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GR0177 #94
Problem
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Quantum Mechanics}Perturbation Theory

The energy for first-order perturbation theory of ( is the known Hamiltonian and is the perturbed Hamiltonian) is given by , where the wave-functions are the unperturbed ones.

Thus, the problem amounts to calculating . This is just raising and lowering operator mechanics.

. But, after bra-ketting, one finds that the expectation value of and are 0, since and , are orthogonal. Thus, the problem becomes,

. Applying the given eigen-equations, one finds that . For , one finds , as in choice (E).

(Note that: and and .)  Alternate Solutions
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sjayne
2015-06-23 11:22:35
I did it the same way. a ended up being |1> and adagger was |3> . Foiling it out any of the terms where |1> and |3> multiply go to 0, and in any of the terms where the kets are the same the kets go to 1. So you are left with V(2+3)=5V risyou
2012-11-07 22:49:18
It just looks like the harmonic oscillator if you have tried to express them with the up down operator.
So I just put n=2 to it.. no answer so I give it up.
I feel so tried after doing 70+ question. Donofnothing
2010-10-08 12:04:06
this doesn't make much sense. I got the same answer from facotring out the (a+adagger) term, ignoring the (a*adagger), and using only a^2 and a^dagger squared. did i just get lucky, or is this alternate?
 keradeek2011-08-26 01:03:37 yeah, you just got super lucky.
 FutureDrSteve2011-11-04 17:34:55 Awesome! My plan is to get lucky on 100 problems in a row.... :S
 yummyhat2017-10-27 05:19:08 same\r\n      LaTeX syntax supported through dollar sign wrappers $, ex.,$\alpha^2_0$produces . type this... to get...$\int_0^\infty\partial\Rightarrow\ddot{x},\dot{x}\sqrt{z}\langle my \rangle\left( abacadabra \right)_{me}\vec{E}\frac{a}{b}\$