HW #2 (129A), due Sep 27, 4pm
1. Dirac introduced a relativistic wave equation for spin 1/2 particle,
i¯h∂
∂t ψ=Hψ =hc~α ·~p +mc2βiψ. (1)
The matrices αand βare given in the lecture notes. Answer the following
questions.
(1) Show that the momentum ~p commutes with the Hamiltonian and hence
is conserved.
(2) Show that the orbital angular momentum ~
L=~x ×~p does not commute
with the Hamiltonian, and hence is not conserved, while the total an-
gular momentum ~
J=~
L+¯h
2~
Σ is.
(3) To label a state, you specify eigenvalues of operators that commute
with each other. Show that ~
Jdoes not commute with the momentum
and cannot be used to label a state together with the momentum.
(4) On the other hand, the angular momentum along the direction of the
momentum can be used. Verify this by calculating the commutator or
~p ·~
Jwith ~p, and by showing the eigenvalues ~p ·~
J=±¯h
2|~p|.
Note The combination h≡~p·~
J
|~p|is called helicity, and its eigenvalue ±¯h
2shows
if the spin is paralell or anti-paralell to its motion. You specify a state
of a free relativistic particle by its three-momentum ~p and its helicity
h. When h=¯h
2, the particle is said to be right-handed, while when
h=−¯h
2left-handed.
2. When the electron moves in a constant magnetic field, show that its
spin and its momentum rotate by 2πwith the same frequency (spin precession
and Larmor frequencies), if g= 2 exactly. It means that the right-handed
electron stays right-handed in cyclotron motion. This is the basis with which
we measure the deviation of gfrom 2.
3. Solve Problem 1.1 from Cahn–Goldhaber.