Minimal Coupling in Wave Mechanics

Minimal coupling is the standard way a nonrelativistic charged particle couples to prescribed electromagnetic fields in wave mechanics. The ordinary momentum operator is replaced by a gauge-covariant momentum, and the scalar electric potential contributes to the energy.

Helpful background. The Time-Dependent Schrödinger Equation identifies the Hamiltonian’s dynamical role; Probability Current supplies the flow interpretation used later; Canonical Commutation Relations supports operator checks.

For a particle of mass $m$ and charge $q$, the Hamiltonian is

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

Here $\Phi(\mathbf r,t)$ is the scalar potential and $\mathbf A(\mathbf r,t)$ is the vector potential. The time-dependent Schrödinger equation is

$$i\hbar\frac{\partial\psi}{\partial t} = \left[ \frac{1}{2m} \left( -i\hbar\nabla-q\mathbf A \right)^2 +q\Phi \right]\psi .$$

This page treats $\Phi$ and $\mathbf A$ as classical background fields. Spin, radiation reaction, pair creation, and quantized electromagnetic fields require additional structure.

Potentials and Fields

The electric and magnetic fields are obtained from the potentials by

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

The potentials are not unique. Many different pairs $(\Phi,\mathbf A)$ describe the same physical fields. This is not a defect of the formalism; it is gauge redundancy.

The Hamiltonian is nevertheless written in terms of potentials because quantum phases are sensitive to them. Magnetic fields enter local forces through $\mathbf B$, but vector potentials enter the wave equation directly. The Aharonov–Bohm phase discussion shows where this distinction becomes unavoidable.

Canonical and Kinetic Momentum

In position representation, the canonical momentum operator is

$$\hat{\mathbf p}=-i\hbar\nabla.$$

Minimal coupling defines the kinetic, or mechanical, momentum operator

$$\hat{\boldsymbol\pi} = \hat{\mathbf p}-q\mathbf A = -i\hbar\nabla-q\mathbf A.$$

The Hamiltonian is then

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

The distinction matters. Canonical momentum is tied to translations and representation conventions. Kinetic momentum is tied to mechanical velocity:

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

In a magnetic field, the kinetic momentum components generally do not commute:

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

This noncommutativity is one of the algebraic seeds of magnetic quantization and Landau levels.

Operator Ordering

The compact expression

$$\left( -i\hbar\nabla-q\mathbf A \right)^2$$

must be read as an operator product. Since derivatives act on everything to their right, $\mathbf A(\mathbf r,t)$ and $\nabla$ do not commute when $\mathbf A$ depends on position.

Acting on a wavefunction,

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

This is why it is safer to keep the gauge-covariant form until a specific gauge has been chosen. Expanding too early is a common source of sign and ordering errors.

Gauge Transformations

A gauge transformation is specified by a scalar function $\chi(\mathbf r,t)$:

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

These changes leave the physical fields $\mathbf E$ and $\mathbf B$ unchanged. The wavefunction must transform at the same time:

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

This equation is often written with the exponent displayed as $iq\chi/\hbar$; the exponential is a phase because $\chi$ is real.

The covariant momentum transforms consistently:

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

Thus the transformed Schrödinger equation has the same physical content. Gauge transformations change the description, not the measured probabilities.

Probability Current

The density remains

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

For the minimally coupled Hamiltonian, the conserved probability current is

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

Equivalently,

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

With this definition,

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

The free-particle current formula is recovered when $\mathbf A=0$. Keeping the free-particle expression after introducing a vector potential gives a non-gauge-covariant current and generally the wrong local conservation law. The derivation and flux interpretation are developed in Probability Current and Continuity Equation.

Phase Form

Write

$$\psi(\mathbf r,t) = R(\mathbf r,t)e^{iS(\mathbf r,t)/\hbar}, \qquad R\ge0.$$

Then the density is $\rho=R^2$, and the current becomes

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

The gauge-invariant local velocity field is therefore controlled by

$$\nabla S-q\mathbf A,$$

not by $\nabla S$ alone. Under a gauge transformation,

$$S'=S+q\chi, \qquad \mathbf A'=\mathbf A+\nabla\chi,$$

so

$$\nabla S'-q\mathbf A' = \nabla S-q\mathbf A.$$

This is the simplest way to see why the vector potential term in the current is not optional.

