# Quantum Mechanics: Raising and Lowering Operators

1. Apr 25, 2015

### Robben

1. The problem statement, all variables and given/known data

Consider a particle in an energy eigenstate $|n\rangle.$

Calculate $\langle x\rangle$ and $\langle p_x\rangle$ for this state.

2. Relevant equations

$x = \sqrt{\frac{\hbar}{2m\omega}}(a+a^{\dagger})$

3. The attempt at a solution

$\langle x\rangle = \sqrt{\frac{\hbar}{2m\omega}}\langle n(a + a^{\dagger})|n\rangle = \sqrt{\frac{\hbar}{2m\omega}}(\sqrt{n}\langle n|n-1\rangle+\sqrt{n+1}\langle n|n+1\rangle).$

But why does it equal zero?

2. Apr 25, 2015

### ShayanJ

The energy eigenstates form an orthonormal set and here is no different. So $\langle n|m\rangle=\delta_{nm}$.

3. Apr 25, 2015

### Robben

So, $\langle n|n+1\rangle = 0$ and $\langle n|n-1\rangle$ equal zero from the property of the dirac function?

4. Apr 25, 2015

### ShayanJ

Yes!

5. Apr 25, 2015

### ShayanJ

And that is Kronecker delta, not Dirac delta!

6. Apr 26, 2015

### Robben

Opps! Thanks for clarifying.