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3/PH/SB Quantum Theory - Week 2 - Dr. PA Mulheran
EXPECTATION VALUE AND UNCERTAINTY
2.1 Introduction

You may be aware of the special role eigenvalue equations play in quantum
mechanics. Many of the important equations you have seen have been in the
form of
Q  n  q n n ,
where Q is an operator (e.g. the Hamiltonian on the left hand side of
Schrodinger’s equations), and qn and n respectively are the eigenvalues and
eigenfunctions of this operator labelled by quantum number(s) n(lm...). Then a
measurement of the quantity represented by the operator Q on the system
whose state is represented by n will definitely result in the answer qn.

What happens if the wave function of the system is not an eigenfunction of the
operator Q ? In this case the answer to the measurement will be one of Q ’s
eigenvalues, but it is not certain which one it will be! There is however a well
tested mathematical procedure for calculating the probability of each possible
answer occurring as we shall see later in the course.

Thus in quantum mechanics we typically calculate the probabilities of results of
measurements. You are already familiar with this idea through the interpretation
of the wave function, where * gives the probability of finding the particle at a
given position. This restriction to probability rather than certainty extends to all
physical measurements, not just position. The only exceptions are when the
wave function representing the state of a system happens to be an eigenfunction
of the operator representing the physical quantity, and in these cases the outcome
of the measurement will be known with probability one.

It is convenient to characterise the distribution of possible results of the
measurements. The first useful statistical quantity is the average value of the
result obtained from a large number of repeated measurements on identical
states of the system. In the quantum mechanics this is called the expectation
 .
value of the operator, and is denoted by  Q

The second statistical characteristic is the standard deviation of the measured
values. It is called the uncertainty in the result q in quantum mechanics and is
denoted by q . This is the quantity that is used in Heisenberg’s uncertainty
relations.
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3/PH/SB Quantum Theory - Week 2 - Dr. PA Mulheran
2.2 Expectation Values

Expectation value for position measurement:
Since the probability of find the particle between x and x+dx is given by
u*(x)u(x).dx, where the spatial wave function has been normalised,

 x   u * ( x ). x. u ( x ). dx

This way of writing the product of u*.u.x = u*.x.u is consistent with the general
formula described below.

Similarly for any function of position, such as potential energy, we have

 V   u * ( x ).V ( x ). u ( x ). dx


To find a suitable expression for other operators which are not simple functions
of position but may involve differentiation, we take guidance from the TISE:
 ( x )  V ( x ). u ( x )  E . u ( x ) ,
Tu
 2 d 2

where T 
is the kinetic energy operator.
2 dx 2
Multiplying by u*(x) and integrating over all space we find

 ( x ). dx   V  E .
 u * (x ). Tu

Now classically the total energy E = T + V, the sum of kinetic and potential
energies. We certainly expect to find the same result from the averages taken
over a large number of measurements on identical systems,
E  T    V  .
We thus identify the expectation value of the kinetic energy operator to be given
by

 ( x ). dx .
 T   u * ( x ). Tu

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3/PH/SB Quantum Theory - Week 2 - Dr. PA Mulheran

We now suppose that there is nothing unique about the kinetic energy operator,
so that the formula for all expectation values is

 ( x ). dx .
 Q   u * ( x ). Qu

N.B. The operator Q operates on the wave function(s) to its right!

Whilst we have looked at the one dimensional case here, generalisation to higher
dimensions is obvious.

  as the average of a large
It is important to remember the definition of  Q
number of measurements taken on identical systems represented by the wave
function u(x). This is not the same as the most likely result of a single
measurement; statistically the mean is not necessarily equal to the mode!
2.3 Uncertainty

The standard deviation in a set of results is a measure of how uncertain we
would be about the value of a single measurement. In statistics it is derived as
the root-mean-square deviation from the average value,
q 

(q   q  ) 2   q 2    q  2 .
This same expression is adopted in quantum mechanics, where the expectation
value formula derived above is used to calculated the uncertainty in the result of
a measurement of a physical quantity represented by the operator Q ,
q   Q 2    Q  2 .

