Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

Appendix A
From two-body to one-body
problem
Let us consider a quantum system formed by two interacting particles of mass
m 1 and m 2 , with no external fields. The interaction potential V (r) depends
only upon the distance r = |r 2 −r 1 | between the two particles. We do not make
any further assumption on the form V (r). For the case of the Hydrogen atom,
V will be of course the Coulomb potential. The Hamiltonian is
H = p2 1
2m 1
+ p2 2
2m 2
+ V (|r 2 − r 1 |)
(A.1)
As in the case of classical mechanics, one can make a variable change to the
two new variables R and r:
R = m 1r 1 + m 2 r 2
(A.2)
m 1 + m 2
r = r 2 − r 1 (A.3)
corresponding to the position of the center of mass and to the relative position.
It is also convenient to introduce
M = m 1 + m 2 (A.4)
m =
m 1 m 2
m 1 + m 2
(A.5)
where m is known as the reduced mass.
By introducing the new momentum operators: P = −i¯h∇ R and p = −i¯h∇ r ,
conjugate to R and r respectively, the Hamiltonian becomes
H = P 2
2M + p2
2m + V (r)
(A.6)
i.e. we have achieved separation of the variables. The center of mass behaves
like a free particle of mass M; the solutions are plane waves. The interesting
part is however the relative motion. The Schrödinger for the relative motion is
the same as for a particle of mass m under a central force field V (r), having
spherical symmetry with respect to the origin.
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