I had thought that the expectation value would be the same....whether you did it in momentum space or position space. Could someone explain what is going on in this particular problem?(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\psi (x) = \sqrt{b} e^{-b |x| + i p_0 x / \hbar }

[/tex]

Taking the Fourier transform, I can get this function in momentum space.

[tex]

\phi (p) = \sqrt{\frac{2b}{\pi \hbar}} \thickspace \frac{\hbar^2 b}{\hbar^2b^2+(p-p_0)^2}

[/tex]

Now, if I compute [itex] <p^2> [/itex], I should get the same value no matter which space I choose to do it in. However, this does not seem to be the case. I computed these by hand...and by Mathematica....so I am fairly confident that they are correct (just confusing).

[tex]

\begin{align*}

<p^2> &= \int \phi^* p^2 \phi \, dp = b^2 \hbar^2 + p_0^2\\

<p^2> &= \int \psi^* \left(\frac{\hbar}{i}\right)^2 \frac{\partial^2 \psi}{\partial x^2} \, dx = -b^2 \hbar^2 + p_0^2

\end{align*}

[/itex]

As you can see...and please carry out the computations if you doubt them...the expectation value seems to depend on the space in which I computed it. I thought that this was NOT supposed to happen. Any ideas?

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# Expectation Value Dependence

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