Using these definitions of uncertainty, Heisenberg’s Uncertainty Principle for
position and momentum reads
x. p x 
1
.
2
In some texts the right-hand-side may be quoted differently, although it will
always involve Planck’s constant. This means that they are using slightly
different definitions of uncertainty to ours and is not a cause for concern. The
principle is the same, that there is limit to how well conjugate variables such as
the x-component of position and the x-component of momentum can be known
simultaneously; more on this later.
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3/PH/SB Quantum Theory - Week 2 - Dr. PA Mulheran
WORKSHOP QUESTIONS
Hand your solutions to the following questions to Dr. Mulheran at the start of the
second workshop in week 3. 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.
2.1
The normalised ground state wave function for the one dimensional harmonic
oscillator is
  
  2 
u0 ( x )  
x ,
 exp 
  
 2 
where as usual  is the particle’s mass and  is the classical angular frequency
of the oscillator.
(a) Calculate the expectation values for position and momentum,  x  and
 Px  , with this wave function and comment on your results.
[2 marks]
Hint: think about the symmetry of the functions you must integrate!
(b) Calculate the expectation values  x 2  and  P 2  .
x
[3 marks]
Hint: the following standard integral will be of help:
1/ 4

x
2
exp(x 2 )dx 

(c)
(d)
2.2

.
2 3/ 2
Hence calculate  T  and  V  , where T is the kinetic energy
operator, and show that the sum of these equals the total energy of the
ground state as required.
[3 marks]
Using the results from (a) and (b), now calculate the uncertainties in the
particle’s position and momentum. Is Heisenberg’s Uncertainty Principle
obeyed?
[2 marks]
A bead which slides freely around a wire hoop (as in question 1.2) has the
normalised wave function
u( ) 
1

sin( ) .
The angular momentum operator for this situation (i.e. one with cylindrical
symmetry) is

L z  i .

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3/PH/SB Quantum Theory - Week 2 - Dr. PA Mulheran
(a)
(b)
(c)
Express u() as a linear superposition of the eigenfunctions of L̂ z .
[2 marks]
2


Calculate  Lz  ,  Lz  and l z for the bead and comment on your
results.
[3 marks]
Hint: the integrals used in the expectation values run from 0 to 2 in this
case.
Teacher’s Pet question:
Bonus marks are available for anyone able to show why the angular
momentum operator for cylindrical systems is as given above. See Dr.
Mulheran for hints!
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3/PH/SB Quantum Theory - Week 2 - Dr. PA Mulheran
WORKSHOP SOLUTIONS
2.1
(a)
u(x )  A exp( x 2 )
 x 

 A exp(
x 2 ). x. A exp( x 2 ) dx  0
since
the

integrand is an odd function of x and we integrate over all x. Similarly
 Px 


 A exp(

 A exp(


d


x 2 ). i . A exp( x 2 ). dx
dx


x 2 ).i 2 xA exp( x 2 ). dx  0
These results are obviously true, since the wave function is evenly spread
about the origin, and the particle is equally likely to be found moving the
right as it is moving to the left.
(b)

 A exp(
 x 
2
x 2 ). x 2 . A exp( x 2 ) dx



 8 3

 A 5/ 2 3/ 2 
.

2 
 32 3 3 2
2
u’=-2ax.u, u’’=-2a.u+4a2x2u,
 Px2    2 ( 2  4 2  x 2 )
4 2 2 

    .
.

2
4
2 
2
1

 T 
 Px2 
2
4
1

 V   2  x 2 
2
4

Clearly  T    V 
which is the ground state energy.
2

x   x 2    x  2 
 0,
2
2
(c)
(d)

p x   Px2    Px  2 
 0.
2
Thus

x. p x  , so Heisenberg’s Uncertainty Principle is obeyed. In fact
2
the ground state wave function of the harmonic oscillator, which is a
Gaussian function, is the minimum uncertainty wave packet possible, and
the equality in the HUP is not normally found even for ground state wave
functions in general systems.
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3/PH/SB Quantum Theory - Week 2 - Dr. PA Mulheran
2.2
(a) The eigenfunctions of L̂ z are
u  
(b)
1
2i 2
1
2
exp im  with integer m. Thus
exp i   exp  i .


1 2
d
i 2
 L z   sin( ). i
sin( )d 
0 sin( ).cos( )d  0
 0
d




2
2
 2 d

1
 2 2 2
2

 Lz   sin( ).  
sin( )d 
sin ( )d   2

2
 0
d
 0


lz  
Using Euler, we see that the wave function is a linear superposition of two
states with angular momentum . Thus the average of a large number of
measurements is zero, and the uncertainty is .
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