Raising and lowering operators of the Hamiltonian

[CAF123]

Homework Statement

a) The operators $a$ and $a^{\dagger}$ satisfy the commutation relation $[a,a^{\dagger}] = 1$. Find the normalization of the state $|\psi \rangle = C (a^{\dagger} )^2 |0\rangle$, where the vacuum state $|0\rangle$ is such that $a|0\rangle = 0$

b)A one dimensional simple harmonic oscillator has the Hamiltonian $$\hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2}mw^2 \hat{x}^2.$$ The operator $\hat{a}$ is defined as $$\hat{a} = \left(\frac{mw}{2\hbar}\right)^{1/2}\hat{x} + \frac{i}{\sqrt{2mw\hbar}} \hat{p}$$

Suppose that the oscillator frequency is changed slightly to $w \rightarrow \sqrt{1+\epsilon} w$ What are the exact new energy eigenvalues? Now calculate the first order perturbation in the energy and compare with the exact answer.

Homework Equations

In a), perhaps that $a|n\rangle = \sqrt{n}|n-1\rangle$ and that $a^{\dagger} |n\rangle = \sqrt{n+1} |n+1\rangle$

The Attempt at a Solution

In a), write out the commutator explicitly $aa^{\dagger} - a^{\dagger}a = 1$. Now I tried various manipulations of this equation to get something that resembled the equation given. Multiply by $(a^{\dagger})^2$ on the left and $|0\rangle$ on the right would give $(a^{\dagger})^2 a a^{\dagger} |0\rangle - (a^{\dagger})^3 a|0\rangle = (a^{\dagger})^2 |0\rangle.$ But this simplifies to LHS=RHS, so it is of no use.

b)New energy eigenvalues would be $E_n = \hbar (\sqrt{1+\epsilon} w) (n+1/2)$ and the shift is $E' = \langle n | \hat{H'} | n\rangle$ where $\hat{H'}$ is the Hamiltonian above but with $w$ replaced with $\sqrt{1+\epsilon}w$ Do we treat $\epsilon$ as a constant here?

Many thanks.

 

[dextercioby]

For a), just compute the vector itself, a†|0> = √0+1 |1> = |1>, etc.

 

[CAF123]

For a), just compute the vector itself, a†|0> = √0+1 |1> = |1>, etc.

Applying a† again would give √2 |2>, not some constant premultiplying the state |0>. So I think somehow we have to incorporate the commutation relation given, which I tried to do in the OP.

 

[vela]

Try using $\|C(a^\dagger)^2\vert 0 \rangle\|^2=1$.

 

[paranormal]

I would first commend the student for their attempt at solving the problems and for recognizing and utilizing relevant equations. I would then provide guidance on how to approach the problems and provide some additional information that may be helpful.

For part a), it may be helpful to use the fact that the operator $a^{\dagger}$ can be expressed in terms of the creation and annihilation operators for the harmonic oscillator, i.e. $a^{\dagger} = \sqrt{2m\omega\hbar}\hat{a}^{\dagger}/\sqrt{\hbar}$. This will allow for the explicit calculation of the normalization constant $C$.

For part b), it is important to note that the perturbation in the energy is given by the expectation value of the perturbation Hamiltonian, which is proportional to $\epsilon$. Therefore, it is reasonable to treat $\epsilon$ as a small constant in this case. Additionally, it may be helpful to use the known expression for the perturbation in the energy of a harmonic oscillator when the frequency is changed slightly.

I would also suggest that the student review the concept of raising and lowering operators and their properties, as well as the calculation of energy eigenvalues and eigenstates for the harmonic oscillator.