3/PH/SB Quantum Theory - Dr. P.A. Mulheran - Week 4
THE POSTULATES OF QUANTUM MECHANICS - PART II
4.1 The Measurement Postulate

It is clear that if the wave function of a system is an eigenfunction u n of a
Hermitian operator Q , then a measurement of the dynamical quantity
represented by Q will yield the corresponding eigenvalue q n with probability
one.

Conversely it is reasonable to expect that if the measurement has resulted in one
particular eigenvalue q n , then the wave function of the system following the
measurement must be the corresponding eigenfunction u n . This expectation is
confirmed in experiments, some of which will be discussed later in this course.

However this means that in general measurements are not passive and
profoundly disturb the state of a system. If the initial wave function of a system
is described as a linear superposition of the eigenfunctions before the
measurement, after the measurement it has been “reduced” or “collapsed” to one
eigenfunction (assuming that we have performed a perfect ‘noise-free’
experiment and found a definite value for the measured quantity).
measure result
q
n u
  
n

This collapse of the wave function is the source of most of the interpretational
problems associated with quantum theory, as will be discussed later in the
course. However most physicists are pragmatists and accept the fact that this rule
works perfectly well in describing the behaviour seen in experiments and they do
not worry too much about what it means!
P4
The probability of finding the result q n when measuring the dynamical quantity
represented by the Hermitian operator Q is | a n |
function can be expressed as
2
where the initial wave
   a n u n , which is a linear superposition of
n
the eigenfunctions u n of Q . Upon finding the result q n the wave function is
reduced to the corresponding eigenfunction u n .
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3/PH/SB Quantum Theory - Dr. P.A. Mulheran - Week 4
4.2 Links with classical mechanics

The development over time of a linear superposition of bound states in a well
often helps us see the link to the classical description. For example, consider a
particle in a simple harmonic oscillator with the wave function
  x, t  
 1
 1
 3

u 0  x  exp   i  c t  
u1  x  exp   i  c t 
2
2
 2

 2

1
The probability   x, t  of finding the particle at position x in the well swings
2
from side to side; in fact <x> oscillates back and forth with the classical
frequency of the harmonic oscillator, as you will be asked to show in the
workshops.

A similar consideration of a linear superposition of states for a Hydrogen atom
shows how an oscillating dipole can arise in this system. This helps to explain
how light of the correct frequency can excite an atom from its ground state to
certain excited states.

This form of a linear superposition that includes the time-dependence, generally
written as
 E 
 r, t    a n u n r  exp   i n t  ,
 

n
where Hˆ u n r   E n un r  , is justified since this wave function satisfies the Time
Dependent Schrodinger Equation.

Amplitudes can be found at any arbitrary time and interpreted in the usual way
as measurement probabilities for the eigenvalues of the Hamiltonian (i.e. the
energy levels of the system). These probabilities do not change over time
 E 
because the time dependent factors exp   i n t  have modulus one. This
 

shows that energy is conserved in these systems.

Another link with classical Newtonian mechanics can be found by considering
the rate of change of the expectation value of momentum in quantum mechanics:
 Pˆx

t
V
x
(This result is an example of the application of the Ehrenfest Theorem). In
particular it shows how F=ma is recovered from the quantum mechanical
equations when the spatial extent of a wave function is much less than the scale
on which the potential energy varies. Thus it appears that quantum mechanics
encompasses classical mechanics, so that people believe the quantum theory is
the true theory that applies to everything from electrons to footballs and beyond!
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3/PH/SB Quantum Theory - Dr. P.A. Mulheran - Week 4
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.
4.1
A particle is confined to the region [0 < x < L] by walls of infinite potential. It
is in the ground state of this well when the wall at x = L is instantaneously
moved to a new position at x = 2L. By assuming that the wave function
remains unchanged by the movement of the wall, show that the probability of
measuring the particle’s energy to be that of the ground state of the wider well
is approximately 0.36.
[5 marks]
Hint: Express the particle’s wave function as a linear superposition of states of
the wider well.
4.2
Consider a harmonic oscillator with the time-dependent wave function
1
 1
 1
 3

  x, t  
u 0  x  exp   i  c t  
u1  x  exp   i  c t  ,
2
2
 2

 2

where u0(x) and u1(x) are the n=0 and n=1 eigenfunctions of the Hamiltonian
and  c is the classical oscillator frequency.
(a)
Show that the expectation value of the oscillator’s position varies with
time as

 x 
cos( c t ) .
2 c
[3 marks]
(b)
Compare this to the classical motion of the oscillator when it has total
energy given by <E> for the above wave function.
[2 marks]
Hint: the normalised eigenfunctions are
  c
u 0  x   




1
4
  c 2 
exp  
x 
 2

and
1
 4  4   c 
u1  x     

    
3
3
4
  c 2 
x exp  
x .
 2

3/PH/SB Quantum Theory - Dr. P.A. Mulheran - Week 4
WORKSHOP SOLUTIONS
4.1
The wave function is
 2
 x 
, 0  x  L

sin 
 ( x )   L  L 

0 otherwise.

The ground state wave function of the new well is
 1
 x 
, 0  x  2 L

sin 
u1 ( x )   L  2 L 

0
otherwise.

Expressing  (x) as a linear superposition of the energy eigenfunctions of the
new well, the amplitude for the first state is given by
L
2  x   x 
. sin 
.dx
a1   u1 * ( x ). ( x ).dx  
sin 
L
 L   2L 
allspace
0
L
  x 
 2   3 x 



 dx

cos

cos

2 L 0   2 L 
 2 L 
L
 1  2 L  3 x  2 L   x 
 


sin 
 sin 
L 2  3
 2L  
 2 L  0
4 2
.
3
Thus probability of measuring the ground state energy is
32
| a1 |2 
 0.36 .
9 2

4.2a
  x, t  
2


1 2
u 0  x   u12  x   2u 0  x u1  x  cos c t 
2

x 
 x
2
dx





 0  0  cos c t   xu0  x u1  x dx  cos c t  

2  c
u12  x dx
which is the answer given since u1(x) is normalised.
4.2b
1 1
3

 c   c    c .

2 2
2

Classical amplitude A when total energy is all potential,
1
E   c2 A 2
2
2
Thus A 
which is twice the amplitude of <x>.
E 
 c
4