# Canonical transformation for Harmonic oscillator

1. Oct 11, 2012

### aaaa202

Find under what conditions the transformation from (x,p) to (Q,P) is canonical when the transformation equations are:
Q = ap/x , P=bx2
And apply the transformation to the harmonic oscillator.
I did the first part and found a = -1/2b
I am unsure about the next part tho:
We have the hamiltonian in (x,p):
H = 1/2m(p2+(m$\omega$x)2 , $\omega$2=k/m
So transforming to (Q,P) whilst setting am=-1 you get the nice equation:
H = P(Q2+$\omega$2)
Should I now use hamiltons equation in the new coordinate basis, i.e.;
dH/dP = Q2+$\omega$2 = dQ/dt
dH/dQ = 2QP = -dP/dt
And solve these differential equations for Q,P and transform back? I think so but these equations are just not very easu. I mean the second one is easy to do by means of substitution but really the first one is just a mess. What is the smartest way to do this? I can't really get the first one solved.

2. Oct 14, 2012

### gabbagabbahey

Do you mean $a=-\frac{1}{2b}$, or $a=-\frac{1}{2}b$? Brackets are important!

Again, brackets are needed if you meant $H=\frac{1}{2m}\left(p^2+(m\omega x)^2\right)$

(1) Why are you setting am=-1?
(2) You don't get that when you set am=-1

Without setting am=-1, you should get $H=-\frac{1}{4m}PQ^2 -am\omega^2P$

The second DE is only easy to solve once you know Q. The first one is seperable, so al that is needed is some striaghtforward integration... try a trig substitution if you are stuck on the integration part