I need to evaluate $\langle n|x|n \rangle$ and $\langle n|x^2|n \rangle$, using the $a^\dagger$ and $a$ operators. This is used to define delta(x). I'm aware that $a|n\rangle = \sqrt{n} |n-1 \rangle$ and that $a^\dagger |n \rangle = \sqrt{n+1}|n+1 \rangle$. I'm unsure how I evaluate these with the position operator $x$ in the equation, beyond expanding x = K(a+a*) and thus Squaring that formula for x^2. Once I obtain those, would I simply apply them to |n> in order [ignoring the + signs] or simply apply them separately to |n> and add the resultant values at a later date? And why?
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3$\begingroup$ Welcome to Physics Stack Exchange. This is a wonderful site for physics questions and answers. We have some rules and guidelines to make sure the quality of the site remains as high as it is. One rule is that questions like this one, asking for help solving a specific problem, must show effort and ask a specific conceptual question. Just asking how to solve a problem without showing your effort and identifying where you're stuck inevitably leads to the question being closed. Can you edit your question to show effort and identify where you need help? $\endgroup$– DanielSankCommented Apr 19, 2018 at 2:33
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$\begingroup$ Doesn't your textbook discuss $\hat x$ and $\hat p$ in context of $\hat a$, $\hat a^\dagger$? $\endgroup$– Kyle KanosCommented Apr 19, 2018 at 10:12
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$\begingroup$ This might be of use to you: physics.stackexchange.com/q/82691/25301 $\endgroup$– Kyle KanosCommented Apr 19, 2018 at 11:20
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I am assuming that the question is in the context of $1D$ simple harmonic oscillator.
If you consult any introductory quantum mechanics textbook, you will see that $\hat{x}$ can be written as $K(\hat{a} + \hat{a}^{\dagger})$, where $K$ is the appropriate constant. Now, just plug in the corresponding expression for $\hat{x}$ in the expecttaion values and you will get the results.
