- #1

chocopanda

- 15

- 1

- Homework Statement
- Verify the proof of the sum rule for the one-dimensional harmonic oscillator:

$$\sum_l^\infty (E_l-E_n)\ | \langle l \ |p| \ n \rangle |^2 = \frac {mh^2w^2}{2} $$

- Relevant Equations
- The exercise explicitly says to use laddle operators and to express $p$ with

$$b=\sqrt{\frac {mw}{2 \hbar}}-\frac {ip}{\sqrt{2 \hbar mw}} $$

$$b^\dagger =\sqrt{\frac {mw}{2 \hbar}}+\frac {ip}{\sqrt{2 \hbar mw}} $$

I'm trying verify the proof of the sum rule for the one-dimensional harmonic oscillator:

$$\sum_l^\infty (E_l-E_n)\ | \langle l \ |p| \ n \rangle |^2 = \frac {mh^2w^2}{2} $$

The exercise explicitly says to use laddle operators and to express $p$ with

$$b=\sqrt{\frac {mw}{2 \hbar}}-\frac {ip}{\sqrt{2 \hbar mw}} $$

$$b^\dagger =\sqrt{\frac {mw}{2 \hbar}}+\frac {ip}{\sqrt{2 \hbar mw}} $$

For p I get $$p=i \sqrt{\frac{\hbar}{2mw}} (b-b^\dagger) $$

To solve the exercise, we need to calculate the left side. I'm still very much a novice and am not very sure how to use the ladder operators... To start, I at least tried to expand the bra-ket:

$$\sum_l^\infty (E_l-E_n)\ \langle l \ |p| \ n \rangle \langle n \ |p| \ l \rangle $$

and tried to insert the p I solved:

$$\sum_l^\infty (E_l-E_n)\ (-\frac{\hbar}{2mw}) \langle l \ |b-b^\dagger| \ n \rangle \langle n \ |b-b^\dagger| \ l \rangle $$

is this correct? If yes, how do I continue? The hint says to probably use $$H|n\rangle=\hbar(n+\frac 12)|n\rangle$$ and I know that $$H|n\rangle=E|n\rangle$$

$$\sum_l^\infty (E_l-E_n)\ | \langle l \ |p| \ n \rangle |^2 = \frac {mh^2w^2}{2} $$

The exercise explicitly says to use laddle operators and to express $p$ with

$$b=\sqrt{\frac {mw}{2 \hbar}}-\frac {ip}{\sqrt{2 \hbar mw}} $$

$$b^\dagger =\sqrt{\frac {mw}{2 \hbar}}+\frac {ip}{\sqrt{2 \hbar mw}} $$

For p I get $$p=i \sqrt{\frac{\hbar}{2mw}} (b-b^\dagger) $$

To solve the exercise, we need to calculate the left side. I'm still very much a novice and am not very sure how to use the ladder operators... To start, I at least tried to expand the bra-ket:

$$\sum_l^\infty (E_l-E_n)\ \langle l \ |p| \ n \rangle \langle n \ |p| \ l \rangle $$

and tried to insert the p I solved:

$$\sum_l^\infty (E_l-E_n)\ (-\frac{\hbar}{2mw}) \langle l \ |b-b^\dagger| \ n \rangle \langle n \ |b-b^\dagger| \ l \rangle $$

is this correct? If yes, how do I continue? The hint says to probably use $$H|n\rangle=\hbar(n+\frac 12)|n\rangle$$ and I know that $$H|n\rangle=E|n\rangle$$