50
2. Electromagnetism as a Gauge Theory
The answer to the question just posed is evidently negative, since it is
clear that the same ‘ψ’ cannot possibly satisfy both (2.29) and the analogous
′
equation with (V, A) replaced by (V , A
′ ). Unlike Maxwell’s equations, the
Schr¨ odinger equation is not gauge invariant. But we must remember that the
wavefunction ψ is not a directly observable quantity, as the electromagnetic
fields E and B are. Perhaps ψ does not need to remain unchanged (invariant) when the potentials are changed by a gauge transformation. In fact,
in order to have any chance of ‘describing the same physics’ in terms of the
gauge-transformed potentials, we will have to allow ψ to change as well. This
is a crucial point: for quantum mechanics to be consistent with Maxwell’s
equations it is necessary for the gauge transformations (2.31) and (2.32) of
the Maxwell potentials to be accompanied also by a transformation of the
quantum-mechanical wavefunction, ψ → ψ
′ , where ψ
′ satisfies the equation
(
)
1
∂ψ
′ (x, t)
′
(−i∇ − qA
′ )
2 + qV ψ
′ (x, t) = i
.
(2.33)
2m
∂t
Note that the form of (2.33) is exactly the same as the form of (2.29) – it is
this that will effectively ensure that both ‘describe the same physics’. Readers
of appendix D will expect to be told that – if we can find such a ψ
′ – we may
then assert that (2.29) is gauge covariant, meaning that it maintains the same
form under a gauge transformation. (The transformations relevant to this use
of ‘covariance’ are gauge transformations.)
′
Since we know the relations (2.31) and (2.32) between A, V and A
′ , V ,
we can actually find what ψ
′ (x, t) must be in order that equation (2.33) be
consistent with (2.29). We shall state the answer and then verify it; then we
shall discuss the physical interpretation. The required ψ
′ (x, t) is
ψ
′ (x, t) = exp[iqχ(x, t)]ψ(x, t)
(2.34)
where χ is the same space–time-dependent function as appears in equations
(2.31) and (2.32). To verify this we consider
(−i∇ − qA
′ )ψ
′
= [−i∇ − qA − q(∇χ)][exp(iqχ)ψ]
= q(∇χ) exp(iqχ)ψ + exp(iqχ) · (−i∇ψ)
+ exp(iqχ) · (−qAψ) − q(∇χ) exp(iqχ)ψ. (2.35)
The first and the last terms cancel leaving the result:
(−i∇ − qA
′ )ψ
′ = exp(iqχ) · (−i∇ − qA)ψ
(2.36)
which may be written using equation (2.30) as:
(−iD
′ ψ
′ ) = exp(iqχ) · (−iDψ).
(2.37)
Thus, although the space–time-dependent phase factor feels the action of the
gradient operator ∇, it ‘passes through’ the combined operator D
′ and converts it into D: in fact comparing the equations (2.34) and (2.37), we see that
2. Electromagnetism as a Gauge Theory
The answer to the question just posed is evidently negative, since it is
clear that the same ‘ψ’ cannot possibly satisfy both (2.29) and the analogous
′
equation with (V, A) replaced by (V , A
′ ). Unlike Maxwell’s equations, the
Schr¨ odinger equation is not gauge invariant. But we must remember that the
wavefunction ψ is not a directly observable quantity, as the electromagnetic
fields E and B are. Perhaps ψ does not need to remain unchanged (invariant) when the potentials are changed by a gauge transformation. In fact,
in order to have any chance of ‘describing the same physics’ in terms of the
gauge-transformed potentials, we will have to allow ψ to change as well. This
is a crucial point: for quantum mechanics to be consistent with Maxwell’s
equations it is necessary for the gauge transformations (2.31) and (2.32) of
the Maxwell potentials to be accompanied also by a transformation of the
quantum-mechanical wavefunction, ψ → ψ
′ , where ψ
′ satisfies the equation
(
)
1
∂ψ
′ (x, t)
′
(−i∇ − qA
′ )
2 + qV ψ
′ (x, t) = i
.
(2.33)
2m
∂t
Note that the form of (2.33) is exactly the same as the form of (2.29) – it is
this that will effectively ensure that both ‘describe the same physics’. Readers
of appendix D will expect to be told that – if we can find such a ψ
′ – we may
then assert that (2.29) is gauge covariant, meaning that it maintains the same
form under a gauge transformation. (The transformations relevant to this use
of ‘covariance’ are gauge transformations.)
′
Since we know the relations (2.31) and (2.32) between A, V and A
′ , V ,
we can actually find what ψ
′ (x, t) must be in order that equation (2.33) be
consistent with (2.29). We shall state the answer and then verify it; then we
shall discuss the physical interpretation. The required ψ
′ (x, t) is
ψ
′ (x, t) = exp[iqχ(x, t)]ψ(x, t)
(2.34)
where χ is the same space–time-dependent function as appears in equations
(2.31) and (2.32). To verify this we consider
(−i∇ − qA
′ )ψ
′
= [−i∇ − qA − q(∇χ)][exp(iqχ)ψ]
= q(∇χ) exp(iqχ)ψ + exp(iqχ) · (−i∇ψ)
+ exp(iqχ) · (−qAψ) − q(∇χ) exp(iqχ)ψ. (2.35)
The first and the last terms cancel leaving the result:
(−i∇ − qA
′ )ψ
′ = exp(iqχ) · (−i∇ − qA)ψ
(2.36)
which may be written using equation (2.30) as:
(−iD
′ ψ
′ ) = exp(iqχ) · (−iDψ).
(2.37)
Thus, although the space–time-dependent phase factor feels the action of the
gradient operator ∇, it ‘passes through’ the combined operator D
′ and converts it into D: in fact comparing the equations (2.34) and (2.37), we see that