Consider two coupled identical harmonic oscillators described by the Hamiltonian
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?
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 ...
A detailed solution is given.