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School of Mathematical and Physical Sciences PHYS1220
PHYS1220 – Quantum Mechanics
Lecture 4
August 27, 2002
Dr J. Quinton
Office: PG 9
ph 49-21-7025
[email protected]
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School of Mathematical and Physical Sciences PHYS1220
The Correspondence Principle
Any theory must match the well-known laws of classical physics if the
conditions match the classical case. This is known as
The Correspondence Principle
Recall from special relativity that when v<<c, the theory must
simplify to Newtonian physics
1 2
2
mv
 eg relativistic kinetic energy KE    1 mc becomes KE 
2
if you take the expansion and make v<<c
In Quantum Mechanics, the same applies in going from microscopic
to macroscopic situations (ie when the system >> de Broglie l)
As the quantum number, n, approaches infinity, any real system
should behave in a way that is consistent with classical physics

Eg in the Bohr model, the discrete energy levels En get closer and closer
together and as n→, they essentially become ‘continuous’
 Z 2e4 m   1 
lim En  En1  0
En    2 2   2 
n
 8 0 h   n 



The same applies for rn and L as n→. The exercise is left to the student
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Quantum Mechanics
Bohr’s model contained a remarkable mixture of classical and
quantum concepts, thus it provoked much thought about the
wave nature of matter, light and the laws of how they interact
with one another
This led to the development of a comprehensive theory to
describe microscopic phenomena, started independently in
(1925) by

Werner Heisenberg (Matrix mechanics, Nobel Prize 1932)
 Heisenberg’s approach employs matrices and matrix functions and
although very powerful, is mathematically complicated and less suitable
for teaching elementary concepts.

Erwin Schrödinger (Wave mechanics, Nobel Prize 1933)
 Schrödinger’s approach uses multivariable functions and operators.

Their approaches were very different from one another but their
theories are fully compatible
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The Uncertainty Principle
Every measurement has an associated uncertainty
According to classical physics, there is no limit to the ultimate
refinement of the apparatus or measuring procedure

It is possible to determine everything to infinite precision
Quantum Theory predicts otherwise. With his matrix mechanics,
Heisenberg showed the existence of what is called the
Heisenberg Uncertainty Principle
If a measurement of position is made with precision x and a
simultaneous measurement of momentum component px is made
with precision px, then
 x px 

It is fundamentally impossible to simultaneously measure the exact
position and exact momentum of a particle
Other uncertainty relations are
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 E t 
  L 
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Uncertainty Principle Example
A ‘thought’ experiment of Heisenberg’s





Suppose you want to simultaneously measure the
position and momentum of an electron as precisely
as possible with a powerful light microscope
In order to determine the electron’s location (ie
making x small ~ l) at least one photon of light
(with momentum h/l must be scattered (as in (a))
But the photon imparts an unknown amount of its
momentum to the electron (as in (b)), thus altering
it’s path and speed! ie p ~ h/l becomes larger!
The very light that allows you to determine the
position changes the momentum by some
undeterminable amount
In making measurements on microscopic scales,
you must now appreciate that you cannot make a
measurement without interacting with the very
thing that you are attempting to measure!
So x p ~ h
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Is the Bohr Model Realistic?
Question: According to the Bohr model, the electron in the
ground state moves in a circular orbit with the Bohr radius
r1=0.529x10-10m, at a speed of 2.2x106 ms-1 (check it for
yourself!). In view of the HUP, is the model realistic?
Answer: Because the model assumes that the electron is located
at r1, the uncertainty r is zero. If the magnitude of the total
momentum of the electron is mv, then the radial component of
momentum must be less than or equal to this value
pr  mv  (9.11x1031 kg )(2.2x106 m.s 1 )  2.0x1024 kg .m.s 1
According to the uncertainty principle, the minimum uncertainty
in the radial position is therefore
rmin
1.05x1034 J .s


 0.525x1010 m
24
1
pr 2.0x10 kg.m.s
which is ~ r1! so the Bohr model is not realistic
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Philosophical Implications
Newtonian physics is completely deterministic.

If you know the configuration of a system at any point in time then you
can predict its future and also extrapolate its past
Quantum mechanics (QM) has drastically altered our viewpoint

Because the wave nature dominates on atomic scales, we must relinquish
determinism and accept a probabilistic approach.
 The expected position of a microscopic particle (such as an electron that is
moving around a nucleus) can only be predicted by calculating a probability,
which in turn indicates an expected statistical average over many
measurements.

