HW- set -3
(spin 1/2  and denstiy operator)

Due date:  Feb. 5, 2009

3.1. Use the commutation relation of angular momentum
operators, i.e., follow p171-174 of Griffths or Merzbacher p.238 for the spin
1/2  systems to derive the matrix representation
of the Pauli matrices as shown in eq. (1) on note  page 4.2.

3.2. Derive the four equations under Eq. (2) on the same
page. For the last equation, find the coefficients for a matrix where A11=1+i
, A12=1+2i; A21=1.0 and A22=2+i.

3.3.  Show that

where is a unit vector. "Remember" this equation
which will be used often.

3.4. This is for a spin 1/2 system. The matrix
representation of any operator that is defined with respect to the base states which are eigenstates of is called standard representation. All the matrice
below will be referred to the standard representation.  Let  be the
polar angles.

(a) Write
down  in terms
of x,y, z components of  Write down
the matrix representation of  in the
standard base.

(b) Show
that given on page 4.3 are indeed the eigenstates of .

(c) If you
measure Sz for a system of particles in the pure state , what values will you get each time, and what is the
average value you will get?

(d) Suppose
that there are 106 particles in part (c) to start with, write down
the matrix representation of the state operator (or density operator) before
the measurement and after the measurement.

(e)  If 30% of the 106 particles
are in the  state, and
70% are in the  state.
What is the expectation value of Sz for such an ensemble. Use the
intuitive approach.

We will next
solve the same problem in (e) using state operator below.

(f) Write
down the state operator in the operator form.

(g) Express
the same operator in matrix form.

(h)
Calculate the expectation value of Sz and compare with your results
from (e).

(i)  Calculate the expectation value of Sy.

3.5.  Do the exercises shown on Pages 4.6,
4.7, 4.9.

Study
Aid—

·
Wikipedia has a good short introduction on
electron spin.  See

http://www.answers.com/topic/spin-quantum-number

Pick up a few phases while you are
there: Levi-Civita symbols;
Uhlenbeck and Goudsmit,
Ehrenfest, Kronig, Lande .
Bohr magneton. Skip the Dirac equation.

·
This link goes to a higher level which is not
needed for us at this point, but you might want to take a quick look  http://www.answers.com/topic/spin

·
You may want to learn a bit about isospin
while we are talking about spin. This part below is adopted from:  http://www.answers.com/isospin

Isospin was introduced by Werner
Heisenberg in 1932[1]
(although it was named by Eugene Wigner in 1937[2]) to
explain symmetries of the then newly discovered neutron:

The mass of the
neutron and the proton are almost identical: they are nearly degenerate,
and both are thus often called nucleons.
Although the proton has a positive charge, and the neutron is neutral, they
are almost identical in all other respects.

The
strength of the strong interaction between any pair of nucleons is the
same, independent of whether they are interacting as protons or as
neutrons.

Thus, isospin was introduced as a concept well before the development in the
1960s of the quark
model which provides our modern understanding.

The nucleons,
baryons of spin 1⁄2,
were grouped together because they both have nearly the same mass and interact
in nearly the same way. Thus, it was convenient to treat them as being
different states of the same particle. Since a spin 1⁄2
particle has two states, the two were said to be of isospin 1⁄2. The proton and
neutron were then associated with different isospin projections Iz = +1⁄2 and −1⁄2 respectively. When
constructing a physical theory of nuclear forces,
one could then simply assume that it does not depend on isospin.

These considerations would also prove useful in the analysis of meson-nucleon
interactions after the discovery of the pions in 1947. The
three pions (π+, π0, π−)
could be assigned to an isospin triplet with I = 1 and Iz = +1,
0 or −1. By assuming that isospin was conserved by nuclear interactions,
the new mesons were more easily accommodated by nuclear theory.

As further particles were discovered, they were assigned into isospin
multiplets according to the number of different charge states seen: a doublet I = 1⁄2 of K mesons,
a triplet I = 1 of Σ baryons, a single I = 0
Λ, four I = 3⁄2
Δ baryons, and so on. This multiplet structure was combined with strangeness in Murray
Gell-Mann's Eightfold way (physics), ultimately leading to the quark model
and quantum
chromodynamics.