Dipole approximation in acceleration form

[kknull]

hi,

I have 2 localized states in a hydrogenic atom.. we're in dipole approimation.
I have to proof that

<b| [p,V] | a> = <b| grad(V) |a>
or equally:
<b| V grad |a> = 0

(this is the dipole approximation in acceleration form)

someone can help me?

you can see http://books.google.it/books?id=1oXrE5YV19IC&pg=PA321&lpg=PA321&dq=dipole+approximation+matrix&source=bl&ots=cEbdhO2-ne&sig=pHxoDCxTrGi6ySPBRAF9tsws4Vc&hl=it&ei=fAd7TtH5GoLP0QXj48SjAw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CB4Q6AEwADgK#v=onepage&q&f=true"

page 323 eq (8.22)

thanks :)

 

[olgranpappy]

first work out the commutators:

[p,x]=?

[p,x^2]=?

[p,x^3]=?

and so on...

Then expand V as a power series and use the above commutators to prove the relation.