# Expectation value <p> of the ground state of hydrogen

## Homework Statement

How should I calculate the expectation value of momentum of an electron in the ground state in hydrogen atom.

## The Attempt at a Solution

I am trying to apply the p operator i.e. $-ihd/dx$ over $\psi$. and integrating it from 0 to infinity. The answer I am getting is $ih/{2(pi)a^3}$

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kuruman
Homework Helper
Gold Member
You are actually trying to calculate the expectation value of px. To do that, you must express the ground state wavefunction explicitly as a function of $x$ so that you can take the derivative. You have to do a 3d integral.

So I did the following to integrate it in 3d:$$\iiint_V \Psi^* (-i\hbar) \frac {d\Psi} {dr} r^2 sin\theta dr d\theta d\phi$$

The final answer i am getting is $\frac {i\hbar}{a_b}$ which does not look right because it is imaginary and momentum operator is hermitean. I can't figure out what mistake I am doing

kuruman
You are assuming that in spherical coordinates $p_r=-i \hbar \frac{\partial}{\partial r}.$ It is not quite that. Do some research on the internet to find out what it is and why.