# 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_π

where H_M = βΒ²/ 2M β_MΒ² = PΒ² / 2M

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

and H_π = βΒ²/ 2π β_πΒ² + V(r) = pΒ²/2π + V(r)

suggest that the motion of the reduced mass system appears as a particle of mass π 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

π(r, R) = π_π (r) π_M (R)

The Schrodinger equation

Hπ = Eπ

H_ππ_π + H_Mπ_M = E_ππ_π + E_Mπ_M

or, H_ππ_π / π_π + H_Mπ_M / π_M = E_π + E_M

or, H_ππ_π = E_π π_π

or, H_Mπ_M = E_Mπ_M

The free particle wave equations,

H_Mπ_M = E_Mπ_M has a plane wave solutions

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

E_Mβ©Ύ 0. The potential problem thus reduces to

For V(r) to centrally symmetric

So we have,

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