Solving for <p>, <x> and <x^2> using raising and lowering operators

[njdevils45]

Homework Statement

A) Show that <x>=<p>=0
hint: use orthogonality
B) Use the raising and lowering operators to evaluate an expression for < x² >

Homework Equations

Also A^- and A⁺ will definitely come in handy

The Attempt at a Solution

I tried setting up the equations for <x> and <p> but I don't know how the operators are meant to be used in this scenario. I think that the integral is meant to be set up as ∫eq1*x_op*the general equation for ψ_n for a harmonic oscillator, however whatever I do I can't get the math to come out to 0 in the end for either.

 

[Orodruin]

Is this the entire problem exactly as stated? Clearly it is possible to find states such that $\langle x\rangle \neq 0$ and the same for $\langle p\rangle$.

 

[njdevils45]

Yes that's the entire problem

 

[Orodruin]

Thereis no mention of what state you should compute the expectation values for?

 

[njdevils45]

None at all. That's why I'm confused, I might just ask the professor for help on the setup to be honest

 

[Orodruin]

If the state is not mentioned the problem statement is misleading at best. Now, it is true for the energy eigenstates so this is presumably the missing assumption. It is generally not true for linear combinations of energy eigenstates that contain adjacent energy states.