18.1 The Helium-Like Atom 765
quantum mechanically, so we must work with approximations. We begin with the
zero-order approximation, which corresponds to ignoring the term that depends onr 12.
This is not a good approximation, but it is a starting point for better approximations.
The zero-order Hamiltonian operator is a sum of two hydrogen-like Hamiltonians:̂H(0)−h ̄2
2 m∇ 12 −
Ze^2
4 πε 0 r 1−
h ̄^2
2 m∇^22 −
Ze^2
4 πε 0 r 2(18.1-3)
ĤHL(r 1 ,θ 1 ,φ 1 )+ĤHL(r 2 ,θ 2 ,φ 2 )ĤHL(1)+ĤHL(2) (18.1-4)whereĤHL(1) andĤHL(2) are hydrogen-like Hamiltonians. In the last equality we
abbreviate the coordinates of a particle by writing only the particle index. These
hydrogen-like Hamiltonians must correspond toZ2 for the helium atom, orZ 3
for the Li+ion, and so on. The zero-order time-independent Schrödinger equation iŝH(0)Ψ(0)(1, 2)[
ĤHL(1)+ĤHL(2)
]
Ψ(0)(1, 2)E(0)Ψ(0)(1, 2) (18.1-5)
where we continue our practice of the last chapter, using a capitalΨfor a multielectron
wave function and a lower-caseψfor a one-electron wave function. Equation (18.1-5)
can be solved by separation of variables, using the trial solution:Ψ(0)(1, 2)ψ 1 (r 1 ,θ 1 ,φ 1 )ψ 2 (r 2 ,θ 2 ,φ 2 )ψ 1 (1)ψ 2 (2) (18.1-6)whereψ 1 andψ 2 are two orbitals. In the second equality we abbreviate the coordinates
by writing only the subscript specifying the electron. A multielectron wave function
that is a product of orbitals is called anorbital wave function.
We substitute the trial solution of Eq. (18.1-6) into Eq. (18.1-5) and use the fact that
ψ 1 (1) is treated as a constant when̂HHL(2) operates andψ 2 (2) is treated as a constant
whenĤHL(1) operates. The result is:ψ 2 (2)̂HHL(1)ψ 1 (1)+ψ 1 (1)ĤHL(2)ψ 2 (2)E(0)ψ 1 (1)ψ 2 (2) (18.1-7)We divide byψ 1 (1)ψ 2 (2):1
ψ 1 (1)ĤHL(1)ψ 1 (1)+^1
ψ 2 (2)̂HHL(2)ψ 2 (2)E(0) (18.1-8)The variables are now separated. That is, each of the terms on the left-hand side of the
equation depends only on a set of variables not occurring in the other term. Because
each set of variables can be allowed to range while the other is fixed, the first term
must equal a constant, which we callE 1 , and the second term must equal a constant,
E 2. These constants must add to the approximate energy eigenvalueE(0):E 1 +E 2 E(0) (18.1-9)
We set the first term in Eq. (18.1-8) equal toE 1 and multiply byψ 1 (1) and carry out
the analogous operations on the second term. We now have two equations:ĤHL(1)ψ 1 (1)E 1 ψ 1 (1) (18.1-10)ĤHL(2)ψ 2 (2)E 2 ψ 2 (2) (18.1-11)