Transcript Slide 1

Part II
deBroglie and his matter waves, and its consequences
for physics and our concept of reality
me vr  n n  1, 2, 3...
  h / 2
an integer number of wavelengths
fits into the circular orbit
n  2r
where
h

p
 is the de Broglie wavelength
Value of 
Particle
Electrons of kinetic energy
Protons of kinetic energy
1 eV
100 eV
10000 eV
12.2 A
1.2 A
0.12A
1 keV
1 MeV
1 GeV
0.009 A
28.6 F
0.73 F
Thermal neutrons (300K)
Neutrons of kinetic energy
He atoms at 300K
me, walking to the student union
for lunch at 2 miles per hour
1 A  108 cm;1 F  10-13 cm  105 A
1.5 A
9.0 F
0.75 A
2.541034 m
Electrons have a wavelength that is much shorter than visible light.
The smallest detail that can be resolved is equal to one wavelength.
Human hair
Table Salt
Red Blood Cells
Backscattered
electrons can be used to
identify elements in the
material.
Auger electrons also can
give compositional
information.
Secondary electrons
are low in energy and
thus can’t escape from
the interior of the
material. They mostly
give information about
the surface
topography.
X-rays are produced
by de-excited atoms.
foil (thin) bulk (thick)
Specimen interactions involved in forming an image
Unscattered electrons are
those which are transmitted
through the material. Since
the probability of
transmission is proportional
to thickness, it can give a
reading of thickness
variations.
Elastically scattered
Bragg electrons give info
about atomic spacing,
crystal orientations, etc.
Loss of energy by
electrons is
characteristic of bulk
composition.
minima occur at
D sin    / 2
electronwavelength:   h / p
for small :
sin    
h
2 pD
•A large number of electrons going through a double
slit will produce an interference pattern, like a wave.
•However, each electron makes a single impact on a
phosphorescent screen-like a particle.
•Electrons have indivisible (as far as we know) mass
and electric charge, so if you suddenly closed one of
the slits, you couldn’t chop the electron in halfbecause it clearly is a particle.
•A large number of electrons fired at two
simultaneously open slits, however, will eventually,
once you have enough statistics, form an
intereference pattern. Their cumulative impact is
wavelike.
•This leads us to believe that the behavior of electrons
is governed by probabilistic laws. --The wavefunction
describes the probability that an electron will be found
in a particular location.
   scatteringangle  
 (h sin  ) /   electronx momentum (h sin  ) / 
 2h
  
px x   sin  
h

 2 sin  
Uncertainty on optics:
x   /(2 sin  )
Given the Uncertainty Principle, how do you write an equation of motion
for a particle?
•First, remember that a particle is only a particle sort of, and a wave sort of,
and it’s not quite like anything you’ve encountered in classical physics.
We need to use Fourier’s Theorem to represent
the particle as the superposition of many waves.
wavefunction
of the
electron
adding varying amounts of
an infinite number of waves sinusoidal expression
for harmonics

( x,0)   a(k )e dk
ikx

amplitude of wave with
wavenumber k=2/
•We saw a hint of probabilistic behavior in the double slit experiment.
Maybe that is a clue about how to describe the motion of a “particle” or
“wavicle” or whatever.
We can’t write a deterministic equation of motion as in
Newtonian Mechanics, however, we know that a large number of
events will behave in a statistically predictable way.
probability for an
electron to be found
between x and x+dx
(x,t)
b
2
dx
P   ( x, t ) dx
2
a
Assuming this particle exists, at any given time it must be
somewhere. This requirement can be expressed mathematically as:
If you search from
hither to yon



( x, t ) dx  1
2
you will find your particle
once, not twice (that would be
two particles) but once.
Let’s try a typical classical wavefunction:
( x, t )  A sin(kx  t )
For a particle propagating in the +x direction.
Similarly, for a particle propagating in the –x direction:
( x, t )  A sin(kx  t )
We also know that if 1 and 2 are both allowed waves,
then 1+2 must also be allowed (this is called the
superposition principle).
1( x, t )   2( x, t )
 A1 sin(kx  t )  A2 sin(kx  t )
 2 sin kx cost
Oops, the particle vanishes at integer multiples of /2, 2/3, etc.
and we know our particle is somewhere.
( x, t )  Aei ( kxt )  A{cos(kx  t )  i sin(kx  t )}
Euler's equation
ei  cos  i sin 
e i  cos  i sin 
Graphical representation of
a complex number z as a
point in the complex plane.
The horizontal and vertical
Cartesian components give
the real and imaginary parts
of z respectively.
Why do we express the wavefunction as a complex number instead
of sines and cosines like classical waves??
Consider another typical classical wavefunction:
( x, t )  A cos(kx  t )
Now imagine k -> -k and  -> -
( x, t )  A cos{(kx  t )}  A cos(kx  t )
Clearly, we need to express this in a different way.


x