Typically, the interaction potential depends only on the vector r ≡ r1 – r2 between the tw...

Typically, the interaction potential depends only on the vector r ≡ r1 – r2 between the two particles. In that case the Schrödinger equation separates, if we change variables from r1, r2 to r and R ≡ (m1r1 + m2r2)/(m1 + m2) (the center of mass).

(a) Show that r1 = R + (μ/m1)r, r2 = R – (μ/m2)r, and ∇1 = (μ/m2)∇R + ∇r, ∇2 = (μ/m1)∇R – ∇r, where

 [5.8]

is the reduced mass of the system.

(b) Show that the (time-independent) Schrödinger equation becomes

(c) Separate the variables, letting ψ (R, r) = ψR(R)ψr(r). Note that ψR satisfies the one-particle Schrödinger equation, with the total mass (m1 + m2) in place of m, potential zero, and energy ER, while ψr satisfies the one-particle Schrödinger equation with the reduced mass in place of m, potential V(r), and energy Er. The total energy is the sum: E = ER + Er. What this tells us is that the center of mass moves like a free particle, and the relative motion (that is, the motion of particle 2 with respect to particle 1) is the same as if we had a single particle with the reduced mass, subject to the potential V. Exactly the same decomposition occurs in classical mechanics;1 it reduces the two-body problem to an equivalent one-body problem.