I have a question about deriving the coordinate representation of momentum operator from the commutation relation, [x,p]=i.
One derivation (ref W. Greiner's Quantum Mechanics: An Introduction, 4th edi, p442) is as following: ⟨x|[x,p]|y⟩=⟨x|xp−px|y⟩=(x−y)⟨x|p|y⟩.
We use (x−y)δ(x−y)=0. Take the derivative with respect to x; we have δ(x−y)+(x−y)δ′(x−y)=0. Thus (x−y)δ′(x−y)=−δ(x−y).
Comparing Eqs. (1) and (2), we identify ⟨x|p|y⟩=−iδ′(x−y).
In addition, we can add αδ(x−y) on the right-hand side of Eq. (3), i.e. ⟨x|p|y⟩=−iδ′(x−y)+αδ(x−y),
My question is, what is the most general expression of ⟨x|p|y⟩? Can we always absorb the additional term into a phase factor like Dirac's quantum mechanics book did?
Thank you very much in advance.
Answer
We start by mentioning a couple of standard formulas
ψ(x) = ⟨x|ψ⟩,
and
⟨x|y⟩ = δ(x−y).
The canonical commutation relation (CCR) is
[ˆx,ˆp] = iℏ1.
The standard Schrödinger position representation reads
ˆx = x,ˆp = −iℏ∂∂x.
We may conjugate the standard Schrödinger position representation (4) by an unitary operator ˆU=e−if(ˆx), where f:R→R is a given differentiable function. In this way we obtain an unitary equivalent position representation
ˆx = x,ˆp = −iℏe−if(x)∂∂xeif(x) = −iℏ∂∂x+ℏf′(x),
of the CCR (3). The standard Schrödinger position representation (4) corresponds to f≡const. For a general irreducible representation of the CCR (3), see the Stone-von Neumann Theorem.
The representation (5) implies
⟨x|ˆp|ψ⟩ = (ˆpψ)(x) = −iℏe−if(x)(eifψ)′(x) = −iℏψ′(x)+ℏf′(x)ψ(x).
From (6) we conclude that the momentum matrix elements reads
⟨x|ˆp|y⟩ = −iℏδ′(x−y)+ℏf′(x)δ(x−y)
in the representation (5).
Finally, here and here are two other Phys.SE posts that also discuss ambiguities in x↔p overlaps.
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