Trouble with Orthogonality in Quantum Mechanics Algebra?

[sxc656]

Quantum Mechanics algebra - time independent peturbation theory

Hi

Homework Statement

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!

Homework Equations

see pic

The Attempt at a Solution

put the i's equal to j (i.e. orthogonality) but don't know what to do next.

Thanks

 

[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...

 

[sxc656]

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...

This was part of a hint for a question on time independent 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

 

[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} into the energy formula? If you get stuck, show your work.

 

[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

 

[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

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

 

[diazona]

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