3/PH/SB Quantum Theory - Week 6 - Dr. P.A. Mulheran
SPIN AND ITS REPRESENTATION
6.1 Matrix Mechanics

Heisenberg’s theory of quantum mechanics, which he discovered in 1925,
focussed on the use of matrices as representations of dynamical variables.
Although complicated to use this method is just as successful as Schrodinger’s
wave mechanics. In fact Pauli derived the energy level spectrum of Hydrogen
atoms in the same year as Schrodinger (1926) using the matrix approach.
Schrodinger himself then showed how the two approaches are entirely
equivalent from a mathematical viewpoint.

Consider an eigenfunction of some Hermitian operator:
Q  (r )  q. (r )
Now expand the wave function in any complete basis set of orthonormal
functions such as the eigenfunctions of some other Hermitian operator used in
quantum theory:
 (r )   a n un (r )
n
so that
a
n
Q u n ( r )  q  a n u n ( r ) .
n
n
Multiplying by u m * ( r ) and integrating over all space we then find
 Qmn a n  q a m
n
where the numbers
Qmn 
u
m
* (r ). Q un (r ). d
all space
are called matrix elements for the obvious reason given below.

This equation is true for all m, so that it can conveniently be expressed in terms
of a matrix equation
Qa  q a
where the elements of the matrix Q are the numbers Qmn and the elements of the
column vector a are the amplitudes an.

This matrix equation is an eigenvalue equation, and the eigenvalues q of the
matrix Q are the same as the eigenvalues of the operator Q . Furthermore the
eigenvectors a of Q are sufficient to provide full knowledge of the wave
function, and can themselves be used to describe the state of the system.

From the definition of the matrix elements it is clear the matrix Q is Hermitian,
and all the results that we have derived for the eigenfunctions of Hermitian
operators apply for the eigenvectors of the Hermitian matrices. Thus the
postulates of quantum theory P1-P4 can be equally well expressed in terms of
Hermitian matrices and their eigenvectors, with the same agreement with the
results of actual measurements.
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3/PH/SB Quantum Theory - Week 6 - Dr. P.A. Mulheran
6.2 Pauli Spin Matrices

In 1921, before Schrodinger and Heisenberg developed their theories of quantum
mechanics, Stern and Gerlach had discovered that the magnetic moment of
electrons is one-half a Bohr magneton, so that the measured z-component of
angular momentum is 
1
 . They found this by passing a beam of silver atoms
2
through an inhomogeneous magnetic field, and observing that the beam splits
into two paths.

It was initially thought that the electron possesses a magnetic moment because it
spins about its axis (rather like the Earth does) thus creating a circulating charge.
However we now know that this model is wrong and that the electron “spin” is
not a simple spatial rotation. Indeed we have seen that the z-component of
orbital angular momentum is quantised as m , where m must be an integer to
keep the wave function exp im  single-valued. Thus the value m 
result from a spatial wave function for the electron’s spin.
1
cannot
2

If a spatial wave function does not exist we clearly cannot use the angular
momentum operators L x etc. to represent spin. However we can use matrices
instead to represent all dynamical variables including angular momentum, and
this does not restrict us to spatial rotations.

The measurement of spin results in two possible values, so we need to use 2x2
matrices to represent spin since these will have two eigenvalues. Indeed the
matrix
sz 
has eigenvalues 
1 1 0 


2  0 1
1
 and so is a suitable matrix to represent the z-component
2
of spin. It has the normalised eigenvectors
 1
az   
 0
 0
bz   
 1
which are the state-vectors when the spin has been measured to be ‘up’ or
‘down’ respectively in the z-direction. These eigenvectors are orthogonal to one
another
a z b z  0
as we expect for the Hermitian matrix sz.
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3/PH/SB Quantum Theory - Week 6 - Dr. P.A. Mulheran

We also need matrices to represent the x and y components of spin, which also
must have eigenvalues of 
1
 . Furthermore experiment shows that the
2
components of spin are incompatible, i.e. that only one component can be
specified at a time, just as with orbital angular momentum. Hence the matrices
for the components of spin must obey the same cyclic commutation relations as
the operators for L x , L y and L z .

Pauli found the matrices that satisfy these demands:
sx 
1  0 1


2  1 0
For example the commutator
s
x
sy 
1  0 i


2  i 0

, s y  s x s y  s y s x  is z
as required.

The normalised eigenvectors of sx are
1 1
1  1
ax 
bx 
 
 
2 1
2   1
which are the state-vectors when the spin has been measured to be ‘up’ or
‘down’ respectively in the x-direction. Again these eigenvectors are orthogonal
to one another ( a x b x  0 ).
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3/PH/SB Quantum Theory - Week 6 - Dr. P.A. Mulheran
6.3 Multiple Spin Measurements

Consider the measurement of the z-component of spin by passing a beam of
electrons along the y-axis to a suitably oriented Stern-Gerlach apparatus,
denoted SGZ. The state vector for the electrons that emerge in the ‘up’ beam is
 1
az   
 0
by the measurement postulate P4.

This beam is now passed through another Stern-Gerlach apparatus SGX oriented
to measure the x-component. We calculate the probability of the spin being
measured ‘up’ or ‘down’ the x-axis by using the superposition principle, where
the state vectors replace the wave functions:
 1 1  1 1  1 
      
 0 2  1 2  1
az 
1
1
ax 
bx
2
2

Hence by the measurement postulate P4 the probability of measuring a particular
eigenvalue is given by the square of the modulus of the corresponding
coefficient in the superposition. Thus the probabilities of measuring the spin
‘up’ or ‘down’ the x-axis is one half for each outcome. This is what we expect,
since there is no reason why ‘x-up’ should be preferred to ‘x-down’ given we
started with ‘z-up’.

