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I need to solve the Hamiltonian of a one-dimensional system:

[tex]H(p, q) = p^2 + 3pq + q^2[/tex]

And I've been instructed to do so using a canonical transformation of the form:

[tex]p = P \cos{\theta} + Q \sin{\theta}[/tex]

[tex]q = -P \sin{\theta} + Q \cos{\theta}[/tex]

And choosing the correct angle so as to the get the Hamiltonian of an harmonic oscillator.

Applying this transformation, I get:

[tex]H(P, Q) = P^2 + Q^2 - 3/2 (P^2 - Q^2) \sin{2 \theta} + 3 P Q \cos{2 \theta}[/tex]

And as far as I can see, no choice of angle will get me to an Hamiltonian of an harmonic oscillator.

Am I correct? Can someone please check my calculation?

Thanks.

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# Homework Help: Canonical transformation

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