Wave properties of particles

• hypothesis of Louis de Broglie (1924): particles may have wave-like properties

• note: it took almost 20 years after noting that waves have particle like properties that particles could also have wave-like properties

• 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 coordinate at time is proportional to (Max Born, 1926):

The to find a particle in an experiment in a volume element at the coordinate 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.

Wave Functions and Wave Packetsλ

Δx

Δx

Δx

• infinite waves

• wave packet

• 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

• 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.

• 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

• patterns of single crystals

Ta97Te60 YbSi1.41

more images• http://www.microscopy.ethz.ch/• http://www.emez.ethz.ch/

• electron diffraction is used to determine crystal structure

• in particular for surfaces

Nobel Prize in Physics (1937)

"for their experimental discovery of the diffraction of electrons by crystals"

b. 1892d. 1975

b. 1881d. 1958

London University London, United Kingdom

Bell Telephone Laboratories New York, NY, USA

United KingdomUSA

1/2 of the prize1/2 of the prize

George Paget ThomsonClinton Joseph Davisson

Other Diffraction Experiments with Massive Particles• Also scattering of other types of particles (Neutrons, Atoms, …) off crystals show this effects.

• Neutron scattering is used routinely to determine the crystal structure.

• In modern experiments the observation particle properties of matter is also pursued.

• 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.

• 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

• Which path did the particle (ball) pass through?• why do we not see interference of footballs or tennis

balls?

interference no interference

R. P. Feynman, R. B. Leighton, M. L. Sands, „The Feynman Lectures on physics“, Vol. III: Quantenmechanik“, Addison Wesley, Reading (Mass.) (1965)

• interference of C60 or C70fullerenes (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)

Experimental Apparatus

• SiN interference grid with pitch of 100 nm and slits of 50 nm width

• single molecule ionization detector

• one molecule interferes with itself

Measurement Result

• spatially resolved interference pattern of bucky balls

• massive particle interfering with itself

• matter waves

Double slit interference with single quanta• [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)

• from micro to macro?

• from classical to quantum?

Wave Packets

• infinite waves

• wave packet

• beating of two waves

λ = 2π/k

Δx

Δx

Δx

Δx = ∞ real space Fourier space

Uncertainty Principle

ψ2: ω + Δω

ψ1: ω

ψ1 + ψ2

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 strict derivation to be presented at later stage

Uncertainty Principle

• before: derived using wave properties of particles

• alternatively: consider particle properties of waves (light)

• here: observation of an electron using scattered light, i.e. determining position and momentum of the electron

Wave Particle Duality

• every photo detector reacts to a single photon

• interference pattern is formed even from accumulation of single particles events

• trying to determine which slit the electron has passed through will necessarily destroy the interference pattern

The Structure of Atoms

• 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.

• 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.

• 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.

• Thomson (1898) model of the atom: homogeneously distributed positively charged matter with interspersed electrons.

• It took another 13 years to put this model to a first test and actually note that it was wrong.

Scattering of Alpha Particles

• Geiger and Marsden (1911) experiment motivated by Rutherford

• Idea: scatter alpha particles of a thin metal foil to probe its atomic structure.

• 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.

• 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

• Rutherford expected the positive charge of the atom to be accumulated in a nucleus in the center of the atom.

• 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 Angledependence 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 EquationConsider 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

• confirms Rutherford model of the atom

• could be seen as the discovery of the atomic nucleus

• strong dependence on atomic number Z scattering angle θ and alpha particle energy Ek

• can be used to determine charge of nucleus

Estimate Size of Nucleusapproximate 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 being the total number of protons and neutrons in the nucleus.

Electron Orbits

• In an atom model in which the negatively charged electrons move around the small positively charged nucleus stable orbits are possible.

• 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: