|San José State University|
& Tornado Alley
of a Deuteron as a Rotor
Consider a system consisting of two particles of equal mass m separated by a variable distance s rotating about their center of mass. The coordinates are the distance r of the masses from their center of mass and the angle θ which the line between the masses makes with an arbitrary line.
Let the rate of rotation be denoted as ω. The angular momentum of the system is then
The radial momentum is
The tangential kinetic energy K of the system is then
The radial kinetic energy J is
The force of attraction F between the particles is a function of the separation distance s=2r. Thus the potential energy of the system is
It is necessary to express the kinetic energies in terms of the momenta. For the radial component this is easy.
For the tangential component note that (ωr)=L/(2mr). Thus
Thus the Hamiltonian function for the system is given by
The Hamiltonian operator for the system is then
The time independent Schroedinger equation for the system is then
Multiplying through by r² gives
Now consider separation of variables; i.e., Ψ=R(r)Θ(θ). This results in the equation
Division by RΘ results in
Gathering all functions of r on the left side of the equation and all functions of θ on the right side results in
This means that each side of the equation must be equal to a constant, say −μ².
This means that the equation
must be satisfied. The solution to this latter equation is
Since Θ must be single valued Θ(2π) = Θ(0) and hence μ must be an integer.
The equation for the radial function R(r) is
For large values of r the terms V(2r) and μ²/r² become insignificant. The equation reduces to
which has the solution
It is then useful to look for a solution of the form
(To be continued.)
HOME PAGE OF Thayer Watkins