Lecture 6. De Broglie Waves

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Lecture 6. De Broglie Waves

• Outline
• The de Broglie Hypothesis
• The Davisson-Germer Experiment
• The Electron Interference Experiment

Is the wave/particle identity crisis only a
problem with light, or it is pertinent to objects
that seem undeniably to be particles?
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The Need for a New Mechanics
If, in some cataclysm, all of scientific
knowledge were to be destroyed, and only one
sentence passed on to the next generation of
creatures, what statement would contain the most
information in the fewest words? I believe it is
the atomic hypothesis that All things are made
of atoms-little particles that move around in
perpetual motion, attracting each other when they
are a little distance apart, but repelling upon
being squeezed into one another.In that one
sentence, you will see, there is an enormous
amount of information about the world, if just a
little imagination and thinking are applied.
Oops! Classical physics cannot explain existence
of stable atoms!
- non-relativistic motion
- power emitted by an accelerated charge
The lifetime of a classical atom
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De Broglie Hypothesis
The e.-m. waves can be described using the
language of quantum particles (photons). Can
particles behave as waves?
De Broglie (1923) suggested that a plane
monochromatic wave is associated with a freely
moving particle
This is a solution of the wave equation in one
dimension
This wave (its phase) travels with the phase
velocity
Well apply the same logic which helped us to
establish the relationship between p and ? for
photons
The phase is a Lorentz-invariant
quantity, the (scalar) product of two 4-vectors
Particle properties
Wave properties
- both the time-like and space-like components
of these 4-vectors should transform under L.Tr.
in a similar way
- de Broglie wavelength p - the objects momentum
Thus, well require
and
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De Broglie Wavelength
- depends on the momentum rather then energy
(e.g., for an object at rest, ? ?)
Compare with Compton wavelength of the particle
- formally speaking, C. wavelength can be
considered as the dB wavelength that corresponds
to the momentum equal to the length of 4-vector
(iE/c,p)
Why is there a kink on this plot?
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Examples
1. What is the de Broglie wavelength of the
charge carriers in a typical metal? The kinetic
energy of charge carriers (conduction
electrons) in metals is of an order of a few eV
(5 eV for Au) its called the Fermi energy, EF
.
Non-relativistic case
Even for this relatively small K, the de Broglie
wavelength of an electron is tiny thus, the
difficulty of constructing a pair of slits for an
electron interference experiment.
For non-relativistic electrons accelerated
through the potential difference V
2. A buckeyball (fullerene) is a large molecule
comprised of 60 carbon atoms (C60) arranged in a
shape somewhat like a hollow sphere 0.71 nm in
diameter. Imagine that we create a beam of
buckeyballs all moving at the same speed v. What
is the maximum value that v can have if the de
Broglie wavelength of the buckeyball beam is to
be at least 2 times the size of the buckeyball?
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Example
What would be the kinetic energy of each electron
in a beam of electrons having a de Broglie
wavelength of 633 nm (the wavelength of light
emitted by the common helium-neon laser)?
This is the kinetic energy of an electron
accelerated from rest through a potential
difference of 3.7 microVolts. For comparison, at
room temperature the kinetic energy of a free
electron is
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Example
Find the kinetic energy of particles in an
accelerator which is used to explore the
structure of matter at a scale of 1 Fermi
(10-15m).
- otherwise, the structure cannot be resolved
If the particles are electrons
If the particles are protons
Large Hadron Collider protons with
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Phase and Group Velocities
- no limitations on the phase velocity, (phase of
a plane wave does not carry any information)
This phase velocity
The observable is the group velocity (the
velocity of propagation of a wave packet or
wave group. Lets consider the superposition of
two harmonic waves with slightly different
frequencies (???, k?k)
The velocity of propagation of the wave packet
fast oscillations within the wave group
envelope wave group
-the group velocity
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Group Velocity of de Broglie waves
- the group velocity of de Broglie waves coincide
with the particles velocity
Periodic processes discrete spectrum (Fourier
series).
Aperiodic processes continuous spectrum
(represented as Fourier integral)
f(t)
t
-1
1
0
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The Davisson-Germer Experiment (1925)
heated cathode
Electron gun preparation of a monoenergetic
(monochromatic) beam of electrons.
anode
Scattering of electrons from metal surfaces (a)
polycrystalline surfaces the number of
scattered electrons is a monotonic function (b)
single-crystal surface at some angles the
electron scattering is enhanced.
electron beam
-

Explanation constructive interference of
electron waves reflected from atomic planes in
the crystal.
inter-plane distance
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Two-Slit Experiment with Electrons
Jönsson, 1961
two slits open
one slit open
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Interference at Low Intensities
Two-dimensional array of small particle/photon
(quanton) detectors
Strictly-wave model smooth oscillating
variations of intensity (the number of particles).
Strictly-particle model discrete events but no
oscillations of intensity.

• Implications
• behavior of individual quantons is not
  deterministic (newtonian)
• each individual quanton knows about both
  slits
• any attempt to conduct which-way experiment
  kills the interference
• neither the particle nor wave models are
  adequate

The wave model describes correctly the
statistical distribution of quanton arrivals, the
particle model describes the interaction of each
individual quanton with a detector (collapse of
the wavefunction in the process of measurement).
Statistical Interpretation of de Broglie Waves
(Max Born) de Broglie wave the wave of
probability, the intensity of dB wave at a given
location is proportional to the probability to
detect the particle at this location - to be
discussed later. The statistical properties can
be studied only if one can repeat the same
experiment with identical particles many times
(or observe many identical particles in identical
conditions at the same time).
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Earlier (incorrect) Interpretation of de Broglie
Waves
Earlier ideas (Schrödinger) particle the wave
group. In favor the group velocity of de Broglie
waves coincide with the particles velocity.
However, the wave packet wouldnt live for a long
time because of the dispersion of de Broglie
waves in vacuum
In general
- no dispersion (c?c(k))
for light in vacuum
Deformation of a 1D wave group, mme (1
Bohr0.053nm, time units h3/mee42.4?10-17s)
Thus, a particle IS NOT the group of de Broglie
waves!
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State-of-the-art detection of single THz photons
Detection of the visible-range photons not a big
deal, the photon energy is sufficient to generate
photoelectrons (the photoelectric effect)
- comparable with the energy gap between the
valence and conduction bands in typical
semiconductors
photomultiplier
This task becomes more challenging at lower
photon energies...
SAFIR
nanostructures at ultra-low T to increase
sensitivity
cold (4K) antenna to reduce photon noise
The energy resolution sufficiently high for
detecting single 0.1-THz photons _at_ 0.1K and 1-THz
photons _at_ 0.3K.
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Particle-Wave Dualism
Conclusion all forms of matter (both particles
and fields) exhibit wavelike aspects.
De Broglies equations
equally apply to particles and photons
The wave-like character of an object becomes more
apparent at low kinetic energies as its de
Broglie wavelength increases it is much easier
to observe interference with visible light than
with electrons.
(Very) roughly speaking
- particle behavior dominates
- wave behavior dominates
Difficulty of interference experiments with
particles it is not easy to prepare
monochromatic beams of slow particles with a
sufficiently large de Broglie wavelength. Difficu
lty of observation of the particle-like nature of
e.-m. waves the energy carried by a single
photon becomes too small for its detection.