Schrodinger equation for a system of two interacting particles

Two particles system is obviously 6 dimensional problem. We can 6 dimensional problem to three dimensional problem by introducing relative and centre of mass coordinates.

For two particles 1 and 2 that interact only through the mutual interaction, the Hamiltonian is

Introducing relative and centre of mass coordinate

Similarly we get

Thus the equation ii can be separated into two independent parts as

H = H_M + H_𝜇

where H_M = ℏ²/ 2M ∇_M² = P² / 2M

suggests that the motion of centre of mass appears as a free particle of mass M

and H_𝜇 = ℏ²/ 2𝜇 ∇_𝜇² + V(r) = p²/2𝜇 + V(r)

suggest that the motion of the reduced mass system appears as a particle of mass 𝜇 in the central potential

Now, the total angular momentum

splits into operators for relative and centre motions as

As the Hamiltonian is a sum of two parts

involving independent coordinates we can write

𝜓(r, R) = 𝜓_𝜇 (r) 𝜓_M (R)

The Schrodinger equation

H𝜓 = E𝜓

H_𝜇𝜓_𝜇 + H_M𝜓_M = E_𝜇𝜓_𝜇 + E_M𝜓_M

or, H_𝜇𝜓_𝜇 / 𝜓_𝜇 + H_M𝜓_M / 𝜓_M = E_𝜇 + E_M

or, H_𝜇𝜓_𝜇 = E_𝜇 𝜓_𝜇

or, H_M𝜓_M = E_M𝜓_M

The free particle wave equations,

H_M𝜓_M = E_M𝜓_M has a plane wave solutions

It simply represents the translation of the system as a whole with K.E

E_M⩾ 0. The potential problem thus reduces to

For V(r) to centrally symmetric

So we have,

This note is a part of the Physics Repository