11.1: Center of Mass Frame and the Two-body Problem

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The problem of a particle in a given potential can be solved classically from Newton’s equations. The Schrödinger equation can be used to describe the behavior of one particle in a field.

The problem of two particles interacting via conservative fields can be reformulated into two parts: the behavior of the center of mass and the behavior of the relative velocities of the particles. If we work in the center of mass frame (COM), then the behavior of the center of mass is trivial, and we need worry only about the relative motions. This can be described by a single effective particle with effective mass \(\mu = \frac{m_1m_2}{m_1+m_2}\). This effective particle can then be treated with one particle equations.

Note

The problem of three interacting particles cannot be reduced in this way. Hence the ‘three-body problem’ is in general insoluble.

The COM transformation allows us to treat the scattering problem as a one body problem. For scattering problems we work in the COM frame, describing two real particles as an effective particle moving in a potential. Do not forget that for any experiment we will have to apply the above transformation to relate theory to the experimental results, though if the target particle is much heavier than the other the transformation may be slight. Note also that this transformation is invalid if there is an external field.