49
2.4. Gauge invariance (and covariance) in quantum mechanics
2.4 Gauge invariance (and covariance) in quantum
mechanics
The Lorentz force law for a non-relativistic particle of charge q moving with
velocity v under the influence of both electric and magnetic fields is
F = qE + qv × B.
(2.27)
It may be derived, via Hamilton’s equations, from the classical Hamiltonian
2
H =
1 (p − qA)
2 + qV.
(2.28)
2m
The Schr¨ odinger equation for such a particle in an electromagnetic field is
(
)
1
∂ψ(x, t)
(−i∇ − qA)
2 + qV ψ(x, t) = i
(2.29)
2m
∂t
which is obtained from the classical Hamiltonian by the usual prescription,
p → −i∇, for Schr¨ odinger’s wave mechanics (ħ = 1). Note the appearance of
the operator combinations
D ≡ ∇ − iqA
(2.30)
D
0
≡ ∂/∂t + iqV
in place of ∇ and ∂/∂t, in going from the free-particle Schr¨ odinger equation
to the electromagnetic field case.
The solution ψ(x, t) of the Schr¨ odinger equation (2.29) describes completely the state of the particle moving under the influence of the potentials
V , A. However, these potentials are not unique, as we have already seen:
they can be changed by a gauge transformation
A → A
′ = A + ∇χ
(2.31)
′
V → V
= V − ∂χ/∂t
(2.32)
and the Maxwell equations for the fields E and B will remain the same.
This immediately raises a serious question: if we carry out such a change
of potentials in equation (2.29), will the solution of the resulting equation
describe the same physics as the solution of equation (2.29)? If it does,
we shall be able to assume the validity of Maxwell’s theory for the quantum world; if not, some modification will be necessary, since the gauge symmetry possessed by the Maxwell equations will be violated in the quantum
theory.
2 We set ħ = c = 1 throughout (see appendix B).
2.4. Gauge invariance (and covariance) in quantum mechanics
2.4 Gauge invariance (and covariance) in quantum
mechanics
The Lorentz force law for a non-relativistic particle of charge q moving with
velocity v under the influence of both electric and magnetic fields is
F = qE + qv × B.
(2.27)
It may be derived, via Hamilton’s equations, from the classical Hamiltonian
2
H =
1 (p − qA)
2 + qV.
(2.28)
2m
The Schr¨ odinger equation for such a particle in an electromagnetic field is
(
)
1
∂ψ(x, t)
(−i∇ − qA)
2 + qV ψ(x, t) = i
(2.29)
2m
∂t
which is obtained from the classical Hamiltonian by the usual prescription,
p → −i∇, for Schr¨ odinger’s wave mechanics (ħ = 1). Note the appearance of
the operator combinations
D ≡ ∇ − iqA
(2.30)
D
0
≡ ∂/∂t + iqV
in place of ∇ and ∂/∂t, in going from the free-particle Schr¨ odinger equation
to the electromagnetic field case.
The solution ψ(x, t) of the Schr¨ odinger equation (2.29) describes completely the state of the particle moving under the influence of the potentials
V , A. However, these potentials are not unique, as we have already seen:
they can be changed by a gauge transformation
A → A
′ = A + ∇χ
(2.31)
′
V → V
= V − ∂χ/∂t
(2.32)
and the Maxwell equations for the fields E and B will remain the same.
This immediately raises a serious question: if we carry out such a change
of potentials in equation (2.29), will the solution of the resulting equation
describe the same physics as the solution of equation (2.29)? If it does,
we shall be able to assume the validity of Maxwell’s theory for the quantum world; if not, some modification will be necessary, since the gauge symmetry possessed by the Maxwell equations will be violated in the quantum
theory.
2 We set ħ = c = 1 throughout (see appendix B).