Lesson Module 4
I.
Heisenberg Uncertainty Principle
A.
Fourier Transform
A common technique for solving differential equations is the Fourier Transform
that converts the solution of a difficult differential equation into an easier to
solve algebraic equation.
The Fourier transform F(k) of a function f(x) is found by
1 
F(k) 
f(x) e i k x dx

2π 
and the original function can be obtained using the inverse Fourier transform
f(x) 
1 
F(k) e - i k x dk

2π 
In addition to the advantage of being able to convert calculus based equations
(differential equations) into easier to solve algebra equations, Fourier transforms
also sometimes are more aligned with the way experimental data can be obtained.
For instance, the frequency response of an electrical circuit is more easily
obtained than its time response.
B.
Uncertainty with the Fourier Transform
Any two variables related by the Fourier transform (example: x and k) obey the
uncertainty relationship
Δx Δk 
1
2
This relationship is due to the fact that the Fourier transform is really equivalent
to an expansion of a function f(x) in terms of a series of sinusoidal waves with
different wave numbers. If f(x) is a single sine wave that starts at negative infinity
and continues to positive infinity then only a single wave is needed. In the case,
the wave number is known with absolute certainty, but the position is totally
uncertain. If we try and restrict the domain of the sinusoid then additional waves
of different wave numbers must be added to destructively interfere in the regions
we wish to exclude. In this case, the uncertainty in the domain is decreased but
the uncertainty in the wave number is increased. This uncertainty relationship was
has important applications in many fields of engineering including electrical
engineering. Since time and frequency are variables related by a Fourier
transform, using a band pass or other type of frequency filter to reduce noise in
your signals will always lead to reduced ability to distinguish between signals
arriving at different times. If you want to design circuits with excellent time
resolution then you must have a very wide pass band for your electronics. Since
all electronic components have built in capacitance and inductance, you have
inherent filters in your designs. Thus, proper selection of component materials is
essential in such circuits. This is uncertainty relationship is called time-frequency
reciprocity by electrical engineers doing filter design.
C.
Quantum Mechanics and Uncertainty
The uncertainty relationship between variables related by Fourier transforms was
well known by Physicists long before quantum mechanics. However, an
uncertainty in the wave number of a baseball had no physical meaning in classical
mechanics. deBroglie's duality relationship however gives the uncertainty
relationship important consequences in quantum mechanics.
1.
Linear Momentum and Position
We start by taking the uncertainty relationship for x and k and multiplying both
sides by h-bar.
 Δk Δx 

2
Since h-bar is a constant, we notice that
Δ p  Δ  k   Δk
Substituting this into our previous equation, we obtain the Heisenberg Uncertainty
relationship for 1-dimension that
Δp Δx 

2
This relationship says that you can not simultaneously know both the linear
momentum and the position of an object at the same time. A good way to think
about the uncertainty principle is to realize that measuring momentum is to
determine the wave number. Thus, you experiment will show the wave properties
of the object but not its particle nature. If you attempt to measure position then
you are measuring the object's particle nature but not its wave nature. Object's
have duality (wave and particle aspects but not simultaneously). It is
impossible to devise an experiment that simultaneously demonstrates the wave
and particle nature at the same time. You might think that you could first measure
the particle nature of the object and then measure its wave properties. However,
the uncertainty principle says that the very act of measuring the wave nature
destroys the object's particle information and vice versa. This was a stunning blow
to the concept of determinism that drove classical physics, but experimental
results continue to confirm the uncertainty principle and demonstrate the error of
determinism. We will see shortly that the momentum operator and the position
operator do not share a common set of eigenfunctions which is alternative way of
viewing the uncertainty principle that will allow us to handle new physical
quantities that had previously not be used with Fourier transform.
Energy-Time Uncertainty
Another set of variables related by a Fourier transform are angular frequency and
time. Following the same algebraic procedure as before and using Einstein's
relationship, we have
 Δω Δt 