It is tempting to think that each individual electron as it passes down the
apparatus already knows whether it is ‘up’ or ‘down’ before the measurement
takes place. However this is not the case. If one of the beams leaving the SGX
apparatus, say the ‘x-up’ beam, is fed into another SGZ apparatus then again
only half the electrons on average will come through the ‘z-up’ channel, despite
the fact that they all emerged through the ‘z-up’ of the first SGZ experiment.
This means that the intermediate measurement of the x-component of the spin
destroyed the previous knowledge of the z-component, and the electrons
approach the third apparatus with the state vector ax. There is no ‘hiddenvariable’ that records the state of the z-component of spin of each electron!
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3/PH/SB Quantum Theory - Week 6 - Dr. P.A. Mulheran
WORKSHOP QUESTIONS
Hand your solutions to the following questions to Dr. Mulheran at the start of the
second workshop in week 7. Some of your solutions will be marked as part of the
continuous assessment of this course which contributes 20% of the overall module
grade. Your solutions must be well presented; untidy work will be penalised.
6.1
(a)
Show that the Hermitian adjoint of a product of matrices is equal to the
product of their adjoints in reverse order:
Q a
(b)

 a  Q .
[2 marks]
Starting from the eigenvalue equation for the Hermitian matrix Q:
Qa  q a
now show that
aQ  aQ  q * a
[1 mark]
(c)
(d)
6.2
Hence show that the eigenvalues of a Hermitian matrix are real;
[1 mark]
and that the eigenvectors of a Hermitian matrix are orthogonal to one
another.
[1 mark]
(a) Find the eigenvalues and normalised eigenvectors (ay and by) of the Pauli
spin matrix
sy 
1  0 i

.
2  i 0
[3 marks]
(b) Show that the matrix sy is Hermitian.
[1 mark]
(c) Show that the eigenvectors ay and by are orthogonal to each other.
[1 mark]
6.3
Express ay as a linear superposition of ax and bx , utilising the orthogonality of
the eigenvectors of a Hermitian matrix. What interpretation can you place on
your result?
[5 marks]
6.4
(a) Show that the matrix representing the square of the total spin is
s 2  s 2x  s 2y  s 2z 
3 2  1 0
 
.
4  0 1
[2 marks]
(b) What are the eigenvalues of s2, and does this result agree with the general
formula  2 ss  1 ?
[1 mark]
(c) Does this matrix commute with the matrices sx, sy and sz? What is the
significance of this result?
[2 marks]
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3/PH/SB Quantum Theory - Week 6 - Dr. P.A. Mulheran
WORKSHOP SOLUTIONS
6.1
(a)
The matrix multiplication is marching the rows down the columns:
c  Qa  cij   Qik a kj .
k
Taking the Hermitian adjoint (transposed complex conjugate) we find
c ji *   Q jk * a ki *   a ik Qkj  c   a  Q 
k
k
as required.
Qa  qa
(b)
Qa  
 q * a
a Q  a Q  q * a 
(c)
(d)
6.2
Qa  qa

aQ a  q aa
aQ  q * a

aQ a  q * aa
The left hand sides are equal, so that the complex conjugate of the
eigenvalue q equals itself and so is real.
Introduce a second eigenvector b with eigenvalue q’:
Qb  q  b
+
Then multiplying with a from the left:
a  Qb  q  a  b .
But multiplying the result from (b) by b from the right we find:
a  Qb  q * a  b  q a  b .
The left hand sides are again equal, so unless q=q’ and the eigenvectors
are degenerate, the right hand sides must be zero and a+b=0.
First find the eigenvalues through the secular equation:
 i
 0  2  1  0    1 .
i

1
.
2
For the eigenvector corresponding to   1 :
 0  i  c 
 c

    1.    c   id
 i 0   d
 d
So the eigenvalues of sy are 
 ay 
1 1
 
2  i
where the eigenvector has been normalised ( a y a y  1 ).
For the eigenvector corresponding to   1 :
 0  i  c 
 c
1  1

    1.     c   id  b y 
 .
 i 0   d
 d
2   i
Check the orthogonality:
 1 1
1
a y b y  1  i   1  1  0 .
2
  i 2
The matrix sy is indeed Hermitian:
i *
1  0
1  0  i
s y  
  
  sy
2  i * 0 
2 i 0 
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3/PH/SB Quantum Theory - Week 6 - Dr. P.A. Mulheran
a y  c1a x  c 2 b x
6.3
where the coefficients
c1  a x a y c1 
1
1
1 1   1  i
2
2
i 
and
c 2  b x a y c 2 
1
1
1  1   1  i .
2
2
i 
Check:
1  1 1  i 1  1  1  i 1  1 
.
.
  
  
  .
2
2
2 i 
2  1
2   1
Interpretation: if a measurement is performed to find the x-component of spin
on an electron prepared in the ‘y-up’ state, then the probability of finding ‘x2 1
2
2 1
2
up’ is c1   and the probability of finding 'x-down' is c 2   .
4 2
4 2
a y  c1a x  c 2 b x 
6.4
sx2 
2
4
 0 1  0 1  2



 1 0  1 0 4
 1 0


 0 1
2
s 
4
 0  i  0  i  2



 i 0  i 0  4
 1 0


 0 1
2
4
1 0 1 0  2



 0  1  0  1 4
 1 0


 0 1
2
y
s x2 
Thus the sum of these is as given (s=1/2 in the formula). The unit matrix
commutes with all other matrices of the same shape. This means that the total
square spin is compatible with any one component of spin.
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