Normalized function and minimum energy

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Discussion Overview

The discussion revolves around proving a mathematical inequality involving a normalized quantum state |Ψ> and the Hamiltonian operator H, specifically addressing the expression <Ψ|H\overline{^}|Ψ>\geqE0. The context is rooted in quantum mechanics, particularly in the framework of eigenstates and their properties.

Discussion Character

  • Homework-related
  • Mathematical reasoning

Main Points Raised

  • One participant requests a proof of the inequality involving the normalized function |Ψ> and the Hamiltonian operator H.
  • Another participant suggests using the hint provided in the exercise, specifically expanding |Ψ> in the energy basis.
  • A different participant expresses uncertainty about how to begin the proof, indicating a lack of familiarity with the problem.
  • There is a question posed regarding the understanding of what an eigenstate is, implying a need for clarification on this concept.
  • A participant provides a translation of the term "eigenstate" into Polish, indicating a potential language barrier in understanding the problem.

Areas of Agreement / Disagreement

The discussion reflects a lack of consensus, with some participants seeking clarification and others providing hints or questions, indicating that multiple viewpoints and levels of understanding are present.

Contextual Notes

Participants have not yet resolved the initial mathematical steps required to approach the proof, and there is a noted uncertainty regarding foundational concepts such as eigenstates.

satach
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please prove this:

<Ψ|H\overline{^}|Ψ>\geqE0
function |Ψ> is normalized.

prompt: (expand function |Ψ> in the eigenbasis of H operator, count average and use ortonormality of the H base)

exercise 5.2.2 , p.163 , "principles of quantum mechanics" Shankar
 
Last edited:
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Have you tried using the hint? What did you get after writing |ψ> in the energy basis?
 
no, i didn't try , because i don't know how to start :D
 
Do you know what an eingenstate is?
 
in polish language is sometnihg like : expand |Ψ> in base of self functions of H ...
 

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