Transcript lecture27

Wave-Particle Duality - the Principle of Complementarity
We have phenomena such as diffraction and interference that show that
light is a wave, and phenomena such as the photoelectric effect that show
that it is a particle.
Which is it?
This question has no answer; we must accept the dual wave-particle
nature of light.
The principle of complementarity states that both the wave and
particle aspects of light are fundamental to its nature.
Indeed, waves and particles are just our interpretations of how
light behaves.
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The photon, the quantum of light
Energy: E  pc  hf 
hf
h

Momentum: p 
c

Wave Nature of Matter
Just as light sometimes behaves as a particle, matter sometimes behaves
like a wave. (Idea of the symmetry in nature - de Broglie 1927)
The wavelength of a particle of matter (De Broglie wavelength) is:
• This wavelength is extraordinarily small.
• The wave nature of matter becomes more important for very light
particles such as the electron.
• Electron wavelengths can easily be on the order of 10-10 m;
electrons can be diffracted by crystals just as X-rays can.
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Example1: What is the de Broglie wavelength of a 0.20kg ball moving with
speed 15m/s?
h
h
6.6 1034 J  s
 

 2.2 1034 m
p mv 0.20kg   15m / s) 
Example2: Determine the wavelength of an electron that has been
accelerated through a potential difference of 100 V.
V  100V
m  9.1110 31 kg
1
2
e  1.60 10 19 C
 ?
2eV
mv  eV  v 
;
m
2
h


mv
h
2eVm
  1.2  10 10 m
Scanning electron
microscopy (SEM)
image
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De Broglie’s Hypothesis Applied to Atoms
The correspondence principle applies here as well – when the differences between
quantum levels are small compared to the energies, they should be imperceptible.
De Broglie’s hypothesis is the one associating a wavelength with the momentum of
a particle. He proposed that only those orbits where the wave would be a circular
standing wave will occur. This yields the same relation that Bohr had proposed.
In addition, it makes more reasonable the fact
that the electrons do not radiate, as one would
otherwise expect from an accelerating charge.
quantization:
de Broglie’s wavelength:
Bohr’s quantization:
2rn  n
h

mv
nh
mvrn 
2
These are circular standing
waves for n = 2, 3, and 5.
About Quantum Mechanics
Quantum mechanics incorporates wave-particle duality, and
successfully explains energy states in complex atoms and molecules,
the relative brightness of spectral lines, and many other phenomena.
It is widely accepted as being the fundamental theory underlying all
physical processes.
Quantum mechanics is essential to understanding atoms and
molecules, but can also have effects on larger scales.
Classical physics
v<<c
xpx  
Classical relativistic physics
xpx  
v~c
Quantum nonrelativistic physics
xp x  
v<<c
Quantum relativistic physics
xp x  
v~c
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The Wave Function and Its Interpretation
Question: An electromagnetic wave has oscillating electric and magnetic
fields. What is oscillating in a matter wave?
Answer: This role is played by the wave function, Ψ.
The square of the absolute value of the wave function at any point is
proportional to the number of electrons expected to be found there.
For a single electron, the wave function is the probability of finding the
electron at that point.
In the classical mechanics we use Newton’s equations of motion to describe
particles positions and velocities, in the classical electrodynamics we use
Maxwell’s equations to describe the electric and magnetic fields.
In quantum mechanics we use Schrödinger equation to describe the
function: Ψ= Ψ(x,y,z;t).
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The Wave Function and the Double-Slit Experiment
The interference pattern is
observed after many electrons
have gone through the slits.
If we send the electrons through
one at a time, we cannot predict
the path any single electron will
take, but we can predict the
overall distribution.
Philosophic Implications - Probability versus Determinism
The world of Newtonian mechanics is a deterministic one. If you know the
forces on an object and its initial velocity, you can predict where it will go.
Quantum mechanics is very different – you can predict what masses of
electrons will do, but have no idea what any individual one will.
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The Heisenberg Uncertainty Principle
Quantum mechanics tells us there are limits to measurement –
not because of the limits of our instruments, but inherently.
This is due to the wave-particle duality, and to interaction
between the observing equipment and the object being observed.
xp x  
h

2
x - the uncertaint y in the position
p x - the uncertaint y in the momentum
Similar:
yp y  
zpz  
The Heisenberg uncertainty principle tells us that the position and
momentum cannot simultaneously be measured with precision.
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Two examples
Example 1: An electron moves in a straight line with a constant speed
v=1.10x106 m/s which has been measured to a precision of 0.10%.
What is the maximum precision with which its position could be
simultaneously measured?
v  10 3 v
p  mv


1.06  10 34 J  s
7
x 



1
.
1

10
m
31
3
6
p mv 9.11  10 kg 10  1.10  10 m / s



Example 2: What is the uncertainty in position, imposed by the uncertainty
principle, on 150–g baseball thrown at (93 ± 2) mph = (42 ± 1) m/s?


1.06  10 34 J  s
x 


 7  10 34 m
p mv 0.150kg 1m / s 
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The uncertainty principle for energy and time
Et  
This says that if an energy state only lasts for a limited time, its energy
will be uncertain.
It also says that conservation of energy can be violated if the time is
short enough.
Example: The Z boson typically decays very quickly. Its average energy is
91.19 GeV, but its short lifetime shows up as an intrinsic width of 2.49 GeV
(rest energy uncertainty). What is the lifetime of this particle?

h
t 

E 2E
4.14  10 15 eV  s
55
t 

2
.
65

10
s
9
2 2.49  10 eV


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Example:
Imagine trying to see an electron with a powerful
microscope.
At least one photon must scatter off the electron
and enter the microscope, but in doing so it will
transfer some of its momentum to the electron.
The uncertainty in the momentum of the electron
is taken to be the momentum of the photon:
p 
h

In addition, the position can only be measured to
about one wavelength of the photon:
x  
Combining, we find the combination of uncertainties:
x 

c
c
hc
E  hf 
t 

xp x  h
Et  h
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