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Two-Particle Systems
Consider a system consisting of two particles, mass
and
,
interacting via the potential
which only depends on the
relative positions of the particles. According to Eqs. (419)
and (426), the Hamiltonian of the system is written
 |
(437) |
Let
 |
(438) |
be the particles' relative position, and
 |
(439) |
the position of the center of mass.
It is easily demonstrated that
 |
|
|
(440) |
 |
|
|
(441) |
Hence, when expressed in terms of the new variables,
and
,
the Hamiltonian becomes
 |
(442) |
where
 |
(443) |
is the total mass of the system, and
 |
(444) |
the so-called reduced mass.
Note that the total momentum of the system can be written
 |
(445) |
The fact that the Hamiltonian (442) is separable when expressed
in terms of the new coordinates [i.e.,
suggests, by analogy with the analysis
in the previous section, that the wavefunction can be factorized: i.e.,
 |
(446) |
Hence, the time-dependent Schrödinger equation (423)
also factorizes to give
 |
(447) |
and
 |
(448) |
The above equation can be solved to give
 |
(449) |
where
,
, and
are constants. It is clear, from Eqs. (445), (446), and (449), that
the total momentum of the system takes the constant value
:
i.e., momentum is conserved.
Suppose that we work in the centre of mass frame of the system, which is characterized by
. It follows that
. In this case, we can write the wavefunction of the system in the form
, where
 |
(450) |
In other words, in the center of mass frame, two particles of mass
and
, moving in the potential
, are equivalent
to a single particle of mass
, moving in the potential
,
where
. This is a familiar result from classical dynamics.
Next: Identical Particles
Up: Multi-Particle Systems
Previous: Non-Interacting Particles
Richard Fitzpatrick
2010-07-20