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Probability of observable unbounded self-adjoint operator.

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Let H be the unbounded self-adjoint operator defined by -d^2/dx^2 (the negative of the second derivative with respect to x) on:

D(H) ={f element of L^2 | Integral( |s^2 F f(s)|^2 )ds element of L^2}

Where "F" denotes the Fourier Transform.


For the state vector h(x) = 1/sqrt(2) if x is in [0,2]
0 if x is not in [0,2]

What is the probability that the observable H will be measured in the interval [1/2,1] ?

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Solution Summary

This is a question regarding probabilities and state vectors. The elements for Fourier Transforms are determined.

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In terms of the Fourier transform, the operator is H = s^2, and the bounds 1/2 < H < 1 in terms of s mean
-1 < s < -1/sqrt(2) and 1/sqrt(2) < s < 1

Therefore, since the given state vector h(x) is normalized, the requested ...

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