# Quantum Mechanics algebra

1. Dec 30, 2009

### sxc656

Quantum Mechanics algebra - time independant peturbation theory

Hi

1. The problem statement, all variables and given/known data
The potential shown is operating on the eigenstate as shown in the pic. I am having trouble getting the second term using orthogonality (got the first term :-) ). Please Help!

2. Relevant equations
see pic

3. The attempt at a solution
put the i's equal to j (i.e. orthogonality) but don't know what to do next.

Thanks
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

#### Attached Files:

• ###### algebra.JPG
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Last edited: Dec 30, 2009
2. Dec 30, 2009

### JSuarez

Just note that, in the sum, the only nonzero term in the second part will be the one where
i + 1 = j, so i will be equal to...

3. Dec 30, 2009

### sxc656

This was part of a hint for a question on time independant pertubation theory (see pic). i was reluctant to post the whole question because i wanted a good crack at it without help but i am now stuck. i can't seem to get the 2nd order term and can't work out what happens with the first order term.

Thank you

#### Attached Files:

• ###### algebra2.JPG
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Last edited: Dec 30, 2009
4. Dec 30, 2009

### diazona

Well, you're given a formula for the energy $E_0$ in terms of $\hat{V}$, and you're given a formula for $\hat{V}$. What happens when you plug $\hat{V}$ in to the energy formula? If you get stuck, show your work.

5. Dec 31, 2009

### sxc656

Thanks for the replies. This is my working so far (see attachment). I have plugged V in but am stuck on the last 2 lines of my working. Using orthogonality i think you get lambda and lambda* for the 1st terms of the second-last and last-lines respectively but what about the second terms in those lines?
Thanks

#### Attached Files:

• ###### algebra3.JPG
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6. Dec 31, 2009

### sxc656

Ah, do they equal zero (the last terms) because i is from 0 to infinity and those two terms require i=-1?

7. Dec 31, 2009

### diazona

Yeah, you can use that argument for the second term on each of the last two lines.