3/PH/SB Quantum Theory - Dr. P.A. Mulheran - Week 3
THE POSTULATES OF QUANTUM MECHANICS - PART I
3.1 Eigenvalue Equations

The recipe book of quantum mechanics is traditionally written in the form of
postulates which describe how calculations are to be performed. Just as in any
physical theory the proof of these recipes lies in their successful prediction of
experimental results. Seven decades of man’s inventiveness has yet to contradict
the postulates of quantum mechanics.

We have noted that the important equations in quantum mechanics are in fact
eigenvalue equations; an operator operates on one of its eigenfunctions and
results in an eigenvalue times the same eigenfunction. For example the Time
Independent Schrodinger Equation is itself an eigenvalue equation. Other
operators whose eigenvalues we have encountered are the momentum operator

P  i , the total energy operator E tot  i , and the angular momentum
t
2
operators L and L z .

Recall that the operator for the x-component of momentum

Px  i
x
was introduced through its operation on the de Broglie wave function for a free
particle with definite momentum p x  k ;
u(x , t )  expi (kx  t  .


We now change our perspective on this situation, and ask the following
question; if the dynamical variable momentum is represented in quantum
mechanics by the operator Px , what are its eigenfunctions and eigenvalues? The
answer of course is that only the de Broglie waves satisfy the eigenvalue
equation of Px , and that the corresponding eigenvalues are p x  k .
Furthermore these eigenvalues are the quantities one measures in an experiment.
Thus the operator Px  i

itself defines its own eigenvalues and
x
eigenfunctions; it is sufficient to specify the form of the operator only, since
the rest follows from the mathematics.

We can view all the other important equations concerning dynamical quantities
in quantum mechanics from a similar perspective. Each dynamical variable is
represented by an operator whose eigenvalues are the possible results of
measurements of this dynamical variable. For example, measuring the energy of
a Hydrogen atom always results in one of the allowed energy levels, which
themselves are the eigenvalues of the Hamiltonian operator for the system (the
left hand side of the TISE).
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3/PH/SB Quantum Theory - Dr. P.A. Mulheran - Week 3
3.2 The First Three Postulates
P1
For every dynamical system there exists a single-valued, normalisable wave
function from which all possible predictions of the physical properties of the
system can be obtained.
P2
Every dynamical variable may be represented by a linear Hermitian operator
whose eigenvalue spectrum is the set of all possible results of a measurement
of that variable.
P3
The operators representing position and momentum of a particle are r and
i respectively for wave functions written as a function of position. The
operators for other dynamical variables can be found from the corresponding
classical relation between P and r (e.g. H  T  V used in the TISE).
3.3 Definition of a Hermitian Operator

The operator Q is Hermitian if and only if it satisfies the following relation:
 ( r ). d   g ( r ). Q * f ( r ). d
 f (r ). Qg
allspace
allspace
for any well-behaved (i.e. physically realistic) functions f(r) and g(r) that
vanish for | r|   .

N.B. The operator operates on the functions to its right, so in general
 ( r )  f ( r ). Q .
Qf

It is easily shown that the operators for position and momentum defined above
are Hermitian.
3.4 Properties of Hermitian Operators

The eigenvalues of Hermitian operators are real, which is why they are so
convenient to use in quantum mechanics; physical measurements produce real
answers!

The eigenfunctions of a Hermitian operator are orthogonal to one another. This
leads to another element of quantum theory, namely the Superposition
Principle, which provides the basis for the quantum theory of measurement.
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3/PH/SB Quantum Theory - Dr. P.A. Mulheran - Week 3
3.5 The Superposition Principle

We will assume that the eigenfunctions of Hermitian operators form complete
sets. This means that any well-behaved, physically reasonable wave function
 ( r ) can be written as a unique linear superposition of eigenfunctions u n ( r ) :
 (r )   a n un (r ) .
n

The coefficients (or amplitudes) in this expansion, a n , can easily be found by
utilising the orthogonality of the eigenfunctions:
a n   un * (r ). (r ). d ,
allspace
where we assume that the wave function  ( r ) and the eigenfunctions u n ( r ) are
all normalised.

  of a Hermitian operator Q can now be evaluated
The expectation value  Q
for a general wave function utilising the superposition principle. The expansion
can be made in the most convenient complete set of states, so in this case we
obviously use the eigenfunctions of this particular operator Q :
 (r )   a n un (r ) ,
n
where
 n (r )  q n un (r ) ,
Qu
so that
 Q   | a n |2 q n .
n

But postulate P2 states that the eigenvalues q n are the only possible results of
measurements of the dynamical variable represented by Q . Thus the
  , which is the average of a large number of
expectation value  Q
measurements, will be given by
 Q   Pn q n ,
n
where Pn is the probability of measuring the nth eigenvalue.

Thus we identify
Pn | a n |2
as the probability of a measurement of the dynamical variable represented by the
operator Q yielding the result q n , where the amplitude a n is calculated as
above using the wave function of the system.

