Harmonic oscillator eigenvector/eigenvalue spectrum

[jtaa]

the problem is attached as an image.

im having troubles with the question. I'm assuming this is an induction question?
i can prove it for the basis step n=0.

but I am having trouble as to what i have to do for n+1 (inductive step).

any help or hints would be great!thanks

 

[George Jones]

What is the commutator of $H$ and $a^\dagger$?

 

[jtaa]

N=a†a

[N,a†]=a†
[N,a]=-a

 

[George Jones]

What about $H$?

 

[jtaa]

[H,a†] = $\hbar$$\omega$a†

[H,a] = -$\hbar$$\omega$a

 

[George Jones]

$\phi_{n+1} = \left( n + 1 \right)^{-\frac{1}{2}} a^\dagger \phi_n$ gives $H \phi_{n+1} = \left( n + 1 \right)^{-\frac{1}{2}} H a^\dagger \phi_n$. Now use the commutator to reorder $H a^\dagger$.

 

[jtaa]

re-order how?

by using: Ha†=$\hbar\omega$a† + a†H
?

 

[George Jones]

Yes.

 

[jtaa]

i then get:

=(n+1)^-1/2(a†$\hbar\omega$+a†H)$\phi$_n
=$\hbar\omega$$\phi$_n+1 + (n+1)^-1/2a†H$\phi$_n

not sure what to do from here..

 

[George Jones]

The inductive step assumes what about $H\phi_n$?

 

[jtaa]

it assumes that Hϕ_n = nϕ_n

i.e. that ϕ_n is an eigenvector of H. n being the eigenvalue


=> = ($\hbar\omega$+n)ϕ_n+1

?

 

[George Jones]

it assumes that Hϕ_n = nϕ_n

i.e. that ϕ_n is an eigenvector of H.

The inductive step assumes that $\phi_n$ is an eigenvector of $H$, but it doesn't assume that the associated eigenvalue is $n$. Energies are eigenvalues of the Hamiltonian, so call the the eigenvalue $E_n$. Maybe $E_n = n$, but maybe it doesn't. Let's find out!

i can prove it for the basis step n=0.

What is the eigenvalue of $H$ associated with the eigenvector $\phi_0$?

 

[jtaa]

Hϕ₀=$\frac{1}{2}$ℏωϕ₀

So, E₀=$\frac{1}{2}$ℏω

=> Hϕ_n+1 = ℏωϕ_n+1+E_nϕ_n+1
?

 

[George Jones]

Right.

 

[jtaa]

can i simply say after that:
the energies are:

E_n=hw(n+$\frac{1}{2}$)

?