Monday, 17 June 2019

quantum mechanics - Position and momentum eigenstates in terms of creation and annihilation operator?



Consider a simple harmonic oscillator; the position operator is ˆx=(a+a)/2 and the momentum operator is ˆp=i(aa)/2.


One may verify that the eigenstates of ˆx and ˆp are |xe2 xa(a)2/2|0

|peip2 a+(a)2/2|0.
My question is: how do I verify that the position eigenstates and momentum eigenstates are orthogonal themselves, and that $$\left\propto e^{ipx} ~~?$$ I'm not able to calculate this inner product using the commutator of a and a.




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