Validity of virial theorem in QM

[Heirot]

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?

 

[DrDu]

If I remember correctly, the relation is based on evaluation of the term
$$\langle [H,\mathbf{pr}]\rangle=(E-E) \langle \mathbf{pr}\rangle$$.
In the case of bound states, (E-E)=0 and $$\langle \mathbf{pr}\rangle$$ is finite. In the case of continuum states, the last average diverges, so that one cannot conclude that the whole expression vanishes.

 

[Heirot]

Oh, I see - Thank you!