Wave properties of particles
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hypothesis of Louis de Broglie (1924): particles may have wave-like properties
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note: it took almost 20 years after noting that waves have particle like properties that
particles could also have wave-like properties
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first experimental proof of concept in 1927 in electron scattering/diffraction
experiments
De Broglie Waves
Wave Functions
The
is the complex valued function describing the matter wave.
The
of finding a particle with wave function
at time is proportional to (Max Born, 1926):
The
at the coordinate
at coordinate
to find a particle in an experiment in a volume element
and time is
The wavelength of this matter wave is given by the de Broglie relation. To find the wave
function describing the particle wave is a more complicated problem that we will solve soon.
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Wave Functions and Wave Packets
infinite waves
wave packet
λ
Δx
Δx
Δx
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beating (interference) of two waves
probability density:
Propagation Speed of De Broglie Waves
Phase and Group Velocity of De Broglie Waves
angular frequency
wave vector
both
and
are functions of the particle velocity
phase velocity:
group velocity:
The group velocity of the wave packet describing the particle corresponds to its velocity.
Electron Scattering
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Experimental verification of de Broglie hypothesis of wave character of particles in
electron scattering experiments by Davisson and Germer and independently by
Thomson (1927)
• classical prediction: electron intensity
distribution should be only weakly
dependent on scattering angle and
energy of incident electrons.
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Observation of strong angle and
energy dependence.
nickel block had been heated up to remove oxide from surface
Experimental Observation
for Ekin = 54 eV
Modern Electron Diffraction Measurements
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patterns of single crystals
Ta97Te60
YbSi1.41
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electron diffraction is used to determine crystal structure
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in particular for surfaces
more images
• http://www.microscopy.ethz.ch/
• http://www.emez.ethz.ch/
Nobel Prize in Physics (1937)
"for their experimental discovery of the diffraction of electrons by crystals"
Clinton Joseph Davisson
1/2 of the prize
George Paget Thomson
1/2 of the prize
USA
United Kingdom
Bell Telephone Laboratories
New York, NY, USA
London University
London, United Kingdom
b. 1881
d. 1958
b. 1892
d. 1975
Other Diffraction Experiments with Massive Particles
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Also scattering of other types of particles (Neutrons, Atoms, …) off crystals show this effects.
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Neutron scattering is used routinely to determine the crystal structure.
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In modern experiments the observation particle properties of matter is also pursued.
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One direction of research concerns the observation wave properties of particles with large mass
to answer how large an object would show wave like properties.
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Another direction researches such interference and diffraction experiments with individual
particles to demonstrate that a particle can interfere with itself (its own probability
amplitudes). No large collections of particles are required to observed interference.
He atom matter wave apparatus
Interference of Matter Waves
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interference of C60 or C70
fullerenes (bucky balls)
deBroglie wave length at T =
900 Kelvin is λ ~ 2.5 10-12 m
enough longitudinal and
lateral coherence length to
observe interference
Buckminster fullerene C60
(Nobel prize in Chemistry 1996)
interference
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no interference
Which path did the particle (ball) pass through?
why do we not see interference of footballs or tennis
balls?
R. P. Feynman, R. B. Leighton, M. L. Sands, „The Feynman
Lectures on physics“, Vol. III: Quantenmechanik“, Addison
Wesley, Reading (Mass.) (1965)
Experimental Apparatus
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SiN interference grid with pitch of 100 nm and slits of 50 nm width
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single molecule ionization detector
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one molecule interferes with itself
Measurement Result
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spatially resolved interference
pattern of bucky balls
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massive particle interfering with
itself
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matter waves
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from micro to macro?
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from classical to quantum?
Double slit interference with single quanta
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[photons] G. I. Taylor, Proc. Cambridge, Phil. Soc. 15, 114 (1909)
[electrons] G. Möllenstedt, C. Jönsson, Z. Phys. 155, 472 (1959)
[atoms] O. Carnal, J. Mlynek, Phys. Bl., Mai 1991, S. 379
[clusters] W. Schöllkopf, J. P. Toennies, Science 266, 1345 (1994)
[bucky balls] M. Arndt et al., Nature 401, 680 (1999)
Wave Packets
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Δx = ∞
real space
λ = 2π/k
infinite waves
Δx
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wave packet
Δx
Δx
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beating of two waves
Uncertainty Principle
ψ1 : ω
ψ2: ω + Δω
ψ1 + ψ 2
Fourier space
Uncertainty Principle
width
of modulation:
The modulation is generated by waves with wave numbers different by an amount
What does this imply for the momentum of the particle as given by the de Broglie wave
lengths ?
