Question

A system consists of two particles of mass mi and m2 interacting with an interaction potential V(r) that depends only on the relative distancer- Iri-r2l between the particles, where r- (ri,/i,21) and r2 22,ひ2,22 are the coordinates of the two particles in three dimensions (3D) (a) /3 pointsl Show that for such an interaction potential, the Hamiltonian of the system H- am▽ri _ 2m2 ▽22 + V(r) can be, put in the form 2M where ▽ and ▽ are the Lulacians referring, respectively, to the center of mass (OM) R (miri + m2r2)/(mı + m2) and the relative r = ri-r2 α)(rdinates, whereas M = m1 + m2 is the total inass and μ TIL 1 Tn2/(m1 + m2) is the reduced mass. Note that the vector R is the mass weighted average of the positions of the two particles. If the 3D coordinates of R are denoted by R (X, Y Z), the corresponding laplacian operator is given by ▽ t OP/aX2 +OP/oya +伊/02°, similarly, for r-x, y, z), the corresponding Laplasian is given by ▽2 /dz2+OP/0y2+OP/az21 Thus the Hamiltonian falls into the sum of two independent parts, one depending only on R and the other - only on r, and hence the system is separable in COM and relative coordinates. (b) [2 points] Write down the time independent Schrödinger equation for the two particle wave function ψ(ri r2) with energy E (recall that E is a separation constant with respect to the coor dinate and time variables, for the time dependent Schrödinger equation in a stationary external potential; here, the role of the external potential is taken by the interaction potential between the two particles), and then look for separable solutions with respect to the COM and relative coordinates: ψ(ri, r2) ψ(M) (R)ψ(u) r). Write down the equations of motions that the functions (M) (R) and ψ(x) r) must separately satisfy; fr)m these equations, one should be able to see that the equation for ψ(M) (R) describes the motion o a re no external potential fictitious particle o mass M whereas the equation for (μ) r is the Schrodinger equation for a single fictitious particle of mass moving in an elfective external potential given by the same teraction potential V(r), i.e., the same probleln as described in your textbook, starting froin Eq. (48), which itself then leads to the problem of motion of an electron in a hydrogen atom Thus, the quantum two particle problem in which the interaction potential between the particles, V(r), depends only on the relative distance between the particles can always be reduced to two independent one particle problems: one being a free particle problem, and the other - particle in a spherically symmetric (also referred to as centrally symmetric) external potential V(r) (c) [2 pointsl What are the center of mass (P) and relative (p) momenta in terms of the momenta pi and pz of the wo particles? Check to confirm that the system Hamiltonian, written down in terms of COM and relative momenta, is the same as the one you derived in part (a) in position basis. Check that the commutation relations between the components of R and P (and between the components of f and p) are the same ones as the canonical commutation relations for the components of the coordinates and momenta of any single particle.

Answer 1

Here we have H = -h^2/2 m_1 nabla^2_r_1 - h^2/2m_2 nabla^2_r_2 + V (r) - (1) Here, r = r_1 - r_2 = relative co-ordinate we then change, variable r_1, r_2 to r and R Where, R = Co - ordinate of the COM So, R = m_1 r_1 + m_2 r_2/m_1 + m_2 - (2) and M = m-1 + m_2 mu = m_1 m_2/m_1 + m_2 = m_1 m_2/ from eq^n (2)- (m_1 + m_2) R = m_1 r_1 + m_2 r_2 = m_1 r_1 + m_2 (r_1 - r) = (m_1 + m_2) r_1 - m_2 r implies r_1 = R + m_2/m_1 + m_2 middot r = R + mu/m_1 r similarily, r_2 = R - mu/m_2 r Let R = (x, y, z) and r = (x, y, z) (nabla_r_1)_x = partial differential/partial differential x_r_1 = partial differential x/partial differential x_r_x middot partial differential/partial differential x + partial differential x/partial differential x_r_1 middot partial differential/partial differential x = m_1/m_1 + m_2 partial differential/partial differential x + partial differential/partial differential x = mu/m_2