Active and passive views of gauge invariance in quantum mechanics

For a quantum mechanical particle in a classical electromagnetic field, the active and passive views of gauge invariance are compared. In the active point of view, both the wavefunction and the potentials are transformed so that the Schrödinger equation is form invariant. In the passive point of view, a change in the representation of the canonical momentum is made which just cancels the change in the gauge of the potentials. The condition that a gauge‐invariant operator has the same expectation value in all gauges gives a functional equation, the solution of which shows that gauge‐invariant operators depend only on the kinetic momentum or coordinates.