Monday, 21 September 2015

quantum mechanics - What is the most general expression for the coordinate representation of momentum operator?



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|xppx|y=(xy)x|p|y.

On the other hand, x|[x,p]|y=ix|y=iδ(xy). Thus (xy)x|p|y=iδ(xy).


We use (xy)δ(xy)=0. Take the derivative with respect to x; we have δ(xy)+(xy)δ(xy)=0. Thus (xy)δ(xy)=δ(xy).


Comparing Eqs. (1) and (2), we identify x|p|y=iδ(xy).


In addition, we can add αδ(xy) on the right-hand side of Eq. (3), i.e. x|p|y=iδ(xy)+αδ(xy),

and [x,p]=i is still satisfied. We can also add β|xy|δ(xy)
on the RHS of Eq. (3). Here α and β are two real numbers.


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(xy).


The canonical commutation relation (CCR) is


[ˆx,ˆp] = i1.


The standard Schrödinger position representation reads


ˆx = x,ˆp = ix.


We may conjugate the standard Schrödinger position representation (4) by an unitary operator ˆU=eif(ˆx), where f:RR is a given differentiable function. In this way we obtain an unitary equivalent position representation


ˆx = x,ˆp = ieif(x)xeif(x) = ix+f(x),


of the CCR (3). The standard Schrödinger position representation (4) corresponds to fconst. For a general irreducible representation of the CCR (3), see the Stone-von Neumann Theorem.


The representation (5) implies



x|ˆp|ψ = (ˆpψ)(x) = ieif(x)(eifψ)(x) = iψ(x)+f(x)ψ(x).


From (6) we conclude that the momentum matrix elements reads


x|ˆp|y = iδ(xy)+f(x)δ(xy)


in the representation (5).


Finally, here and here are two other Phys.SE posts that also discuss ambiguities in xp overlaps.


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