Question

(a) Suppose you have solved the Schrodinger equation for the singlyionized helium atom and found a set of eigenfunctions \(\phi_N(r)\).

(1) How do the \(\phi_N(r)\) compare with the hydrogen atom wave functions?

(2) If we include a spin part \(\sigma ^+ (or\,\sigma^-)\) for spin up (or spin down), how do you combine the \(\psi's\) and \(\sigma's\) to form an eigenfunction of definite spin?

(b) Now consider the helium atom to have two electrons, but ignore the electromagnetic interactions between them.

(1) Write down a typical two-electron wave function, in terms of the \(\psi's\) and \(\sigma's\), of definite spin. Do not choose the ground state.

(2) What is the total spin in your example?

(3) Demonstrate that your example is consistent with the Pauli exclusion principle.

(4) Demonstrate that your example is antisymmetric with respect to electron interchange.

Answer 1

(a) (1) The Schrodinger equation for singly-charged He atom is the same as that for H atom with e² → Ze², where Z is the charge of the He nucleus. Hence the wave functions for hydrogen-like ion are the same as those for H atom with the Bohr radius replaced:

µ being the reduced mass of the system. For helium Z = 2.

(2) As \(\phi_N\) and \(\sigma^\pm\) belong to different spaces we can simply multiply them to form an eigenfunction of a definite spin.

(b) (1), (2) A He atom, which has two electrons, may be represented by a wave function

if the total spin is 1.

(3) If \(\sigma^+\) = \(\sigma^-\), \(\phi_{N_1} = \psi_{N_2}\), the wave functions vanish, in agreement with the Pauli exclusion principle.

(4) Denote the wave functions by \(\psi\)(1, 2). Interchanging particles 1 and 2 we have

\(\psi(2,1) = - \psi(1,2)\).