Perturbation Treatment of the Helium-Atom Ground State

3 PERTURBATION TREATMENT OF THE HELIUM-ATOM GROUND STATE 

ITie helium atom has two electrons and a nucleus of charge +2e. We shall consider the nucleus to be at rest (Section 6.6) and place the origin of the coordinate system at the nucleus. The coordinates of electrons 1 and 2 are Zi) and (X2,y2, Z2), see Fig. 9.1. 

If we take the nuclear charge to be +Zc instead of +2e, we can treat heliumlike ions such as H, LF, Be. TTie Hamiltonian operator is 

The Schrodinger equation involves six independent variables, three coordinates for each electron. In spherical coordinates, ijf = (rj, dj, f i, t2,62, fc). 

Because of the Vryi term, the Schrodinger equation for helium cannot be separated in any coordinate system, and we must use approximation methods. To use the perturbation method, we must separate the Hamiltonian (9.39) into two parts, and i, where is the Hamiltonian of an exactly solvable problem. If we choose 

3 Perturbation Treatment of the Helium-Atom Ground State 

The unperturbed system is a helium atom in which the two electrons exert no forces on each other. Although such a system does not exist, this does not prevent us from applying perturbation theory to this system. 

---

Section 9,3 Perturbation Treatment of the Helium-Atom Ground State 253... 

We now reconsider the helium atom from the standpoint of electron spin and the Pauli principle. In the perturbation treatment of helium in Section 9.3, we found the zeroth-order wave function for the ground state to be 15(1)15(2). To take spin into account, we must multiply this spatial function by a spin eigenfunction. We therefore consider the possible spin eigenfunctions for two electrons. We shall use the notation a(l)a(2) to indicate a state where electron 1 has spin up and electron 2 has spin up a(l) stands for a(Wji). Since each electron has two possible spin states, we have at first sight the four possible spin functions ... 

Suppose we take the interelectronic repulsions in the li atom as a perturbation on the remaining terms in the Hamiltonian. By the same steps used in the treatment of helium, the unperturbed wave functions are products of three hydrogenlike functions. For the ground state,... 

---