Purchase Solution

Coupled harmonic oscillators

Not what you're looking for?

Ask Custom Question

Consider two coupled identical harmonic oscillators described by the Hamiltonian
H= p1^2+p2^2/2m+1/2mw^2x1^2+1/2mw^2x2^2+gx1x2
1- What is the lowest energy of the system?
2- What is the ground state eigenfunction?
3-What is the energy and the eigenfunction for the first excited state?

Purchase this Solution

Solution Summary

A detailed solution is given.

Solution Preview

We can write the Hamiltonian as that of two decoupled oscillators as follows. If we rotate in the x1, x2 plane then we keep p1^2 + p2^2 = -hbar^2 (d/dx1^2 + d/dx2^2) invariant, also we keep x1^2 + x2^2 invariant. We can then choose the rotation angle such that x1 x2 gets decoupled. So, we put:

x1 = y1 cos(theta) - y2 sin(theta)

x2 = y1 sin(theta) + y2 cos(theta)

Then we have:

p1^2 + p2^2 = -hbar^2 (d/dy1^2 + d/dy2^2)
x1^2 + x2^2 = y1^2 + y2^2

x1 x2 = 1/2 (y1^2 - y2^2) sin(2 theta) + y1 y2 cos(2 ...

Purchase this Solution

Free BrainMass Quizzes
Intro to the Physics Waves

Some short-answer questions involving the basic vocabulary of string, sound, and water waves.

Classical Mechanics

This quiz is designed to test and improve your knowledge on Classical Mechanics.

Introduction to Nanotechnology/Nanomaterials

This quiz is for any area of science. Test yourself to see what knowledge of nanotechnology you have. This content will also make you familiar with basic concepts of nanotechnology.

The Moon

Test your knowledge of moon phases and movement.

Variables in Science Experiments

How well do you understand variables? Test your knowledge of independent (manipulated), dependent (responding), and controlled variables with this 10 question quiz.