Showing a strange wavefunction satisfies the TDSE.

[Robsta]

Homework Statement

The wavefunction ψ(x,t) obeys the time-dependent Schrödinger equation for a free particle of mass m moving in one dimension.

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

Homework Equations

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

Hφ = -ħ²/2m * ∂²φ/∂x²

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.

 

[SammyS]

Homework Statement

The wavefunction ψ(x,t) obeys the time-dependent Schrödinger equation for a free particle of mass m moving in one dimension.

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

Homework Equations

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

Hφ = -ħ²/2m * ∂²φ/∂x²

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.

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

 

[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.

 

[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)e^i(ax-bt)ψ(x-vt, t) + e^i(ax-bt)∂ψ(x-vt, t)/∂t
But I can't really deal with the second term of that eqn