2
Δ ωΔt 
ΔE Δt 

2

2
This says that if the energy level in an atom had no uncertainty then it would
never make a transition (had to be there forever). If a transition between two
energies occurs then the more rapid the transition the greater the uncertainty in the
energy emitted. The uncertainties in energy widths of atomic transitions are
usually to small to detect, but energy widths of nuclear levels have been measured
in numerous cases. Mossebaur spectroscopy combines the concepts of the
Doppler effect and the uncertainty principle. It is a powerful technique for
characterizing materials in engineering and archeological applications as well as
being a basic research tool for physicists and chemists. In addition to being of
enormous use in explaining the nature of forces in terms of the lifetimes of force
mediating particles in high energy physics experiments, the above relationship
contains time-frequency reciprocity from electrical engineering.
II.
Classical Waves
In classical physics, a wave is a function of space and time which is the solution
of the wave equation. For example, in one-dimension the function f(x,t) is a wave
if it is the solution of the equation
 2f
1  2f

0
 x 2 v2  t 2
where v is the wave speed. deBroglie hypothesis states that all matter may show
either wave or particle phenomena depending on the type of experiment
performed. What does it mean for a baseball to behave like a wave? What is
actually waving?
III.
Wave Function
The mathematical function that completely describes a quantum system is called
the wave function. This function is also sometimes called the state vector. An
alternative way of interpreting quantum mechanics is to consider the wave
functions as residing in a vector space. We will talk more about this later.
for
The wave function serves in quantum mechanics a similar role that the position
vector serves in classical physics. In classical physics, we can apply operators
(derivatives) to the position vector to obtain important information about how the
system changes (i.e. its velocity, acceleration, etc.). Thus, the main problem is
usually to solve Newton's II Law or another equivalent formulation given the
forces applied to the system to solve for the position vector. Applying suitable
operators to the wave function, values for the position, momentum, energy, etc.
the system can be obtained. The main problem in QM is to take the potential
energy function and solve Schrodinger's wave equation to find the wave function.
We are still left with our two questions: 1) What is actually waving? 2) What is
the Schrodinger wave equation?
IV.
Properties of the Wave Function -  (x, t)
A.
The telltale experimental evidence of waves is the property of interference. By
considering the results of electron scattering experiments like the demonstration
in class, Max Born realized that the interaction between a viewing screen and an
individual electron showed a particle nature (localized spot). The interference
pattern was formed due to the distribution of the spots when a large number of
electrons were scattered. Thus, he hypothesized that the wave nature is manifested
in the probability density function that is equal to the square of the wave function.
This is known as the Born Postulate and it forms the basis of the standard
interpretation of Quantum Mechanics (Copenhagen Interpretation).
Mathematically, the probability of finding a particle between x and x+dx is given
by

Ρ(x, t)   (x, t) (x, t) dx  (x, t) dx .
2
It should be noted that the wave functions for physical systems are complex.
However, the probability density is the square of the modulus of the complex
function (i.e. the square of the magnitude of the vector in the complex plane)
which is a real number. If you are rusty on complex numbers, I suggest that you
consult Arfkin.
B.
Other Restrictions on the Wave Function
Born's probability interpretation places other restrictions on the wave function.
1.
The wave function must be finite.
2.
The wave function must be continuous.
3.
The derivative of the wave function must be continuous if the potential
function is finite (a requirement always met in real world problems).
4.
The wave function must be normalized. This means that the total
probability of finding the particle somewhere in space is 100%.

1     (x, t)  (x, t) dx

V.
Developing a Wave Equation
A.
During the question portion of a talk by deBroglie, Erwin Schrodinger objected to
deBroglie's duality work by noting that if physical system's also showed wave
phenomena then you had to have a wave equation. Schrodinger then set about
finding the equation. You can't derive the Schrodinger equation any more than
you can derive Newton's Second Law. We believe in the Newton II because it
properly predicts the outcome of experiments involving mechanics. In the same
manner, the Schrodinger equation correctly describes the outcomes of quantum
experiments and gives classical results when the number of atoms involved are
large.
B.
We begin our development of the Schrodinger equation by considering the
experimental requirements that our new wave equation must meet.
1.
It must be compatible with the deBroglie and Einstein relations
p  k 
h

E ω hf
2.
Its solution must be waves.
3.
For simple classical systems, we know that the Hamiltonian is the total
energy of the system.
H E T  V
4.
From our classical wave analogy, we expect that this wave equation will
involve both spatial and temporal partial derivatives.
One way to ensure that item 2 is met is to take a known classical wave as a
solution. We then look at what happens to the wave when we take spatial and
temporal partial derivatives. Let us use a plane wave traveling in the positive x
direction as the wave function.

