- #1
gulsen
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Homework Statement
(Sakurai 1.27)
[...] evaluate
[tex]\langle \mathbf{p''} | F(r) | \mathbf{p'} \rangle[/tex]
Simplify your expression as far as you can. Note that [itex]r = \sqrt{x^2 + y^2 + z^2}[/itex], where x, y and z are operators.
Homework Equations
[tex]\langle \mathbf{x'} | \mathbf{p'} \rangle = \frac{1}{ {(2 \pi \hbar)}^{3/2} }exp(i \mathbf{p'} \cdot \mathbf{x'} / \hbar)[/tex],
[tex]F(r) | \mathbf{x'} \rangle = F(|\mathbf{x'}|) | \mathbf{x'} \rangle[/tex]
and
[tex]\langle \mathbf{x''} | \mathbf{x'} \rangle = \delta(\mathbf{x''} - \mathbf{x'})[/tex]
The Attempt at a Solution
Using the resolution of identity, I wrote
[tex]\langle \mathbf{p''} | F(r) | \mathbf{p'} \rangle = \int \int \langle \mathbf{p''} | \mathbf{x''} \rangle \langle \mathbf{x''} |F(r) | \mathbf{x'} \rangle \langle \mathbf{x'} | \mathbf{p'} \rangle d \mathbf{x''} d \mathbf{x'}[/tex]
Using the above "relevant equations", I get
[tex]\frac{1}{(2 \pi \hbar)^3}\int exp(i (\mathbf{p'} - \mathbf{p''}) \cdot \mathbf{x'} / \hbar) F(|\mathbf{x'}|) d \mathbf{x'}[/tex]
Is this the correct answer? (wish that answers were avaiable at the backside of the book...)
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