Noninteracting Particles and Separation of Variables

The eigenfunctions of are the spherical harmonics Yf B, f ), and since does not involve r, we can multiply Y by an arbitrary function of r and still have eigenfunctions of and L. Therefore, 

Using (6.16) in (6.15), we then divide both sides by Y P to get an ordinary differential equation for the unknown function R r)  

We have shown that, for my one-particle problem with a spherically symmetric potential-energy function Vif), the stationary-state wave functions are = R r)Yf(6, f ), where the radial factor R r) satisfies (6.17). By using a specific form for V(r) in (6.17), we can solve it for a particular problem. 

Up to this point, we have solved only one-particle quantum-mechanical problems. The hydrogen atom is a two-particle system, and as a preliminary to dealing with the H atom, we first consider a simpler case, that of two noninteracting particles. 

Suppose that a system is composed of the noninteracting particles 1 and 2. Let and Qi symbolize the coordinates (jc,y, z,) and (X2,y2, Z2) of particles 1 and 2. Because the particles exert no forces on each other, the classical-mechanical energy of the system is the sum of the energies of the two particles E = E E2 = Vi T2 V2, and 

Since Hi involves only the coordinate and momentum operators of particle 1, we have Hi[Gi(qi)G2iq2)] = G2 q2)HiGi qi), since, as far as Hi is concerned, G2 is a constant. Using this equation and a similar equation for H2, we find that (6.20) becomes 

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