How to "derive" momentum operator in position basis using STE?

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Discussion Overview

The discussion revolves around the derivation of the momentum operator in the position basis using a specific form of the Schrödinger time evolution (STE). Participants explore different approaches and challenges related to this derivation, including the use of equations and the motivation behind the operator's form.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant expresses difficulty in using LaTeX for posting equations and shares a handwritten screenshot instead.
  • Another participant suggests troubleshooting steps for LaTeX issues, such as logging out and clearing cookies.
  • A third participant notes that the derivation by integration requires the Hamiltonian to include the kinetic energy term, specifically mentioning the form involving the Laplacian operator.
  • A participant asks for clarification on the abbreviation "STE," indicating a lack of understanding of the term used in the discussion.

Areas of Agreement / Disagreement

The discussion does not appear to have a consensus, as participants are exploring different aspects of the derivation and addressing technical issues without resolving the main question.

Contextual Notes

There are limitations regarding the clarity of the derivation due to the use of screenshots instead of LaTeX, and the specific form of STE being discussed is not universally understood among participants.

LightPhoton
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TL;DR
I ask about how one can use generalized STE to motivate momentum operator in position basis using the approach of Griffiths and Schroeter
I am not able to use Latex for some reason. It is very glitchy and if I do one backspace then it fills my whole screen with multiple copies of the same equation. Thus I am pasting a screenshot of handwritten equations instead. Apologies for any inconvenience.

1714255503282.png




In Introduction to Quantum Mechanics by Griffiths and Schroeter, the IMO motivates the form of momentum operator in position basis in a very nice manner. However, the problem is that they use a very specific form of STE (1)

Instead, I want to work in a much more general setting by writing STE as (2)

Now, the authors motivate it by taking time derivative of the expectation value of the position, which leads me to (3).

However, I am not sure how to proceed from here.
 
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LightPhoton said:
I am not able to use Latex for some reason.
You might try logging out, clearing cookies, and then logging in again.

You might also try a different browser.
 
For the derivation by integration you would need the fact that H includes "kinetic energy" part of ##-\hbar^2/2m \ \nabla_x^2##.
 
Sorry, what does it mean STE ?
 
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