Simple Limits and Examples

If $\mathbf A=0$ and $\Phi$ is time independent, minimal coupling reduces to the ordinary scalar-potential Hamiltonian

$$\hat H = \frac{\hat{\mathbf p}^2}{2m} +q\Phi(\mathbf r).$$

The potential energy is $V=q\Phi$.

For a uniform magnetic field

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

two common choices are the Landau gauge

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

and the symmetric gauge

$$\mathbf A = \frac{1}{2}\mathbf B\times\mathbf r.$$

Both give the same magnetic field:

$$\nabla\times\mathbf A = B\hat{\mathbf z}.$$

The wavefunctions and conserved-looking quantum numbers differ between gauges, but gauge-invariant energies and measurable densities agree. This is why magnetic-field problems should be phrased in terms of fields, fluxes, currents, and kinetic momenta, not only in terms of one chosen vector potential.

Common Mistakes

• Writing $q\Phi$ but forgetting the $-q\mathbf A$ shift in momentum.
• Treating $\hat{\mathbf p}$ and $\hat{\boldsymbol\pi}$ as the same observable in a magnetic field.
• Expanding $\left(\hat{\mathbf p}-q\mathbf A\right)^2$ as if $\nabla$ did not act on $\mathbf A$.
• Calling a gauge-dependent wavefunction phase unphysical without checking gauge-invariant interference effects.
• Using the free-particle probability current after turning on $\mathbf A$.
• Thinking that a particular gauge choice is the physical magnetic field.

Exercises

1. Verify the gauge covariance of the kinetic momentum operator.

Solution

Let

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

Then

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

Therefore

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

2. Starting from $\psi=Re^{iS/\hbar}$, derive the current $\mathbf j=\rho(\nabla S-q\mathbf A)/m$.

Solution

The covariant momentum acting on $\psi$ is

$$\left( -i\hbar\nabla-q\mathbf A \right)\psi = e^{iS/\hbar} \left[ -i\hbar\nabla R +R\nabla S -qR\mathbf A \right].$$

Multiplying by $\psi^*=Re^{-iS/\hbar}$ gives

$$\psi^* \left( -i\hbar\nabla-q\mathbf A \right)\psi = -i\hbar R\nabla R +R^2 \left( \nabla S-q\mathbf A \right).$$

The first term is imaginary, so the real part is

$$\rho \left( \nabla S-q\mathbf A \right).$$

Dividing by $m$ gives the stated current.

3. Check that the Landau gauge $\mathbf A=Bx\,\hat{\mathbf y}$ gives $\mathbf B=B\hat{\mathbf z}$.

Solution

In Cartesian components,

$$A_x=0, \qquad A_y=Bx, \qquad A_z=0.$$

The curl has

$$B_z = \frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y} =B-0=B.$$

The other components vanish:

$$B_x=0, \qquad B_y=0.$$

Thus

$$\nabla\times\mathbf A = B\hat{\mathbf z}.$$

Core Connections

• Landau Levels apply minimal coupling to a uniform magnetic field.
• The Aharonov–Bohm phase discussion shows how vector potentials produce gauge-invariant interference phases around excluded flux.

Further Connections

These pages develop the current, gauge symmetry, magnetic models, and applications:

• Probability Current for gauge-covariant current and flux.
• Gauge Transformations: First Encounter for the gauge phase and covariance rules.
• Particle in a Uniform Magnetic Field for magnetic oscillator reduction.
• Magnetic Translations for translation symmetry after minimal coupling.
• Magnetic Moments from Orbital Motion for the orbital Zeeman term.
• Continuity Equation states local probability conservation for the minimally coupled Hamiltonian.
• Dimensionless Variables and Scaling introduces the magnetic length used in uniform-field problems.
• Momentum Operator records the warning that canonical and kinetic momentum differ in electromagnetic fields.
• Why Symmetry Becomes Central explains why gauge transformations are redundancies rather than ordinary physical symmetries.
• From Phase Symmetry to Gauge Theory connects the local phase rule used here to the QFT gauge-theory bridge.
• London Theory applies the same gauge-covariant momentum to a charged condensate and derives magnetic screening.
• Ginzburg–Landau Theory makes that condensate amplitude spatially dynamical and derives critical fields and vortex cores.

References

• 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.
• 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.