Even macroscopic objects that are made up of many atoms are governed
by probability rather than strict determinism.
 eg QM predicts a finite (though negligibly small) probability that an thrown
object (comprising many atoms) will suddenly curve upward rather than follow
a parabolic trajectory

However when large numbers of objects are present in a statistical situation,
deviations from the most probable approach zero, and thus obey classical laws with
very high probability, giving rise to an apparent ‘determinism’
Many people opposed this but ultimately had to accept it. At the time
Einstein believed that “God does not play dice with the universe”
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The Wave Function and its Interpretation
In Quantum Mechanics, each object (particle or the system itself)
is represented by a ‘matter wave’ and is described by a (unique)
wave function, (x,y,z,t). The wavelength is established by
de Broglie, but what is the physical meaning of the amplitude?
One way to interpret the wave function is that it plays the same
role that the electric field vector plays in the wave theory of light

Recall that Intensity

(Amplitude)2
In a similar way, 2 represents an ‘intensity’ or alternatively, the
probability of detecting the object with wave function 

The magnitude of the wave function  (itself generally a complex
quantity) may vary in x,y,z or t, but the probability of detecting the
particle will be greater where and/or when the amplitude is large.
If  represents a single electron (say in an atom) then the value
of ||2dV at a certain point in space and time represents the
probability of finding the electron within the volume dV about the
given position at that time – Max Born, 1928 (Nobel Prize 1954)
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Properties of Wave Functions
Wave functions of particles must possess certain
properties to be useful quantum mechanically.
The function must be continuous
The function must be differentiable
the particle exists and so the the probability of finding it
throughout all of space must be equal to 1. When this is
the case, the function is said to be ‘normalised.’ A
function must be normalised for the probability to make
sense.

eg the probability of detecting an electron with a wave
function y between x=a and x=b is determined by


 dV  1
2

b

 dV
2
a
Expectation values, < >. The expectation value of any
quantity is the statistical average after many
measurements of the quantity are made. For example,
the expectation value of position <x> is given by
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
 x 
x

10
2
dx
School of Mathematical and Physical Sciences PHYS1220
Eigenvalues and Eigenfunctions
Consider
f ( x)  Aekx
d
g ( x) 
f ( x)  k Ae kx  k f ( x)
dx
In this example, the derivative is an operator on the function
f(x). Because the function f(x) is returned (multiplied by a
constant) after it is acted on by the derivative operator, f(x) is
said to be an eigenfunction of the derivative operator

Or more specifically in this case, the exponential function is an
eigenfunction of the derivative operator
The constant that is returned as a multiplier of an eigenfunction
is called its eigenvalue
Here, the constant k is an eigenvalue of the eigenfunction f (x).
Question: is f ( x)  Aekx an eigenvector of the 2nd derivative?
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Waves Revisited
Recall from wave theory that any travelling wave can be
represented mathematically by y ( x, t )  A sin(kx  t   )
0
where k is the angular wavenumber,  the angular frequency,
A is the amplitude and f0 is an initial phase
Now, noticing that
and
then the wave can
be expressed by
h 2
p 
 k
 2 
h
h
E  hf 
2 f  
2
E
p

y ( x, t )  A sin  x  t  0 


Therefore both momentum and energy are contained in the
terms describing the wave
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The Schrödinger Equation
The Schrödinger equation cannot be derived from first principles.
It appears as a postulate, just as Newton’s second law does
We will consider only one dimensional, steady state problems
(where  and the potential U are only a function of spatial x and
independent of time). The 1-D Time-Independent Schrödinger
Equation (T.I.S.E.) is
d 2 ( x)

 U ( x) ( x)  E ( x)
2
2m dx
2
where m is the mass of the particle
The equation is essentially the total energy of the particle
2 2
p2
k
Etot  KE  PE 
U 
U  E
2m
2m
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The Schrödinger Equation II
The T.I.S.E. is based upon conservation of energy, so all 1-D
time-independent systems must obey it
Note that the T.I.S.E. is an operator equation, however.
The wave function of any real object must be an eigenfunction of
Schrödinger’s equation, with its corresponding eigenvalue equal
to the object’s energy, E.

In other words, once you know the eigenfunction of a particle (or its
state) you can just substitute it into Schrödinger’s equation to
calculate the energy
Well that is quite a bit of theory, now let’s use it for some simple
situations
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Free Particles
The simplest wave function to describe is that of a free particle. It
does not have any potential acting on it and therefore no forces.
2
d 2 ( x)
The T.I.S.E. is therefore

Which can be written
2
 E ( x)
2m dx
d 2 ( x) 2mE
 2 0
2
dx
This is the second order differential equation for a harmonic
oscillator with general solution
 ( x)  A sin kx  B cos kx, k 
2mE
2 2
1 2 p2
k
E  mv 

2
2m 2m
Note that k can have any value (ie the energy can be chosen from
a continuous range). Note also that px is zero, so x →
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Free Particles II
Given the wave function, let’s substitute it back into the T.I.S.E.
 ( x)  A sin kx  B cos kx
2
d 2 ( x)
2



k
  A sin kx  B cos kx 
2
2m dx
2m
2

So E 
2
2m
2
2m
k 2 ( x)  E ( x)
k 2 , or alternatively k 
2mE
If we had guessed the wave function, we could have computed
the energy of the free particle (and how it depends upon k)
That is a simple example of the power of Schrödinger’s equation
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Wave Packets
To represent a particle that is well localised (ie its
position is known to be within a small region of
space), we use the concept of a wave-packet
To describe this requires a wave function that is
the sum of many sinusoidal plane waves of
slightly different wavelengths (cf beats).

The smaller the value of x, the more terms are
needed in the sum.
Because each term in the sum has a unique
wavelength (and therefore momentum), the sum
does not have a definite momentum. Rather,
it has a range of momenta, so px is non-zero
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