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I have a wave function, say $$ \begin{align} \Psi(x) = xe^{ikx} \end{align} $$

And I want to find the expectation value for momentum, $<p>$

It's not working out to an integral I can evaluate though $$ \begin{align} <p> &= \int \limits_{-\infty}^\infty dx \Psi^*(x)(-ih\frac{d}{dx})\Psi(x) \\ & = hk \int \limits_{-\infty}^\infty dx~xe^{-ikx} e^{ikx} (1 + x) \\ & = hk \int \limits_{-\infty}^\infty dx(x^2 + x)=\infty. \end{align} $$

Does this have a special meaning? Or am I doing something terribly wrong

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There’s a problem in your wave function: it diverges at infinite far. In real physical system there’s no infinitely big probability density, and the sum of all probabilities is 1. Therefore, the wave function you posted isn’t normalized.

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  • $\begingroup$ Normalization is done with a constant I think. Since the integral doesn't evaluate, I can't find the constant. Does this mean that it isn't possible to normalize this function? $\endgroup$ – Alter Nov 1 '18 at 8:25
  • $\begingroup$ @JOE How about free particle eigenstate $e^{ikx}$ with momentum eigenvalue $\hbar k$. But it is not normalizable $\endgroup$ – K_inverse Nov 1 '18 at 8:32
  • $\begingroup$ @Alter yep. The function u posted is impossible to normalize cuz it diverges as x increases. Still, you can add a finite interval to x, and do the integral in this specific interval, thus get the value of expected momentum $\endgroup$ – JOE Nov 1 '18 at 8:32
  • $\begingroup$ @K_inverse I think $e^{ikx}$ is normalized $\endgroup$ – Alter Nov 1 '18 at 8:44
  • $\begingroup$ @K_inverse u got me in the box there😂well idk if it’s normalizable cuz u can add an infinite small constant ahead. Yet anyway it converges at infinite far (integrable), so there’s a corresponding momentum. $\endgroup$ – JOE Nov 1 '18 at 8:47

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