- #1
Chen
- 977
- 1
Hello,
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.
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.