- #1
element_zero
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Hey guys,
maybe you can help me with the following problem. I have to calculate the commutator relations in position representation:
a) [V,ρ]
b) [p,ρ]
c) [p^2,ρ]
Note that <q'|ρ|q>=ρ(q',q) is a matrix element of the density operator
I already solved the first one. You just have to apply the potential operator on a matix element of the density operator.
[V,ρ]=<q'|Vρ|q>-<q'|ρV|q>=...=(V(q')-V(q))*ρ(q',q)
The rest however is more tricky as the momentum operator is not diagonal in this domain.
[p,ρ]=<q'|pρ|q>-<q'|ρp|q>=...?
I got the hint that I should try an integration over an auxiliary variable which should lead to something like <q'|ρ|q''>~δ(q'-q'') (Delta functions)
A Fourier transformation is NOT necessary as far as I know.
The result of c should be something like:
-(d^2/dq'^2-d^2/dq^2)*ρ(q',q)
Thanks a lot for your help!
maybe you can help me with the following problem. I have to calculate the commutator relations in position representation:
a) [V,ρ]
b) [p,ρ]
c) [p^2,ρ]
Note that <q'|ρ|q>=ρ(q',q) is a matrix element of the density operator
I already solved the first one. You just have to apply the potential operator on a matix element of the density operator.
[V,ρ]=<q'|Vρ|q>-<q'|ρV|q>=...=(V(q')-V(q))*ρ(q',q)
The rest however is more tricky as the momentum operator is not diagonal in this domain.
[p,ρ]=<q'|pρ|q>-<q'|ρp|q>=...?
I got the hint that I should try an integration over an auxiliary variable which should lead to something like <q'|ρ|q''>~δ(q'-q'') (Delta functions)
A Fourier transformation is NOT necessary as far as I know.
The result of c should be something like:
-(d^2/dq'^2-d^2/dq^2)*ρ(q',q)
Thanks a lot for your help!