Griffiths Introduction to Elementary Particles · Problem 5.12

The Feynman Calculus — David J. Griffiths

Physics — QFT · expectation-value-tensor-operator

Problem

Suppose

\mathbf{A}

and

\mathbf{B}

are two fixed vectors. Show that the expectation value of

Q \equiv [3(\mathbf{A}\cdot\hat{r})(\mathbf{B}\cdot\hat{r}) - (\mathbf{A}\cdot\mathbf{B})]

is zero in any spherically symmetrical state. [Hint: First convince yourself that

\langle Q \rangle

must be proportional to

(\mathbf{A}\cdot\mathbf{B})

\langle Q \rangle = \alpha(\mathbf{A}\cdot\mathbf{B})

where

\alpha

, the proportionality constant, is independent of

\mathbf{A}

and

\mathbf{B}

. To evaluate

\alpha

, take the case where

\mathbf{A}

and

\mathbf{B}

are parallel, and choose the

z

axis to lie along this direction. Then

\alpha = \langle (3\cos^2\theta - 1) \rangle

Now perform the

\theta

integral. [Notice that the argument is not affected if an arbitrary function of

r

is thrown into the expectation value. In particular, the expectation value of the first term in equation (5.59) must vanish for an electron in the

S

state.]]