Consider a simple harmonic oscillator; the position operator is ˆx=(a†+a)/√2 and the momentum operator is ˆp=−i(a−a†)/√2.
One may verify that the eigenstates of ˆx and ˆp are |x⟩∝e√2 xa†−(a†)2/2|0⟩
|p⟩∝eip√2 a†+(a†)2/2|0⟩.
My question is: how do I verify that the position eigenstates and momentum eigenstates are orthogonal themselves, and that $$\left
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