Tuesday, 2 December 2014

quantum mechanics - Prove that this operator is unitary


ˆO(1/2π)eiNzdz


ˆO(1/2π)eiNxdx


We have the operator ˆO and its Hermitian adjoint ˆO, in the one dimensional space where x is position. I am trying to prove that this is a unitary operator. I'm told that N does not necessarily equal N. So when I tried the old ˆOˆO=ˆI, I got:


ˆOˆO=(1/2π)ei(NN)xdxdx


I did the double integral and the answer does not turn out nice. I know the periodicity of the function is 2π, but I'm not sure how that helps cancel the denominator. Also confused on what I'm supposed to do with the N terms.


Also tried using ˆO1=ˆO. That did not turn out well either.



How should I go about proving that ˆO is unitary?



Answer



This is a well known definition of a delta function:


δ(xα)=12πeip(xα) dp


therefore:


ˆOˆO=(1/2π)ei(NN)xdxdx=δ(NN)dx=1

for N = N'.


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