# Homework Help: Harmonic oscillator eigenvector/eigenvalue spectrum

1. Jun 5, 2012

### 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 im having trouble as to what i have to do for n+1 (inductive step).

any help or hints would be great!thanks

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2. Jun 5, 2012

### George Jones

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

3. Jun 5, 2012

N=a†a

[N,a†]=a†
[N,a]=-a

4. Jun 5, 2012

### George Jones

Staff Emeritus
What about $H$?

5. Jun 6, 2012

### jtaa

[H,a†] = $\hbar$$\omega$a†

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

6. Jun 6, 2012

### George Jones

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

7. Jun 6, 2012

### jtaa

re-order how?

by using: Ha†=$\hbar\omega$a† + a†H
?

8. Jun 6, 2012

### George Jones

Staff Emeritus
Yes.

9. Jun 6, 2012

### 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..

10. Jun 6, 2012

### George Jones

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

11. Jun 6, 2012

### jtaa

it assumes that Hϕn = nϕn

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

=> = ($\hbar\omega$+n)ϕn+1

?

12. Jun 6, 2012

### George Jones

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

13. Jun 7, 2012

### jtaa

0=$\frac{1}{2}$ℏωϕ0

So, E0=$\frac{1}{2}$ℏω

=> Hϕn+1 = ℏωϕn+1+Enϕn+1
?

Last edited: Jun 7, 2012
14. Jun 7, 2012

### George Jones

Staff Emeritus
Right.

15. Jun 7, 2012

### jtaa

can i simply say after that:
the energies are:

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

?