This identification of | a n | as a probability is consistent with | (r )| being the
probability of measuring the particle at a certain position, and it is also
consistent with the sum of probabilities being unity since  (r ) is normalised.
2
2
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3/PH/SB Quantum Theory - Dr. P.A. Mulheran - Week 3
WORKSHOP QUESTIONS
Hand your solutions to the following questions to Dr. Mulheran at the start of the
next-but-one workshop. 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.
3.1
Write down the eigenvalue equation for the total energy operator

,
E tot  i
 t
and solve this equation for both the eigenfunctions and the eigenvalues.
Comment on how your solution relates to de Broglie waves for free particles,
and how it is used in transforming the Time Dependent Schrodinger Equation
into the Time Independent form.
[5 marks]
3.2
Find the eigenvalues and eigenfunctions of the momentum operator
P̂  i .
[5 marks]
Hint: Use Cartesian co-ordinates.
3.3
(a)
(b)
(c)
3.4
 . Is
Show that if the operator Q is Hermitian then so is the operator Q
the kinetic energy operator Hermitian?
[3 marks]
Show that any real function of position V(r) is Hermitian.
[1 mark]
Hence show that the Hamiltonian operator is Hermitian.
[1 mark]
2
A particle is in a one-dimensional infinite square well of width L. At some
moment in time its wave function is
 2
 3 x 
 4 x 
4
 
, 0  x  L,

sin 
sin 
u( x )   3L  L 
.
3L  L 

0 otherwise .

(a) Evaluate  H  and the uncertainty in the energy E using the integral
formulae given in week 2 of this course.
[3 marks]
(b) What are the possible results of measuring the particle’s energy, and
what are the probabilities of these results occurring? Do your answers
agree with those of part (a)?
[2 marks]
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3/PH/SB Quantum Theory - Dr. P.A. Mulheran - Week 3
WORKSHOP SOLUTIONS
3.1
i

v( t )  E . v( t )
 t
The solutions are that the eigenfunctions v( t )  exp( i t ) and the
eigenvalues E   .
This is the time-dependent part of the de Broglie wave exp i ( kx   t ) for


a free particle with definite momentum and energy. It is also the time
dependent part of the wave function for any particle with definite total energy
E   .
The Time Dependent Schrodinger Equation is

H  (r , t )  i
 (r , t ) ,
 t
where the spatial and time dependencies are separable. Thus the wave function
can be written as a product of an r-dependent function and a t-dependent
function,
 (r , t )  u(r ). v(t ) ,
 and v(t) is an eigenfunction of the total
where u(r) is an eigenfunction of H
energy operator as given above.
3.2
 



Pˆ   
i
j
k
2 
 y
 z 
 x
Thus the eigenvalue equation
Pˆ u  x, y , z   p.u  x, y , z 
is separable and the wave function can be expressed as a product of one
dimensional functions obeying
d
 i
u1 ( x )  p1u1 ( x ) ,
dx
and similar for the other co-ordinates. This last equation has the solution
u1 ( x )  exp(ik 1 x ) as its eigenfunctions and p1  k1 as its eigenvalues.
Thus the total solutions for the eigenvalue equation for the momentum operator
are the eigenfunctions
u( x , y , z)  u1 ( x ). u 2 ( y).u 3 ( z)  exp(ik 1 x ). exp(ik 2 y). exp(ik 3 z)
 expi ( k 1 x  k 2 y  k 3 z)  exp(ik . r )
and eigenvalues


p   k1 i  k 2 j  k 3 k   K .
3.3
(a) Put

f (r ). Q . Q . g(r ). d 
allspace
 f (r ). Q . h(r ). d
allspace
 (r ) , i.e. the function resulting from Q operating on g(r).
where h(r )  Qg
Now use the definition of the Hermitian operator on the right hand side:
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3/PH/SB Quantum Theory - Dr. P.A. Mulheran - Week 3
 (r ). d   h(r ). Q * f (r ). d   h(r ). i (r ). d ,
 f (r ). Qh
 * operating on f(r). Changing the
where the function i(r) is the result of Q
order in which the product of functions is written in the right hand integral we
find
 (r ). d   g(r ). Q * i (r ). d
 i(r ). h(r ). d   i(r ). Qg
where the definition of a Hermitian operator has been used again. Finally
writing out i(r) in full, we have shown that
  (r ). d   g(r ). Q * Q * f (r ). d ,
 f (r ). QQg
allspace
allspace
   Q is also Hermitian.
i.e. that QQ
2
We know Px is Hermitian, hence so is Px . Similarly for the y and z
components of the kinetic energy operator.
(b) The product of three functions is independent of the order in which they are
written, thus
2
 f (r ).V (r ). g (r ). d

allspace
 g (r ).V (r ). f (r ). d
allspace

 g (r ).V * (r ). f (r ). d
allspace
for any real function of position V(r).
  T  V (r ) clearly Hermitian from above.
(c) H
3.4
(a) u( x ) 
1
3
u3 ( x) 
2
u4 ( x)
3
   2 d 2 u( x ) 
 Hˆ   u * ( x ). 
.dx
2
 2  dx 
0
L
E 3 2 E 4  2 2  3 2 2.4 2  41 2 2


  3 
3
3
2 L2  3
6L2

since the eigenfunctions u1 and u2 are orthogonal. Similarly
E 2 2 E 42
 4 4  34 2.4 4  593 4 4
.
 Hˆ 2  3 

  3 
2 4
3
3
4 2 L4  3
12

L

Thus
 2 2 593 41
 2 2
E   Hˆ 2    Hˆ  2 


6
.
53
.
12
6
L2
L2
(b) Individual measurements result in energy E 3 with probability 1/3, and

energy E 4 with probability 2/3, using the fact that u(x) is a linear superposition
1
of the eigenfunctions of H with amplitudes a 1 
and a 2 
3
6
2
.
3