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blue_leaf77

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## Homework Statement

I'm curious in proving that expectation value of momentum for any bound state is zero. So the problem is how to prove this.

## Homework Equations

$$ \langle \mathbf{p_n} \rangle \propto \int \psi^*(\mathbf{r_1}, \dots ,\mathbf{r_N}) \nabla_n \psi(\mathbf{r_1}, \dots ,\mathbf{r_N}) d^3\mathbf{r_1} \dots d^3\mathbf{r_N} $$

where ## n = 1,2, \dots N ##.

## The Attempt at a Solution

If I assume that ## \psi(\mathbf{r_1}, \dots ,\mathbf{r_N}) ## is either even or odd then its first derivative will be its counterpart, so ## \nabla_n \psi(\mathbf{r_1}, \dots ,\mathbf{r_N}) ## has opposite parity in ## \mathbf{r_n} ##. Thus the integral will result zero.

My question is that is this true that all bound states can only be even or odd? If yes why?

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