# Showing a strange wavefunction satisfies the TDSE.

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1. May 12, 2015

### Robsta

1. The problem statement, all variables and given/known data
The wavefunction ψ(x,t) obeys the time-dependent Schrodinger equation for a free particle of mass m moving in one dimension.

Show that a second wavefunction φ(x,t) = ei(ax-bt)ψ(x-vt , t) obeys the same time dependent schrodinger equation, provided a = ħa2/2m and v = ħa/m

2. Relevant equations

The time dependent schrodinger equation is iħ∂φ/∂t = Hφ

Hφ = -ħ2/2m * ∂2φ/∂x2
3. The attempt at a solution

Taking the partial derivative of φ with respect to t, and using the product rule, you have to take ∂ψ(x-vt, t)/∂t and I'm not sure how to do this.

2. May 12, 2015

### SammyS

Staff Emeritus
Which thing are you having trouble with? the product rule? or taking a partial derivative?

3. May 12, 2015

### Robsta

I guess it's taking the partial derivative. I don't know how I can relate ∂ψ(x-vt, t)/∂t to ∂ψ(x,t)/∂t if that's what needs doing.

4. May 12, 2015

### Robsta

I've taken ∂φ/∂t and I'm trying to show it equals Hφ/iħ because then φ satisfies the TDSE.

∂φ/∂t = (-iħa2/2m)ei(ax-bt)ψ(x-vt, t) + ei(ax-bt)∂ψ(x-vt, t)/∂t
But I can't really deal with the second term of that eqn