Required math: calculus
Required physics: Schrödinger equation
Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 5.11.
The crude model of the helium atom ignored the interaction between the two electrons. As a result, the ground state energy of as predicted by the model was quite far off the measured value of . One way of computing a correction to the model is to take the interaction-free wave function and use it to calculate the mean value of the interaction term. That is, we take the ground state wave function to be
The two functions on the RHS aren’t quite the same as the hydrogen ground state functions, however. Recall that the ground state of hydrogen is
where is the Bohr radius:
The in the denominator comes from the interaction energy between the single electron in hydrogen and the single proton in the nucleus. For helium, each electron interacts with the two protons in the nucleus, so the ‘Bohr radius for helium’ has a factor of in place of the for hydrogen. Thus we must replace by to get the helium wave function:
We can now use this function to calculate the mean value of the interaction term . That is, we want the integral:
Isolating the mean value factor and expanding the modulus term in the denominator, we get, if we take to be along the axis:
Each integration element has the form
so we get
The two integrals just give a factor of , so we can do those and then the integral over . This gives
We can now do the remaining integrals in the order , and , where the integral is done in two parts: the first from to and the second from to . The result is
The correction to the energy is then
This adjusts the model energy to which is much closer to the experimental value of .