Quantum Mechanics problem: Determine the value of the constant

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Homework Help Overview

The problem involves quantum mechanics, specifically focusing on the relationship between operators and their commutators within the context of canonical quantization. The original poster seeks assistance in understanding how the commutator relates to a specific identity involving energy eigenstates.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster expresses uncertainty about how to begin the problem and requests hints, particularly regarding the connection between the commutator and the identity. They also inquire about a specific expectation value involving the commutator.
  • Another participant emphasizes the importance of respectful dialogue while addressing the request for hints.
  • The original poster later shares their progress after attempting calculations related to the problem, indicating a shift in their understanding.

Discussion Status

The discussion reflects a mix of initial confusion and subsequent clarity as the original poster engages with the problem. Some guidance has been offered regarding the commutator, and the original poster has made progress in their understanding, although no consensus or final solution has been reached.

Contextual Notes

The problem statement includes specific assumptions about the Hamiltonian's form and requires the determination of a constant related to energy eigenstates. The original poster's request for hints suggests they are navigating complex concepts within quantum mechanics.

Ineedhelpimbadatphys
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Homework Statement
The problem states work for word.

Using canonical quantization relation, prove that
sum operator ((E_n -E_0)) |< E_n | X | E_0 >|^2) = constant

Where E_0 is the energy corresponding to the eigenstate | E_0 >. Determine the value of the constant. Assume the hamiltonian had a general form H = P/2m +V(X)

Hint: One way to proof this is to think how [H, X], X] is connected to the obove identity.
Relevant Equations
all equations i have are in the statement.
I have no idea where to start with this problem. I am interested in any hints, or ways to proof this. But i would especially like to know how the commutator is connected to the identity.
 

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Please, everyone, be respectful of poster asking for a hint about one specific aspect of this problem.
 
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Ineedhelpimbadatphys said:
Homework Statement:: The problem states work for word.

Using canonical quantization relation, prove that
sum operator ((E_n -E_0)) |< E_n | X | E_0 >|^2) = constant

Where E_0 is the energy corresponding to the eigenstate | E_0 >. Determine the value of the constant. Assume the hamiltonian had a general form H = P/2m +V(X)

Hint: One way to proof this is to think how [H, X], X] is connected to the obove identity.
Relevant Equations:: all equations i have are in the statement.

I have no idea where to start with this problem. I am interested in any hints, or ways to proof this. But i would especially like to know how the commutator is connected to the identity.
What is ##< E_0 \mid [H,X],X]] \mid E_0 >##?

-Dan
 
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Thank you so much. I did actually manage to figure it out. I had tried calculatibg that, and got stuck at < E_0 | XHX | E_n > and assumed I was wrong.

After seeing this, I just kept trying and got it. thank you.
 
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