Annihilation and Creation operator problem

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SUMMARY

The discussion centers on the application of annihilation and creation operators in quantum mechanics to compute the matrix element \langle 1 \mid x^2 \mid 2 \rangle. The participant identifies the position operator x as x = \sqrt{\hbar / 2mw}(a^\dagger + a) and seeks guidance on how to proceed with squaring x and applying it to the states. Key insights include recognizing that the states 1 and 2 are orthonormal and correspond to different quantum states, leading to the conclusion that the matrix element evaluates to zero.

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Trogdor27
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I've looked through all my course notes, but I just don't even know where to start with this problem.

The problem:

\langle 1 \mid x^2 \mid 2 \rangle

Use the annihilation and creation form for x to obtain the above matrix element.

What I do know:

I know that x = \sqrt{\hbar / 2mw}(a^\dagger + a)But where do I go from here? Any help or pointers are much appreciated!
 
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So...square that x...and then apply it to your states. What is the effect of the creation operator on the state 2? The effect of the annihilation operator? Lastly, consider that your basis states should be orthonormal.
 
Ah - I didn't realize that 1 & 2 correspond to different states, I thought they were just numbers. It all makes sense now, and if I'm right the answer should be zero.

Thanks!
 

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