Problems
111
4.6 The Galilean transformation (non-relativistic boost) is defined by
′
′
x = x − vt, t = t.
Show that the free-particle time-dependent Schr¨ odinger equation is covariant
under this transformation if the wavefunction transforms according to the rule
′
ψ
′ (x , t
′ ) = exp[if (x, t)]ψ(x, t), where f (x, t) satisfies the condition
∂f
1
i
i
−
− v · ∇f + iv · ∇ =
(∇f )
2
−
∇
2 f − ∇f · ∇.
∂t
2m
2m
m
Find constants a and b such that the function f = at + b · x satisfies this
condition. Show that the resulting transformation rule is consistent with the
way you expect a plane wave solution to transform.
111
4.6 The Galilean transformation (non-relativistic boost) is defined by
′
′
x = x − vt, t = t.
Show that the free-particle time-dependent Schr¨ odinger equation is covariant
under this transformation if the wavefunction transforms according to the rule
′
ψ
′ (x , t
′ ) = exp[if (x, t)]ψ(x, t), where f (x, t) satisfies the condition
∂f
1
i
i
−
− v · ∇f + iv · ∇ =
(∇f )
2
−
∇
2 f − ∇f · ∇.
∂t
2m
2m
m
Find constants a and b such that the function f = at + b · x satisfies this
condition. Show that the resulting transformation rule is consistent with the
way you expect a plane wave solution to transform.