Transition Radiation rates of Hamiltonian

[unscientific]

Homework Statement

Part (a): Show the Commutation relation [x, [H,x] ]
Part (b): Show the expression by taking expectation value in k^th state.
Part (c): Show sum of oscillator strength is 1. What's the significance of radiative transition rates?

Homework Equations

The Attempt at a Solution

Part (a)

Manged to show.

Part (b)

$$\langle H \rangle = \langle k|\frac{p^2}{2m} + V|k\rangle$$
$$\frac{1}{2m}\langle k|p^2|k\rangle + \langle k|V|k\rangle$$

Not sure what to do at this point - it looks nothing like the answer.

 

[TSny]

Instead of taking the expectation value of equation (2.2), take the expectation value of the commutation relation that you showed in part (a).

 

[unscientific]

Instead of taking the expectation value of equation (2.2), take the expectation value of the commutation relation that you showed in part (a).

I tried and that leads to nowhere..

$$\langle \left[x,[H,x]\right] \rangle$$
$$= \langle k|\left[ x, [H,x] \right] |k\rangle$$
$$= \langle k | [x,Hx] - [x,xH]|k\rangle$$

 

[TSny]

I tried and that leads to nowhere..

$$= \langle k | [x,Hx] - [x,xH]|k\rangle$$

Keep going. Write out [x,Hx] and [x,xH]. Then judiciously insert the identity operator in the form $1 = \sum_n |n\rangle \langle n|$

 

[unscientific]

Keep going. Write out [x,Hx] and [x,xH]. Then judiciously insert the identity operator in the form $1 = \sum_n |n\rangle \langle n|$

$$= \langle k | [x,Hx] - [x,xH]|k\rangle$$
$$= \langle k | [x,H]x - x[x,H] |k\rangle$$
$$= \langle k | xHx - Hx^2 -x^2H + xHx|k\rangle$$

 

[TSny]

$$= \langle k | [x,Hx] - [x,xH]|k\rangle$$
$$= \langle k | [x,H]x - x[x,H] |k\rangle$$
$$= \langle k | xHx - Hx^2 -x^2H + xHx|k\rangle$$

Take $\langle k | xHx |k\rangle$ and insert the identity: $\langle k | x H \hat{1} x |k\rangle$

 

[unscientific]

Take $\langle k | xHx |k\rangle$ and insert the identity: $\langle k | x H \hat{1} x |k\rangle$

Yeah got it!