Showing a strange wavefunction satisfies the TDSE.

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Homework Help Overview

The problem involves demonstrating that a modified wavefunction φ(x,t) satisfies the time-dependent Schrödinger equation (TDSE) for a free particle. The original wavefunction ψ(x,t) is known to obey the TDSE, and the task is to show that φ(x,t) does as well under certain conditions related to parameters a and v.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss taking the partial derivative of φ with respect to time and the challenges associated with applying the product rule. There is uncertainty about how to relate the derivative of ψ(x-vt, t) to ψ(x,t). Some participants express confusion specifically regarding the second term in the derivative expression.

Discussion Status

The discussion is ongoing, with participants sharing their attempts to differentiate the wavefunction and questioning the steps involved. There is no explicit consensus, but guidance is being offered regarding the application of the product rule and the nature of partial derivatives.

Contextual Notes

Participants are operating under the constraints of the TDSE and the specific forms of the wavefunctions provided. There is a focus on ensuring that the conditions for a and v are met, which may influence the approach taken in the discussion.

Robsta
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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) = ei(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.
 
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Robsta said:

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) = ei(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?
 
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.
 
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
 

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