Canonical Transformation and harmonic-oscillator

[Shafikae]

Show that the transformation
Q = p + iaq , P = (p-iaq)/2ia
is canonical and find the generating function. Use the transformation to solve the harmonic-oscillator problem.

I was able to determine if the transformation is canonical, and it is. However, when it came to finding the generating function, I wasnt getting it right. And how do we use this transformation to solve the harmonic oscillator?

 

[gabbagabbahey]

Well, what do you know about generating functions?...What does the Hamiltonian become under this transformation?

 

[Shafikae]

I don't know how to obtain the Hamiltonian. But i know that to check whether the transformation is canonical, we can show that the Poisson brackets are invariant. Which I have shown they are. I'm not getting the right generating function, or is the generating function really the hamiltonian??

 

[gabbagabbahey]

I'm not getting the right generating function, or is the generating function really the hamiltonian??

Well, what are you doing to obtain the (incorrect) generating function?

I don't know how to obtain the Hamiltonian.

You don't know what the Hamiltonian of the one-dimensional harmonic oscillator is?

 

[Shafikae]

Ok so I take the poisson bracket and obtain 1, therefore its canonical. I have an example similar to this, so i take q = - partial F / partial p , so i solve from the original equation of Q = p + iaq, i solve for q and then take the integral and solve for F.

 

[Shafikae]

Would the hamiltonian of a harmonic oscillator be H = p²/2 +(1/2)kx²

 

[Shafikae]

sorry i meanH = p²/2m +(1/2)kx²

 

[Shafikae]

H = p²/2m +(1/2)kq²