Saturday, July 28, 2018

quantum mechanics - What is the physical interpretation of the density matrix in a double continuous basis |alpharangle, |betarangle?


(a) Any textbook gives the interpretation of the density matrix in a single continuous basis |α:




  • The diagonal elements ρ(α,α)=α|ˆρ|α give the populations.





  • The off-diagonal elements ρ(α,α)=α|ˆρ|α give the coherences.




(b) But what is the physical interpretation (if any) of the density matrix ρ(α,β)=α|ˆρ|β for a double continuous basis |α, |β?


I know that when the double basis are position and momentum then ρ(p,x) is interpreted as a pseudo-probability. I may confess that I have never completely understood the concept of pseudo-probability [*], but I would like to know if this physical interpretation as pseudo-probability can be extended to arbitrary continuous basis |α, |β for non-commuting operators ˆα, ˆβ and as probability for commuting ones.


[*] Specially because ρ(p,x) is bounded and cannot be 'spike'.




EDIT: To avoid further misunderstandings I am adding some background. Quantum averages can be obtained in a continuous basis |α as


A=dαα|ˆρˆA|α



(a) Introducing closure in the same basis |α


A=dαdαα|ˆρ|αα|ˆA|α=dαdαρ(α,α)A(α,α)


with the usual physical interpretation for the density matrix ρ(α,α) as discussed above.


(b) Introducing closure in a second basis |β, we obtain the alternative representation


A=dαdβα|ˆρ|ββ|ˆA|α=dαdβρ(α,β)A(β,α)


When the two basis are momentum |p and position |x the density ρ(p,x) is the well-known Wigner function whose physical interpretation is that of a pseudo-probability. My question is about the physical interpretation of ρ(α,β) in two arbitrary basis |α, |β.




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