4 Home exercise sheet

4
Home exercise sheet
Exercise 4.1: Commutation relation
Find the commutation relation [φ, Lz ] where Lz is the angular momentum in the z direction
and φ is any function, which is spherically symmetrical about the origin, of the coordinates
and momentum of a particle.
1. Start by finding out what does φ depend on. (i.e. φ(..., ..., ..., ...))
2. Express Lz as a function of ri and pi .
3. Calculate the commutation relation between φ and Lz
4. Could you have guessed the solution with out doing the math?
Exercise 4.2: Spring and moving wall
A mass m is connected to a wall by a horizontal spring with spring constant k and relaxed
length l0 . The wall is arranged to move back and forth with position Xwall = A sin ωt. Let
z measure the stretch of the spring. Find the Hamiltonian in terms of z and its conjugate
momentum, and then write down Hamilton’s equations. Is H the energy? Is H conserved?
Exercise 4.3: The motion of Vortices
Consider the motion n line vortices in the plane with potions ri = (xi , yi ), and with a
strength γi . The equations of motion are:
X
yi − yj
(1)
γi ẋi = −
γi γj
|ri − rj |2
i6=j
X
xi − xj
γi ẏi =
γi γj
(2)
|ri − rj |2
i6=j
where there is no sum over i in these equations. The Hamiltonian of the system is:
X
H=−
γi γj log |ri − rj |
i<j
1
(3)
1. Use the above relations to identify canonical positions qi and momenta pi . Hint - these
need not correspond to the physical position and momentum.
2. Use the canonical variables that you have found to define the Poisson bracket structure:
X ∂f ∂g
∂f ∂g
−
(4)
{f, g} ≡
∂qi ∂pi ∂pi ∂qi
i
3. Verify using the Poisson brackets defined above that the Hamiltonian reproduce the
equations of motion.
P
4. Show that the total momentum defined a P = i pi is a conserved.
5. Show that the following quantity is conserved:
n
1X
J =−
γi x2i + yi2
2 i=1
6. Are tehre any other conserved quantities?
7. Solve the equations of motion in the case of two vortices.
2
(5)