In studying the hyper-fine structure of the hydrogen atom at the 2n level in my notes the following is stated ;

$$\langle W_{mv} \rangle _{2s}=\langle n=2, l=0|- \frac{\hat{P^4}}{8m_e^3c^2}| n=2, l=0\rangle=\frac{-13}{128}m_ec^2\alpha^4$$

and no other explanation as to how this was computed.

for the 2p level there is even less detail.

$$\langle W_{mv}\rangle_{2p}=\frac{-7}{384}m_ec^2\alpha^4$$

and no other explanation as to how this was computed was given either.I'm particularly confused as to why l=0 and l=1,0,-1 give different results ( although i think i understand it theoretically ) as there doen't seem to be a dependance on l in the equation for $W_{mv}$, at least not explicitly.

I can't figure out how these were found could anyone go into some more detail on how to compute them?

I don't really need too much theory , there's 100's of pages of that in the notes. Just a way to compute this expectation value.

  • $\begingroup$ This isn't a homework question, We've finished assignments for the summer( note: this comment is in relation to an edit that added the homework tag) $\endgroup$
    – bhapi
    May 15, 2018 at 18:23
  • 1
    $\begingroup$ Hi exodius. If you haven't already done so, please take a minute to read the definition of when to use the homework-and-exercises tag, and the Phys.SE policy for homework-like problems. $\endgroup$
    – Qmechanic
    May 15, 2018 at 18:43
  • $\begingroup$ @Qmechanic hey Qmechanic , just read the link you sent . I see know that this does constitute a homework-and-exercises question. I had taken that it literally meant a question from an assignment that one needed help on. I'll edit my post to add the tag back in :) $\endgroup$
    – bhapi
    May 15, 2018 at 18:54

1 Answer 1


The expression given is equivalent to $$\int \psi_{2,0,0}^*(r,\theta,\phi) \left( -{ \hat p^4 \over 8 m^3c^2} \right)\psi_{2,0,0}(r,\theta,\phi) \, {\rm d}V$$

$\hat p^4$ is $\hbar^4 (\nabla^2)^2$. You know the formula for $\nabla^2$ in spherical polars, or you can look it up.

You also know the hydrogen atom wave function for $n=2,\ell=0$, or can look that up. Apply the operator to the wave function. There's a lot of differentiation but it's straightforward.

Multiply that by the wave function. In principle this is complex-conjugated but for $m=0$ it's real so that doesn't matter.

Integrate that over all space by multiplying by $4\pi r^2$ and doing the integral over $r$ from 0 to infinity. (if you want to do the $\ell=1$ case you need to do the theta integral, but for $\ell=0$ there is no angular dependence.) This gives you an integral involving powers of $r$ and $e^{-r/2a_0}$ which is messy (I'm not surprised your professor didn't drag you through it) but perfectly straightforward.

  • $\begingroup$ Thank you for your answer. This makes a lot more sense now. My lecturer often, I think, assumes that we have absorbed more from earlier lectures than we usually have and so doesn't often refer back to concepts mentioned in earlier material when explaining material in later lectures (probably because he's such a smart man and is so used to this stuff himself). I see know though how to work through the problem and do recall a formula similar to the one you describe much further back . $\endgroup$
    – bhapi
    May 15, 2018 at 19:00
  • $\begingroup$ P.s. I just looked up the hydrogen wavefunctions. I know you can not speak for my lecturer but do you feel these would be given in an exam or should I try to memorise them ? $\endgroup$
    – bhapi
    May 15, 2018 at 19:02
  • $\begingroup$ I think you should know that they involve exponentials and polynomials in r, and spherical harmonics in theta and phi. But I (personally) see no point in memorising the details. $\endgroup$ May 15, 2018 at 19:16

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