Let us consider the action of Galilean transformations on a quantum mechanical wave function. We restrict ourselves here to the ``proper'' Galilean Transformations: (i) translations; (ii) velocity boosts; (iii) rotations. We shall consider a transformation to be acting on the state (not on the observer). Thus, a translation by x0 on a state localized at x1 produces a new state, localized at x1+x0. In ``configuration space'', we have a wave function of the form ψ(x,t). A translation T(x0) by x0 of this state yields a new state:
ψ′(x)=T(x0)ψ(x,t)=ψ(x−x0,t).(1.122)
Consider a free particle of mass m. The momentum space wave function is
ψ^(p,t)=f^(p)exp(−2mitp2),(1.123)
where p=∣p∣. The configuration space wave function is related by the (inverse) Fourier transform:
ψ(x,t)=(2π)3/21∫(∞)d3(p)eix⋅pψ^(p,t).(1.124)
Obtain simple transformation laws, on both the momentum and configuration space wave functions, for each of the following proper Galilean transformations:
(a) Translation by x0: T(x0) (note that we have already seen the result in configuration space).
(b) Translation by time t0: M(t0).
(c) Velocity boost by v0: V(v0). (Hint: first find
ψ^′(p,0)=f^′(p)=V(v0)f^(p),(1.125)
then
ψ^′(p,t)=f^′(p)e−itp2/2m,(1.126)
etc.)
(d) Rotation about the origin given by 3×3 matrix R: U(R).
Make sure your answers make sense to you in terms of classical correspondence.