# center of mass Hamiltonian of a Hydrogen atom

I'm working through Mattuck's "A Guide to Feynman Diagrams in the Many-Body Problem", but I'm stuck on a bit which I feel should be trivial.

In section 3.2 (p 43 in the Dover edition) he gives a Hydrogen atom as an example of a system which can be considered to have a p-dependent potential term.

In his words,

"the Hamiltonian of the center of mass motion of a Hydrogen atom is $H = p^2 / (m + m_e)$ where $m$ = proton mass and $m_e$ = electron mass.

(First question: shouldn't there be a factor 2 in the denominator here?)

This may be broken up into

$$H = \frac{p^2}{2m} - \frac{m_e}{(m_e+m)m}p^2$$

and the second term treated as if it were a perturbating potential."

Second question: I can't for the life of me figure out how he breaks it up like that, whether I include the aforementioned factor 2 or not..

Both multiplying by $\frac{m - m_e}{m - m_e}$ and ignoring terms quadratic in $m_e$, and deriving the first order taylor expansion around $m$ (in both cases inclluding the factor 2 which I believe should be there) give

$$\frac{p^2}{2m} - \frac{p^2m_e}{2m^2}$$

which looks kind of close, but not quite it.

Of course this is not a hugely significant part of the text but it's really bugging me so a bit of help would be greatly appreciated...

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You are right; Hamiltonian for center of mass of hydrogen atom should be :

$H=\displaystyle\frac{p^2}{2(m+m_e)}$

Where $p$ is momentum of Hydrogen atom (please check what is $p$ in your book).

Now you can also write it as :

$H=\displaystyle\frac{p^2}{2m}(\frac {m}{m+m_e})$

$=\displaystyle\frac{p^2}{2m}(\frac {(m+m_e)-m_e}{m+m_e})$

$=\displaystyle\frac{p^2}{2m}(1-\frac {m_e}{m+m_e})$

$=\displaystyle\frac{p^2}{2m}-\frac {m_e}{2m(m+m_e)}p^2$

So there must be some typos in your book.

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ya, now I have removed it :) – user10001 Jul 11 '12 at 5:12
Ok, already +1'd it, thanks. – Ron Maimon Jul 11 '12 at 5:56
Thanks a lot, yes $p$ is the momentum of the center of mass of the atom... So there's no ignoring of terms even, and the author has forgotten the factor 2 twice. – mszep Jul 11 '12 at 8:44