Coupled Oscillators
المؤلف:
Sidney B. Cahn, Gerald D. Mahan And Boris E. Nadgorny
المصدر:
A GUIDE TO PHYSICS PROBLEMS
الجزء والصفحة:
part 2 , p 58
14-8-2016
1684
Coupled Oscillators
Two identical harmonic oscillators in one dimension each have mass m and frequency ω. Let the two oscillators be coupled by an interaction term Cx1x2, where C is a constant and x1 and x2 are the coordinates of the two oscillators. Find the exact spectrum of eigenvalues for this coupled system.
SOLUTION
The Hamiltonian of the system is
(1)
The problem is easily solved in center-of-mass coordinates. So define
(2)
These new coordinates are used to rewrite the Hamiltonian. It now decouples into separate x- and y- parts:
(3)
(4)
(5)
The x-oscillator has a frequency 
and eigenvalues hΩx (nx + 1/2), where nx is an integer. The y-oscillator has a frequency of 
and eigenvalues hΩy (ny + 1/2), where ny is an integer.
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