QuantumPhysics.dvi

Under gauge transformations,

A→A′=A+∇θ Φ→Φ′= Φ−

∂θ

∂t

(7.7)

the fieldsBandEare invariant (i.e. B→B′ =BandE→E′ =E) for an arbitrary

functionθ=θ(r,t). The classical Lagrangian transforms as follows,

L→L′=L+e

(

r ̇·∇θ+

∂θ

∂t

)

=L′+

d(eθ)

dt

(7.8)

As a result, the classical actionS=

∫

dtLis invariant (except for boundary effects) and hence

the classical mechanics of a charged particle is invariant under suchgauge transformations.

This is of course well-known, since the equations of motion reduce to

m ̈r=eE(r,t) +er ̇×B(r,t) (7.9)

which are manifestly gauge invariant.

The Schr ̈odinger equation (7.5) is nowinvariant since the fieldsAand Φ are clearly

changed under the transformation, but (7.5) is insteadcovariantprovided we also transform

the wave function by,

ψ(r,t)→ψ′(r,t) =eiγθψ(r,t) (7.10)

for some constantγ, which remains to be determined. To see how this works, we use the

gauge transformation rules of (19.6) and (7.10) to derive the following intermediate formulas,

(

−i ̄h∇r−eA′

)

ψ′ = eiγθ

(

−i ̄h∇r−eA

)

ψ+eiγθ( ̄hγ+e)(∇θ)ψ

(

i ̄h∂t−eΦ′

)

ψ′ = eiγθ

(

i ̄h∂t−eΦ(r,t)

)

ψ+eiγθ( ̄hγ+e)(∂tθ)ψ (7.11)

For the value

γ=−

e

h ̄

(7.12)

the second terms on the right hand side cancel, and we see that these particular combinations

(referred to asgauge covariant derivatives) transform exactly in the same manner that the

wave functionψdid in (19.6). As a result, the full Schr ̈odinger equation transforms just as

ψdid, and this property is referred to ascovarianceof the equation.