Sign Issue with Zitelli Quantum Mechanics 3.6

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Homework Help Overview

The discussion revolves around a quantum mechanics problem involving a particle in an infinite potential well. Participants are tasked with proving a relationship between the time derivative of momentum and the gradient of the probability current density vector.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants are examining the equation dp/dt + ∇J(x,t) = 0, noting discrepancies in their results where the two terms appear equal and opposite. They express curiosity about the underlying reasons for this outcome and seek suggestions for improvement.

Discussion Status

Some participants have shared their work and are actively seeking feedback. There is a suggestion to use LaTeX for clarity, indicating a potential path forward in resolving the confusion regarding the negative sign issue. However, there is no explicit consensus on the correct interpretation or resolution of the problem.

Contextual Notes

Participants mention that they have checked their findings with multiple people, indicating a collaborative effort to understand the problem. There is also a note about the legibility of shared work, which may be affecting the discussion.

katieranthony
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Homework Statement
A particle of mass m, in an infinite potential well of length a, has the following initial wave function at t = 0:
Relevant Equations
probability current density vector and partial time derivative of p
I will include a photo. Essentially, we need to prove that dp/dt + the gradient of J(x,t) = 0.
We are getting that the two are instead equal and opposite, and have checked with multiple people.
 
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katieranthony said:
Homework Statement: A particle of mass m, in an infinite potential well of length a, has the following initial wave function at t = 0:
Relevant Equations: probability current density vector and partial time derivative of p

I will include a photo. Essentially, we need to prove that dp/dt + the gradient of J(x,t) = 0.
We are getting that the two are instead equal and opposite, and have checked with multiple people.
 
1707977324631.png
 
Here is our work. Please let us know if you have any suggestions! We are genuinely very curious.
 

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katieranthony said:
Here is our work. Please let us know if you have any suggestions! We are genuinely very curious.
Your work in the photo is illegible. It would be best if you used LaTeX. When you do so, maybe you will resolve the negative sign problem. It's probably some silly mistake that you keep repeating because you are convinced that it's correct (it has happened to me too). Start fresh and don't copy from the board.
 
katieranthony said:
Homework Statement: A particle of mass m, in an infinite potential well of length a, has the following initial wave function at t = 0:
Relevant Equations: probability current density vector and partial time derivative of p

We are getting that the two are instead equal and opposite, and have checked with multiple people.
What two things are equal and opposite?? If ##x_1## and##~x_2## are equal and opposite , don't they sum to zero?? I am confused.
 

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