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Consider a simple harmonic oscillator; the position operator is $\hat{x}=(a^\dagger+a)/\sqrt{2}$ and the momentum operator is $\hat{p}=-i(a-a^\dagger)/\sqrt{2}$.

One may verify that the eigenstates of $\hat{x}$ and $\hat{p}$ are $$\left|x\right>\propto e^{\sqrt{2}~xa^\dagger-(a^\dagger)^2/2}\left|0\right>$$ $$\left|p\right>\propto e^{ip\sqrt{2} ~a^\dagger+(a^\dagger)^2/2}\left|0\right>.$$ My question is: how do I verify that the position eigenstates and momentum eigenstates are orthogonal themselves, and that $$\left<x|p\right>\propto e^{ipx} ~~?$$ I'm not able to calculate this inner product using the commutator of $a$ and $a^\dagger$.

Cosmas Zachos
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Nahc
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  • @knzhou what do you mean by that? Yes I want a direct approach. – Nahc Feb 14 '17 at 04:13
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    is this correct? what is $z$ in your first exponential? Moreover, is $p$ in your second exponential an operator or a number? – ZeroTheHero Feb 14 '17 at 07:08
  • BCH formula and friends are what you want to commute exponentials of operators. What's the conceptual question here? What have you tried? – ACuriousMind Feb 14 '17 at 12:58
  • @ACuriousMind Hi, of course I know I can use BCH formula to do that calculation. The problem is that when you use BCH formula, you need to calculate infinite number of commutators, and because the $a^2$ and $(a^\dagger)^2$, they don't vanish after a few order. – Nahc Feb 14 '17 at 15:33
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    Without an i your normalization of your p is toxically antihermitean, and prevents your from simple implementation of adjoint symmetries involved. Your combinatorics would be vastly expedited by utilizing the isomorphism $a^\dagger \to y$ and $a \to \partial_y$, after you normalize your a's properly, provided you understand translation operators. – Cosmas Zachos Feb 14 '17 at 16:17
  • Formally related question 292899 . – Cosmas Zachos Feb 14 '17 at 16:32
  • @CosmasZachos Hi, this is part of the problem 14.4 in Schwartz's QFT book. My final goal is to understand the orthonormality of the eigenstates of a quantum field, see this problem http://physics.stackexchange.com/q/312006/. But to solve that problem, I think I first need to understand how to show that $\left<x|p\right>=e^{ipx}$. About the quantum field eigenstate problem, I couldn't find any books or papers talking about it. Thank you so much! – Nahc Feb 16 '17 at 22:43
  • Yes, the QFT theory problem is a straightforward extension of this one. But unless reopened, I can't help you here. You have to tweak the standard techniques of Fisher, Nieto, & Sandberg. – Cosmas Zachos Feb 16 '17 at 22:48
  • @CosmasZachos Thank you for the paper, it seems pretty relevant! I'm not able to reopen this problem. – Nahc Feb 16 '17 at 22:52

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