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    Coupled harmonic oscillators

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    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?

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    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 ...

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