# Canonical Transformation and harmonic-oscillator

1. Dec 10, 2009

### 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?

2. Dec 11, 2009

### gabbagabbahey

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

3. Dec 11, 2009

### Shafikae

I dont 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??

4. Dec 11, 2009

### gabbagabbahey

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

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

5. Dec 11, 2009

### 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.

6. Dec 11, 2009

### Shafikae

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

7. Dec 11, 2009

### Shafikae

sorry i meanH = p2/2m +(1/2)kx2

8. Dec 11, 2009

### Shafikae

H = p2/2m +(1/2)kq2