51
2.4. Gauge invariance (and covariance) in quantum mechanics
D
′ ψ
′ bears to Dψ exactly the same relation as ψ
′ bears to ψ. In just the
same way we find (cf equation (2.30))
(iD
0′ ψ
′ ) = exp(iqχ) · (iD
0 ψ)
(2.38)
where we have used equation (2.32) for V
′ . Once again, D
0′ ψ
′ is simply related
to D
0 ψ. Repeating the operation which led to equation (2.37) we find
1 (−iD
′ )
2 ψ
′
= exp(iqχ) ·
1 (−iD)
2 ψ
2m
2m
= exp(iqχ) · iD
0 ψ
(using equation (2.29))
= iD
0′ ψ
′
(using equation (2.30)).
(2.39)
Equation (2.39) is just (2.33) written in the D notation of equation (2.30),
so we have verified that (2.34) is the correct relationship between ψ
′ and
ψ to ensure consistency between equations (2.29) and (2.33). Precisely this
consistency is summarized by the statement that (2.29) is gauge covariant.
Do ψ and ψ
′ describe the same physics, in fact? The answer is yes, but it
is not quite trivial. It is certainly obvious that the probability densities |ψ|
2
and |ψ
′
|
2 are equal, since in fact ψ and ψ
′ in equation (2.34) are related by
a phase transformation. However, we can be interested in other observables
involving the derivative operators ∇ or ∂/∂t – for example, the current, which
is essentially ψ
∗ (∇ψ) − (∇ψ)
∗ ψ. It is easy to check that this current is
not invariant under (2.34), because the phase χ(x, t) is x-dependent. But
equations (2.37) and (2.38) show us what we must do to construct gaugeinvariant currents: namely, we must replace ∇ by D (and in general also
∂/∂t by D
0 ) since then:
ψ
∗′ (D
′ ψ
′ ) = ψ
∗ exp(−iqχ) · exp(iqχ) · (Dψ) = ψ
∗
Dψ
(2.40)
for example. Thus the identity of the physics described by ψ and ψ
′ is indeed
ensured. Note, incidentally, that the equality between the first and last terms
in (2.40) is indeed a statement of (gauge) invariance.
We summarize these important considerations by the statement that the
gauge invariance of Maxwell equations re-emerges as a covariance in quantum
mechanics provided we make the combined transformation
A → A
′ = A + ∇χ
′
V → V = V − ∂χ/∂t
(2.41)
ψ → ψ
′ = exp(iqχ)ψ
on the potential and on the wavefunction.
The Schr¨ odinger equation is non-relativistic, but the Maxwell equations are
of course fully relativistic. One might therefore suspect that the prescriptions
discovered here are actually true relativistically as well, and this is indeed
2.4. Gauge invariance (and covariance) in quantum mechanics
D
′ ψ
′ bears to Dψ exactly the same relation as ψ
′ bears to ψ. In just the
same way we find (cf equation (2.30))
(iD
0′ ψ
′ ) = exp(iqχ) · (iD
0 ψ)
(2.38)
where we have used equation (2.32) for V
′ . Once again, D
0′ ψ
′ is simply related
to D
0 ψ. Repeating the operation which led to equation (2.37) we find
1 (−iD
′ )
2 ψ
′
= exp(iqχ) ·
1 (−iD)
2 ψ
2m
2m
= exp(iqχ) · iD
0 ψ
(using equation (2.29))
= iD
0′ ψ
′
(using equation (2.30)).
(2.39)
Equation (2.39) is just (2.33) written in the D notation of equation (2.30),
so we have verified that (2.34) is the correct relationship between ψ
′ and
ψ to ensure consistency between equations (2.29) and (2.33). Precisely this
consistency is summarized by the statement that (2.29) is gauge covariant.
Do ψ and ψ
′ describe the same physics, in fact? The answer is yes, but it
is not quite trivial. It is certainly obvious that the probability densities |ψ|
2
and |ψ
′
|
2 are equal, since in fact ψ and ψ
′ in equation (2.34) are related by
a phase transformation. However, we can be interested in other observables
involving the derivative operators ∇ or ∂/∂t – for example, the current, which
is essentially ψ
∗ (∇ψ) − (∇ψ)
∗ ψ. It is easy to check that this current is
not invariant under (2.34), because the phase χ(x, t) is x-dependent. But
equations (2.37) and (2.38) show us what we must do to construct gaugeinvariant currents: namely, we must replace ∇ by D (and in general also
∂/∂t by D
0 ) since then:
ψ
∗′ (D
′ ψ
′ ) = ψ
∗ exp(−iqχ) · exp(iqχ) · (Dψ) = ψ
∗
Dψ
(2.40)
for example. Thus the identity of the physics described by ψ and ψ
′ is indeed
ensured. Note, incidentally, that the equality between the first and last terms
in (2.40) is indeed a statement of (gauge) invariance.
We summarize these important considerations by the statement that the
gauge invariance of Maxwell equations re-emerges as a covariance in quantum
mechanics provided we make the combined transformation
A → A
′ = A + ∇χ
′
V → V = V − ∂χ/∂t
(2.41)
ψ → ψ
′ = exp(iqχ)ψ
on the potential and on the wavefunction.
The Schr¨ odinger equation is non-relativistic, but the Maxwell equations are
of course fully relativistic. One might therefore suspect that the prescriptions
discovered here are actually true relativistically as well, and this is indeed