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Expectation value <p> of the ground state of hydrogen

  • #1

Homework Statement


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

Homework Equations


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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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Answers and Replies

  • #3
kuruman
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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.
 
  • #4
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
 
  • #5
kuruman
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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.
 

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