Validity of virial theorem in QM

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SUMMARY

The discussion centers on the validity of the virial theorem in quantum mechanics (QM), specifically addressing its derivation under the assumption of bound states. It highlights that the relation = n/(n+2) fails for the continuous spectrum of the Coulomb potential, where > 0 and > 0, leading to n/(n+2) = -1. The divergence of the average momentum in continuum states is identified as a critical point that invalidates the conclusion drawn from the standard derivation.

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  • Understanding of quantum mechanics principles
  • Familiarity with the virial theorem
  • Knowledge of Coulomb potential in quantum systems
  • Concept of bound versus continuum states in QM
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Quantum physicists, students of quantum mechanics, and researchers exploring the foundations of quantum theory and the virial theorem's applications.

Heirot
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In the standard derivation of the virial theorem, one assumes to be working in the energy basis. One then gets <T> = n/(n+2) <H>. This relation doesn't hold for the continuous spectrum of Coulomb potential where <T> > 0, <H> > 0, n/(n+2) = -1. So, where in the derivation did we use the fact we were dealing with bound states?
 
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If I remember correctly, the relation is based on evaluation of the term
[tex]\langle [H,\mathbf{pr}]\rangle=(E-E) \langle \mathbf{pr}\rangle[/tex].
In the case of bound states, (E-E)=0 and [tex]\langle \mathbf{pr}\rangle[/tex] is finite. In the case of continuum states, the last average diverges, so that one cannot conclude that the whole expression vanishes.
 
Oh, I see - Thank you!
 

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