particles in uniform fields, Landau levels, the Aharonov–Bohm effect, orbital magnetism, and gauge-covariant translation symmetry.

Common Mistakes

Section titled “Common Mistakes”
  • Expanding (p^−qA)2\left(\hat{\mathbf p}-q\mathbf A\right)^2 while forgetting that derivatives act on A\mathbf A and on everything to their right.
  • Treating p^\hat{\mathbf p} as the mechanical momentum in a magnetic field.
  • Changing gauge potentials without transforming the wavefunction phase.
  • Forgetting the scalar potential when the gauge function depends on time.
  • Assuming minimal coupling automatically includes all spin or relativistic effects.
  • Calling A\mathbf A directly observable rather than using gauge-invariant fields, currents, and loop phases.

References

Section titled “References”
  • P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford University Press, 1958.
  • 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.
  • C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
  • Y. Aharonov and D. Bohm, “Significance of electromagnetic potentials in the quantum theory,” Physical Review 115, 485-491, 1959.

Exercises

Section titled “Exercises”
  1. Show that the pure-gauge potentials A=∇χ\mathbf A=\nabla\chi and Φ=−∂tχ\Phi=-\partial_t\chi give zero electromagnetic fields.
Solution

For the magnetic field,

B=∇×A=∇×∇χ=0.\mathbf B = \nabla\times\mathbf A = \nabla\times\nabla\chi = \mathbf 0.

For the electric field,

E=−∇Φ−∂tA=−∇(−∂tχ)−∂t(∇χ)=∇∂tχ−∂t∇χ=0.\begin{aligned} \mathbf E &= -\nabla\Phi-\partial_t\mathbf A \\ &= -\nabla(-\partial_t\chi) - \partial_t(\nabla\chi) \\ &= \nabla\partial_t\chi - \partial_t\nabla\chi = \mathbf 0. \end{aligned}
  1. Starting from π^i=−iℏ∂i−qAi\hat\pi_i=-i\hbar\partial_i-qA_i, compute [π^i,π^j][\hat\pi_i,\hat\pi_j].
Solution

Acting on a test wavefunction and using that the AiA_i commute as multiplication operators,

[π^i,π^j]=[−iℏ∂i,−qAj]+[−qAi,−iℏ∂j]=iℏq(∂iAj−∂jAi).\begin{aligned} [\hat\pi_i,\hat\pi_j] &= [-i\hbar\partial_i,-qA_j] + [-qA_i,-i\hbar\partial_j] \\ &= i\hbar q(\partial_iA_j-\partial_jA_i). \end{aligned}

Since

∂iAj−∂jAi=∑kϵijkBk,\partial_iA_j-\partial_jA_i = \sum_k\epsilon_{ijk}B_k,

we get

[π^i,π^j]=iℏq∑kϵijkBk.[\hat\pi_i,\hat\pi_j] = i\hbar q \sum_k\epsilon_{ijk}B_k.
  1. Explain why adding the Pauli spin term does not contradict the name “minimal coupling.”
Solution

Minimal coupling names the gauge-covariant replacement of canonical momentum by p^−qA\hat{\mathbf p}-q\mathbf A and the scalar-potential term qΦq\Phi. A spinor particle has additional internal structure, so its Hamiltonian may include magnetic-moment coupling such as −μ⋅B-\boldsymbol\mu\cdot\mathbf B. That term is compatible with gauge invariance but is not part of the scalar minimal replacement itself. In effective descriptions, further allowed terms are called nonminimal couplings.