Local potential and diagonal

Hello,

in my lecture notes I have made a note that if V is a local potential, then <r|V|r'> is diagonal. Is this true, and how come?

It popped up with the Lippmann-Schwinger equation, and I know it's important that V is local, but does really locality implicate diagonality?

Best regards,
Jonas
 
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1
jonas_nilsson said:
Hello,

in my lecture notes I have made a note that if V is a local potential, then <r|V|r'> is diagonal. Is this true, and how come?

It popped up with the Lippmann-Schwinger equation, and I know it's important that V is local, but does really locality implicate diagonality?

Best regards,
Jonas
I think no. Take the classical schroedinger equation with an external electromagnetic field (A,V). The interaction 1/2m(A.P+P.A+A^2) (e=c=1) is not diagonal in |r>.
However, you may say that the interaction is not a potential and a potential is of the form V(r) (grad V(r)= f(r) for a local local interaction) hence the diagonality of a potential.

Therefore it is a matter of words.

Seratend.
 

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