Quantum Mechanics: Raising and Lowering Operators

[Robben]

Homework Statement

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

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

Homework Equations

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

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?

 

[ShayanJ]

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

 

[Robben]

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

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

 

[ShayanJ]

Yes!

 

[ShayanJ]

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

And that is Kronecker delta, not Dirac delta!

 

[Robben]

And that is Kronecker delta, not Dirac delta!

Opps! Thanks for clarifying.