Following the sketch given in this answer, I hoped to solve the 1+1 dimensional Schrodinger equation under a potential f(t)x by using a time dependent boost.
(−ℏ22m∂2∂x2+f(t)x)Ψ(x,t)=iℏ∂Ψ∂t
When I attempt to apply a boost in the form
Ψ(x,t)=e−iF(t)ℏx˜Ψ(x,t)
where F(t) is the antiderivative of f(t), I get
−ℏ22m(∂2˜Ψ∂x2−2iF(t)ℏ∂˜Ψ∂x−F(t)2ℏ2˜Ψ2)=iℏ∂˜Ψ∂t
but I am unsure of how to proceed. If I attempt to find a separable solution in the form ˜Ψ(x,t)=X(x)Φ(t), I cannot resolve the cross term.
−ℏ22m(¨XX−2iF(t)ℏ˙XX)=iℏ˙ΦΦ−F(t)22m
Did I choose the wrong ansatz for a boost? Or, is my derivation mistaken?