- #1

thecourtholio

- 19

- 1

## Homework Statement

Consider a charge ##q##, with mass ##m##, moving in the ##x-y## plane under the influence of a uniform magnetic field ##\vec{B}=B\hat{z}##. Show that the Hamiltonian $$ H = \frac{(\vec{p}-q\vec{A})^2}{2m}$$ with $$\vec{A} = \frac{1}{2}(\vec{B}\times\vec{r})$$ reduces to $$ H(x,y,p_x,p_y) = \frac{(p_x+\frac{1}{2}qBy)^2}{2m} + \frac{(p_y-\frac{1}{2}qBx)^2}{2m}$$

Demonstrate

$$ Q = \frac{(p_x+\frac{1}{2}qBy)}{qB} \qquad \qquad P = (p_y-\frac{1}{2}qBx) $$

are conjugate dynamic variables, given ##x, p_x, y, p_y## are, then express $$H(Q,P)$$ in terms of ##m## and the cyclotron frequency, ##\omega \frac{qB}{m}##

Show next that

$$ P' = \frac{(p_x-\frac{1}{2}qBy)}{qB} \qquad \qquad Q' = (p_y+\frac{1}{2}qBx) $$

Are yet another, linearly-independent, conjugate pair whose brackets with ##Q,P## necessarily vanish, i.e.

$$ [Q,Q'] = [Q,P'] = [P,Q'] = [P,P'] = 0 $$

Argue from the foregoing that ##Q',P'## must be constants of the motion

## Homework Equations

Most are listed in problem statement. Definition of poisson bracket (PB): $$ [Q,P] = \frac{\partial Q}{\partial q}\frac{\partial P}{\partial p} - \frac{\partial Q}{\partial p}\frac{\partial P}{\partial q} $$

Fundamental PBs: ## [q_i,q_k] = [p_i,p_k] = 0, \ \ [q_i,p_k] = \delta_{ik}##

## The Attempt at a Solution

My main question is, how exactly do I show that ##Q,P## are conjugate dynamical variables? Is this just by evaluating the PBs ##[Q_i, Q_k], [P_i, P_k], \text{ and } [Q_i,P_k]## and proving they preserve the fundamental PBs? So far, I have found

$$ [Q_i, Q_k]_{q,p} = 0 $$

And

$$ [P_i,P_k]_{q,p} = 0$$

But for ##[Q_i,P_k]_{q,p}## I find

$$ [Q_i,P_k]_{q,p} = \frac{1}{2}+\frac{1}{2}qB \neq 1 (or \neq \delta_{ik})$$

So is this not what I need to be doing or am I just evaluating it wrong?

And as for expressing the Hamiltonian as functions of ##Q## and ##P##, what do I need to do? My prof kind of worked out finding ##H(q(Q,P), p(Q,P))## for a harmonic oscillator by seeking a transformation of the form ##H(q(Q,P),p(Q,P)) = \frac{f^2(P)}{2m}##, but I'm having trouble figureing out how to do it in the reverse, like ##H(Q(q,p), P(q,p))## for this problem.