(x, t)  A e i k x  ω t

If we take the spatial partial derivative, we get


  k x  ω t 
 A ei k x  ω t i
i k 
x
x
If we now multiply both sides by h-bar and do some algebra, we get
 
 k  
i x
 i

 k  
x
Using the deBroglie relation, we get
 i

p
x
By using deBroglie's relationship, we ensure that it is built into our final wave
equation. This math equation belongs to a mathematical class of equations called
eigenvalue equations that are very important in quantum mechanics. The left-hand

side of the equation is an operator,   i

 
 , that is acting upon a function.
 x 
The result according to the right hand side of the equation is the function
multiplied by a number, p that corresponds to the magnitude of the linear
momentum.
The importance of eigenvalue equations to quantum mechanics is expressed in
Liboff by the first postulate of quantum mechanics in Section 3.1
Quantum Mechanics Postulate 1
To any self-consistently and well defined observable in physics (call it A), such as
linear momentum, energy, mass, charge, etc. there corresponds an operator (call it
 ) such that measurement of A yields values (call these measured values a) that
are eigenvalues of  . That is the values, a, are those values for which the
eigenvalue equation
 f  a f
has a solution f. The function f is called the eigenfunction of  corresponding to
the eigenvalue a.
Eigen is German for proper. Not all functions are solutions to our eigenvalue
equation nor will the operator produce just any eigenvalue. For instance, the
derivative of an exponential function is another exponential function, but the
derivative of the sine function is not the sine function but the cosine function.
Linear Momentum Operator
Our results suggest that in quantum mechanics, linear momentum is represented
in 1-dimensional problems by the operator
p̂   i 

x
with the eigenfunction and corresponding eigenvalue of
 
  A e i k x and p   k
Let us now look at the temporal partial derivative of our trial wave function.


 
 A e i k x  ω t i k x  ω t   i ω 
t
t
Again multiplying by h-bar and using some algebra, we get

 
 ω  
i t
We now ensure that our final wave equation will be compatible with Einstein's
relationship by employing the relationship in our work (actually it follow from
our previous use of deBroglie's relationship).

 
E
i t
Again, we have an eigenvalue equation except that this operator produces energy
eigenvalues instead of linear momentum eigenvalues.
Energy Operator
Our results suggest that in quantum mechanics, energy is represented by the
Hamiltonian operator
Ĥ  i 

t
with the eigenfunction and corresponding energy eigenvalue of
  A e- i
 ωt
and E   ω
We have now included all of our requirements for our wave equation except for
the requirement that the Hamiltonian is given by E = T + V.
Using this requirement and the fact that kinetic energy for non-relativistic
2
p
particles is given by T 
, we have
2m
p2
H  (x, t) 
 (x, t)  Vx, t  (x, t)
2m
We now replace with operators and we have
p̂ 2
Ĥ  (x, t) 
 (x, t)  V̂x, t  (x, t)
2m
Using our results for the energy and linear momentum operator, we obtain the
time dependent Schrodinger equation in 1-dimension as follows:
i
  (x, t) 1 
 
 
  i 
   i 
 (x, t)  V̂x, t  (x, t)

t
2m 
 x 
 x 
1-D Time Dependent Shrodinger Wave Equation
 (x, t)   2  2 (x, t)
i

 V̂x, t (x, t)
t
2m  x 2
The solution to the Schrodinger equation (i.e. the system's wave function) can't be
found until you specify the potential energy operator for the problem in the same
way that Newton's Second Law can't be solved until you specify the forces acting
upon the system.
You have already worked several simple quantum problems including the particle
in a box and the hydrogen atom in modern physics. This semester we will review
these problems and learn additional techniques for making these calculations
easier.