: uncertainty in momentum
: uncertainty in space
Uncertainty Principle
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before: derived using wave properties of
particles
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alternatively: consider particle
properties of waves (light)
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here: observation of an electron using
scattered light, i.e. determining
position and momentum of the electron
strict derivation to be presented at later stage
Wave Particle Duality
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every photo detector
reacts to a single
photon
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interference pattern is
formed even from
accumulation of single
particles events
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trying to determine
which slit the electron
has passed through
will necessarily
destroy the
interference pattern
The Structure of Atoms
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In the 19th century it was known that matter was made of different chemical elements
consisting of individual atoms. Not very much was known about the constituents of the
atoms.
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With the discovery of the electron, it became clear that atoms would contain negatively
charged electrons and that some other part of the atom would need to contain positive
charges to realize a neutral atom.
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Realizing that electrons where much lighter than any atoms it was found that most of
mass of the atom should be carried by its positively charged components.
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Thomson (1898) model of the atom: homogeneously
distributed positively charged matter with interspersed
electrons.
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It took another 13 years to put this model to a first test
and actually note that it was wrong.
Scattering of Alpha Particles
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Geiger and Marsden (1911) experiment
motivated by Rutherford
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Idea: scatter alpha particles of a thin
metal foil to probe its atomic structure.
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Alpha particles are doubly ionized
Helium atoms (He2+) that are
generated in nuclear decays of
radioactive materials. They have a
large mass and large energy and
momentum.
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Scattered alpha particles are detected
by light emission of fluorescent film.
Expectation from the Thomson model: Most particles should go straight through the metal
foil because the electrons are only light particles to scatter on and the positive charge was
expected to be homogenously distributed over atom.
Rutherford Model
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Rutherford expected the positive charge of the atom to be
accumulated in a nucleus in the center of the atom.
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To prove this idea he analyzed the scattering of alpha
particles from such a nucleus.
Rutherford’s assumptions:
• The nucleus and the alpha particle can be
considered as point-like charged particles.
• The nucleus is much heavier than the alpha
particle and thus can be considered to remain
at rest in the problem.
• The electrostatic interaction is mediated by a
1/r potential (1/r2 force) leading to a
hyperbolic path of the alpha particle with the
nucleus in the outer focal point.
• b is the impact parameter and θ the scattering
angle
kinetic energy of particle remains
constant because nucleus is assumed to
remain at rest, therefore
is the
particle velocity in the distance
the change of momentum
is therefore
momentum transfer due to the force acting on the alpha particle during the scattering
with:
constant angular momentum:
Scattering Angle
dependence on impact parameter b and kinetic energy Ek of alpha particle
Unfortunately, for a single scattering event the relation between b and θ cannot be determined
experimentally. Therefore we will determine the number of particles that pass by the nucleus
closer than an impact parameter 0f b and thus will be scattered by an angle of at least θ.
Rutherford Scattering Equation
Consider scattering of alpha particles from a metal foil of thickness and an irradiated
area and atoms per unit volume. Thus the fraction of alpha particles scattered by at
least an angle is:
In actual experiments the detector measures the number
of particles scattered in a range of angles
around
the angle .
total surface into which the particles can be scattered
Rutherford Scattering Equation
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confirms Rutherford model of the atom
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could be seen as the discovery of the atomic nucleus
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strong dependence on atomic number Z scattering
angle θ and alpha particle energy Ek
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can be used to determine charge of nucleus
Estimate Size of Nucleus
approximate maximum kinetic energy of natural alpha particles:
Consider head on collision with impact parameter
. All kinetic energy is transformed
into potential energy when the alpha particle reaches closest distance from nucleus .
for gold (Au, Z= 79)
more accurate result for radius of nucleus from high energy (several GeV) electron scattering
With mass (nucleon) number
nucleus.
being the total number of protons and neutrons in the
Electron Orbits
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In an atom model in which the negatively charged electrons move around the small
positively charged nucleus stable orbits are possible.
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Consider the simple example of an atom with a nucleus of charge of +e and one electron
with charge –e on an orbit around it (like in the hydrogen atom).
centrifugal force:
electrostatic force:
stability criterion: