Quantum Mechanics: Harmonic Oscillator

[Robben]

Homework Statement

A particle of mass m in the one-dimensional harmonic oscillator is in a state for which a measurement of the energy yields the values $\hbar\omega/2$ or $3\hbar\omega/2$ each with a probability of one-hald. The average value of the momentum $\langle p_x\rangle$ at time $t=0$ is $\sqrt{m\omega\hbar/2}$. What is this state and what is $\langle p_x\rangle$ at time $t$?

Homework Equations

None

The Attempt at a Solution

The solution states that since $|\psi\rangle$ is the superposition of $n=0$ and $n=1$ then $|\psi\rangle = c_1|0\rangle +c_2|1\rangle$ but why is that? What information specifies the state of the particle?

It goes on by calculating $$|psi\rangle =
\frac{1}{\sqrt{2}}(|0\rangle+e^{i\phi}|1\rangle)$$ $$\langle p_x\rangle=-i\sqrt{m\omega\hbar}/2\langle\psi|(a-a^{\dagger})|\psi \rangle$$ $$=\frac{-i}{2}\sqrt{\frac{m\omega\hbar}{2}}(e^{i\phi}\langle0|a|1\rangle-e^{-i\phi}\langle1|a^{\dagger}|0\rangle)$$ but why does $\langle0|a|1\rangle$ and $\langle1|a^{\dagger}|0\rangle$ equal one?

 

[Orodruin]

What information specifies the state of the particle?

The part saying that is in a superposition of two of the energy eigenstates.

but why does ⟨0|a|1⟩\langle0|a|1\rangle and ⟨1|a†|0⟩\langle1|a^{\dagger}|0\rangle equal one?

What are the properties of the raising and lowering operators?

 

[Robben]

The part saying that is in a superposition of two of the energy eigenstates.

What are the properties of the raising and lowering operators?

Oh, I see for the second part of my question. Thank you. For the first part I am still not sure how they got $|\psi\rangle = c_1|0\rangle +c_2|1\rangle$.

 

[DrClaude]

is in a state for which a measurement of the energy yields the values $\hbar\omega/2$ or $3\hbar\omega/2$

Which states have an energy of $\hbar\omega/2$ and $3\hbar\omega/2$?