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I have a question whether eigenvalue(corresponding physical quantity) is up to coordinate

we establish to solve certain problem.

For example, I have state function combined by momentum eigen states such as [itex] \Psi = \sqrt{\frac{1}{2}}ae^{ikx} + \sqrt{\frac{1}{2}}ae^{-ikx} [/itex]

a is normalization factor of each eigenstate and [itex] \sqrt{\frac{1}{2}} [/itex] means

probability each eigenstate is chosen after momentum measurement is the same.

I think expactation value of momentum of that state is 0 because momentum we can get

from each eigenstate is the same with opposite sign and probability is the same, according

to [tex]<a> = \sum p_{i}a_{i}[/tex], [itex] p_{i} [/itex] is probability [itex] a_{i} [/itex] can be obtained.

I attempted to get expectation value with firsthand calculation with [itex] <\Psi|\stackrel{{^}}{p}\Psi >[/itex]

I establish integral range from 0 to a, [0,a] and i get fianl integral form [tex] \sqrt{\frac{1}{2}}\hbar ka\int_0^{a}2isin(2kx)dx [/itex]

but its calculation is not zero except for case in which integral range is [-a/2,a/2] or

[itex][-\infty,\infty] [/itex]

i can not accept physical quantity is up to on what coordinate we solve problem.

and i have a very important question, orthogonality [tex] <\varphi_{k} | \varphi_{k^{'}} >=0[/tex] only if integral range is [itex][-\infty,\infty] [/itex] ?

Please remove my confusing.

Thank you for pay attention to my question

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# [Q]Is physical quantity depending on coordinate on which we solve problem?

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