The Hamiltonian for the system contains a term describing the repulsion of the electrons -- so, to do it properly, you need to consider both electrons at the same time.
As a first approximation, you can assume that this term is negligible. In that case, the Hamiltonian splits into 2 independant hydrogen Hamiltonians (with a nuclear charge of 2e instead of e, of course), and an exact solution can be found. This solution turns out to be a product of hydrogenic wave functions. Since the He atom is just double the number of protons and electrons of H, this makes sense.
To do the analysis properly, we cannot ignore the electron-electron interaction, in which case the math gets messy. A common way to perform the analysis is to use the variational method. This involves using a trial function with adjustable paramaters to get an approximation for the ground state energy. Using more and more complicated functions, the ground state can be approximated extremely well (of course, we know the real value from experiments).
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