Download Fulltext PDF

Classroom
In this section of Resonance, we invite readers to pose questions likely to be raised
in a classroom situation. We may suggest strategies for dealing with them, or invite
responses, or both. "Classroom" is equally a forum for raising broader ·issues and
sharing personal experiences and viewpoints on matters related to teaching and
learning science.
Kovid Goyal
Department of Physics
Matrix Magic: Spin Half Systems
St. Xavier's College
Mahapalika Marg
Mumbai 400001, India.
Keywords.
Quantum mechanics .
Figure 1. The Stern-Gerlach filter for spin one-half
particles.
A beam of spin half particles entering a Stern-Gerlach
(S-G )filter gets split into two beams. The upper with
spin + 1/2 and the lower with spin -1/2. Either of the
two beams can be blocked, so that the device acts as
a filter and the output contains particles of only one
type. The arrow in the upper right corner indicates
the direction of increasing magnetic field gradient and
defines an axis for the device.
The problem is to carry out a quantum mechanical analysis of this device. Specifically, to calculate what the
probabilities are for a particle in a known initial state
to be in the upper or lower beam, and how these probabilities change when the device is rotated.
The Solution
The device can be analysed quantum mechanically, by
considering a single particle at a time. First, we identify
(
y
~
-76-----------------------------~-----------R-ES-O-N-A-N-C-E--I-F-eb-r-ua-r-y-2-0-0-3
CLASSROOM
each spin state (+ or -) with a vector in a Hilbert space.
Then, the device with one of the beams blocked, is a
projective Von Neumann measurement on the Hilbert
space. We further recognise that the Hilbert space is of
dimension two since a particular particle can be in at
most two mutually orthogonal spin states.
The Feynman lectures, Vol. 3,
Ch.6.
This identification of a single spin half particle as the
quantum system of interest, with the device as a projective measurement immediately leads to the conclusion
that every Stern-Gerlach filter defines its own basis for
the Hilbert space of the particle. Two different S-G filters can differ only in their orientations with respect to
some fixed co-ordinate system. Thus, a rotation of one
filter into another corresponds to a unitary transformation on the Hilbert space of the particle.
This greatly clarifies the problem. The question of finding the probability of a particle that exits from one filter
being in the upper or lower beam of the next simply reduces to finding a unitary transformation (a rotation)
that relates the two filters. Indeed, this is so simple
that it can be accomplished for spin half particles simply
from the properties of space and the axioms of quantum
mechanics 1 .
Let 1+ Z) or 1- Z) denote the spin states of a particle
that has passed through the upper or lower beam respectively of a S-G filter with its axis along the Z-direction.
Then, for a rotation R( B) about the X -axis:
I+Z')
cos ~I+Z) -
I-Z')
-iSin~I+Z) + cos~I-Z),
2
isin~I-Z)
2
(1)
(2)
where Z' is obtained by rotating Z by B about the Xaxis.
-E-S-O-N-A-N-C-E-I-F-e-br-Ua-r-Y-2-0-0-3---------'~-----------------------------n
R
CLASSROOM
For a rotation R( 8) about the Y-axis:
I+Z')
I-Z')
cos ~I+Z)
8
+ sin~I-Z)
8
- sin 21+Z ) + cos 21-Z),
(3)
(4)
where Z' is obtained by rotating Z by 8 about the Yaxis.
For a rotation R( 8) about the Z-axis:
I+Z')
I-Z')
e- iOj2 1+Z)
eiOj2 1_Z),
(5)
(6)
where Z' is obtained by rotating Z by 8 about the Zaxis.
Thus, we have obtained the necessary transformation
laws that govern the rotation of Stern-Gerlach filters
for spin half particles. However, these laws are still in
abstract form and cannot be used conveniently for actual
calculation. In order to perform actual calculations, we
must select a particular representation for the Hilbert
space of a spin half particle.
A convenient choice is the space of 2 x 1 column matrices
over the field C (Complex numbers). Then we make the
convenient identification:
I+Z)
(7)
I-Z)
(8)
Substituting these into the transformation laws, with a
little algebra the following matrix relations can be obtained:
I+Z')
I-Z')
R(8)I+Z)
R(8)I-Z),
(9)
(10)
--------~-------RESONANCE I February 2003
78
CLASSROOM
where the kets are now column matrices and R(B} is a
2 x 2 unitary matrix that represents a rotation of the
filter by B about a particular axis.
The following unitary matrices were obtained:
R(B z ) = (
e-iO/2
0
R(B ) = (COSB/2
-i sin B/2
x
R( B ) =
0)
eifJ/2
(11)
-isinB/2)
cos B/2
(12)
(COS B/2
- sin B/2)
(13)
sin B/2 cos B/2
where R( Bi ) denotes a rotation of B about the i axis.
Y
To demonstrate the power of this formulation, a quick
example:
Suppose the output from a S-G filter A in which the
lower beam is blocked is fed into another filter B, which
is different from A by a rotation of a about the X -axis.
We wish to calculate the probability that the particle
will emerge in the upper beam of B. Then,
= [(
(+BI+A) = (R(ax) + AI+A)
(14)
_~oS::~~2 -~:~:/~2) x (~)] t x (~)
(15)
(cos cr/2 i sincr/2) x (~)
(16)
=
= cos a/2.
(17)
Thus the required probability is cos 2 a. This method
can be easily extended. For compound rotations, simply
multiply the corresponding elementary rotation matrices. For a series of n filters, multiply the corresponding
probability amplitudes from last to first.
This illustrates the power of formulating a problem precisely. It also demonstrates the great utility of the matrix method of doing quantum mechanics.
-Fe-b-rU-ar-Y-2-0-0-3--------~--------------------------7-9
-RE-S-O-NA-N-C-E--'