# 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

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

#### 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??

#### gabbagabbahey

Homework Helper
Gold Member
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 dont 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

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