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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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    – DanielSank
    Apr 19, 2018 at 2:33
  • $\begingroup$ Doesn't your textbook discuss $\hat x$ and $\hat p$ in context of $\hat a$, $\hat a^\dagger$? $\endgroup$
    – Kyle Kanos
    Apr 19, 2018 at 10:12
  • $\begingroup$ This might be of use to you: physics.stackexchange.com/q/82691/25301 $\endgroup$
    – Kyle Kanos
    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.

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