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when I was new into quantum mechanics, I thought we can describe helium atom by two wavefunctions - one for every electron. After some time I discovered how wrong I was - first, because electrons are indistinguishable we can't say 'this' electron now is here and after time it will be here, second - we take wavefunction and antisymetrize it to obtain the Pauli Exclusion Principle.

Okay, that was only an introduction. What should we do if we would like to get a full description of hydrogen atom, let's say in Dirac's equation formalism (assuming proton is elementary particle, we don't want to mess with QCD yet - if somebody has problem with that he can use positron or antimuon)? Of course the standard procedure is to get only an electron and potential from proton, maybe if we want to be super-detalic we can write another Dirac equation for proton (still spin 1/2) and perturb it by attraction from electron. But it's still not full description.

Any ideas, answers, links to handbooks? Or is it open problem? I'm not looking for calculation method, only for theoretical proposition for equation.

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    $\begingroup$ You do know about tensor products don't you? Why not just take the two Hilbert spaces for the two particles and take a tensor product? $\endgroup$ – Martin Feb 1 '15 at 16:33
  • $\begingroup$ To be clear: are you asking for the "usual" (i.e. with the proton considered as fixed) solution for the hydrogen atom in the relativistic case, where the Dirac equation holds? Or are you asking for what happens if you try to solve the hydrogen atom without decoupling nucleus and electron (which can be done even in the non-relativistic approximation)? $\endgroup$ – glS Feb 1 '15 at 16:56
  • $\begingroup$ decoupling nucleus and electron. If I would want to find solution for hydrogen atom in relativistic description, believe me, i have handbooks for that :) $\endgroup$ – Cheshire Cat Feb 1 '15 at 17:39
  • $\begingroup$ @Martin - well, I can try with tensor products. $\endgroup$ – Cheshire Cat Feb 1 '15 at 17:44
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In nonrelativistic quantum mechanics you set up a wavefunction that is a function of configuration space, $\Psi(x_1,y_1,z_1, \dots , x_n,y_n,z_n,t)$. Where $(x_i,y_i,z_i)$ is the position of the i-th particle. If any particles are identical then you make sure that the wavefunction is (anti)symmetric in those coordinates. Then if you are doing a simple scalar potential you make it a function of the same configuration space, and solve the Schrödinger (or Schrödinger-Pauli) equation.

For example with hydrogen, you can have $\Psi(\vec{r}_1,\vec{r}_2,t)$ with $\vec{r}_1$ being the position of the electron and $\vec{r}_2$ being the position of the proton. No antisymmetry required. And we can have a scalar potential energy of $V(\vec{r}_1,\vec{r}_2,t)=-ke^2/|\vec{r}_1-\vec{r}_2|$. To solve it, you often use separation of variables. Most commonly (for this problem) you look for solutions of the form $\Psi(\vec{r}_1,\vec{r}_2,t)$ =$$A(x_1-x_2,y_1-y_2,z_1-z_2)C(\frac{x_1m_e+x_2m_p}{m_e+m_p},\frac{y_1m_e+y_2m_p}{m_e+m_p},\frac{z_1m_e+z_2m_p}{m_e+m_p})T(t),$$ where $A$ is a solution to the hydrogen atom with reduced mass for the relative separation of the electron and the proton, $C$ is a free particle solution for the center of mass, and $T$ is the time dependence determined by the energy.

You can include spin and do the Schrödinger-Pauli equation. You can include magnetic forces, you can include more accurate terms for the kinetic energy to get closer to relativisticaly correct solutions. All without being very different in overall approach. The above is a common approach, detailed for instance, in Griffiths' Introduction to Quantum Mechanics.

However, if you go to full relativistic quantum mechanics, you can include more esoteric effects, like vaccuum polarization and corrections to small effects. But these corrections are based on the fact that it's never truly just two particles isolated from everything, there is a quantum vaccuum, and an entire electron-positron field so the one electron orbiting the proton has to deal with how virtual electrons and virtual positrons and virtual photons mutually interact with it, each other, and the proton.

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