A typical interaction potential depends only on the vector displacement r≡r1−r2 between the two particles: V(r1,r2)→V(r). 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
μ≡m1+m2m1m2
is the reduced mass of the system.
(b) Show that the (time-independent) Schrödinger equation (5.7) becomes
−2(m1+m2)ℏ2∇R2ψ−2μℏ2∇r2ψ+V(r)ψ=Eψ.
(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; it reduces the two-body problem to an equivalent one-body problem.