wave properties of particles

8/13/2019 Wave properties of particles

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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 thatparticles could also have wave-like properties first experimental proof of concept in 1927 in electron scattering/diffractionexperiments

De Broglie Waves


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Wave Functions

The

is the complex valued function describing the matter wave.

The of finding a particle with wave function at coordinateat time is proportional to (Max Born, 1926):

The to find a particle in an experiment in a volume elementat the coordinate and time is

The wavelength of this matter wave is given by the de Broglie relation. To find the wavefunction describing the particle wave is a more complicated problem that we will solve soon.

Wave Funct ions and Wave Packets

xx

x

infinite waves

wave packet

beating interference) of two waves

probability density:


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Propagation Speed of De Broglie Waves

Phase and Group Velocity of De Brogl ie 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.


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Electron Scattering

Experimental verification of de Broglie hypothesis of wave character of particles inelectron scattering experiments by Davisson and Germer and independently byThomson 1927) classical prediction: electron intensitydistribution should be only weakly

dependent on scattering angle andenergy of incident electrons. Observation of strong angle andenergy dependence.

nickel block had been heated up to remove oxide from surface

Experimental Observation

for Ekin = 54 eV


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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. 1892

d. 1975

b. 1881

d. 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


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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 individualparticles to demonstrate that a particle can interfere with itself its own probabilityamplitudes). 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 tennisballs?

interference no interference

R. P. Feynman R. B. Leighton M. L. Sands , The FeynmanLectures on physics, Vol. III: Quantenmechanik, AddisonWesley, Reading Mass.) 1965)

interference of C60 or C70fullerenes bucky balls) deBroglie wave length at T =900 Kelvin is ~ 2.5 10-12 m enough longitudinal andlateral coherence length toobserve interference

Buckminster fullerene C60Nobel prize in Chemistry 1996)


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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 interferencepattern of bucky balls massive particle interfering withitself matter waves

Double slit interference with single quanta [photons] G. I. Taylor, Proc. Cambridge, Phil. Soc. 15, 114 1909) [electrons] G. Mllenstedt C. Jnsson, Z. Phys. 155, 472 1959) [atoms] O. Carnal J. Mlynek , Phys. Bl., Mai 1991, S. 379 [clusters] W. Schllkopf 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?


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Wave Packets

infinite waves

wave packet

beating of two waves

= 2/k

x

x

x

x = real space Fourier space

Uncertainty Principle

2:

1:

1 2


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width of modulation:

The modulation is generated by waves with wave numbers different by an amountWhat does this imply for the momentum of the particle as given by the de Broglie wave

lengths ?

: uncertainty in momentum

: uncertaint in spacestrict derivation to be presented at later stage


before: derived using wave properties ofparticles alternatively: consider particleproperties of waves light) here: observation of an electron usingscattered light, i.e. determiningposition and momentum of the electron


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Wave Particle Duality

every photo detectorreacts to a singlephoton interference pattern isformed even fromaccumulation of singleparticles events trying to determinewhich slit the electronhas passed throughwill necessarilydestroy theinterference pattern

The Structure of Atoms

In the 19th century it was known that matter was made of different chemical elementsconsisting of individual atoms. Not very much was known about the constituents of theatoms. With the discovery of the electron, it became clear that atoms would contain negativelycharged electrons and that some other part of the atom would need to contain positivecharges to realize a neutral atom. Realizing that electrons where much lighter than any atoms it was found that most ofmass of the atom should be carried by its positively charged components.

Thomson 1898) model of the atom: homogeneouslydistributed positively charged matter with interspersedelectrons. It took another 13 years to put this model to a first testand actually note that it was wrong.


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Scattering of Alpha Particles

Geiger and Marsden 1911) experimentmotivated by Rutherford Idea: scatter alpha particles of a thinmetal foil to probe its atomic structure. Alpha particles are doubly ionizedHelium atoms He2+) that aregenerated in nuclear decays ofradioactive materials. They have alarge mass and large energy andmomentum. Scattered alpha particles are detectedby light emission of fluorescent film.

Expectation from the Thomson model: Most particles should go straight through the metalfoil because the electrons are only light particles to scatter on and the positive charge wasexpected to be homogenously distributed over atom.

Rutherford Model

Rutherford expected the positive charge of the atom to beaccumulated in a nucleus in the center of the atom. To prove this idea he analyzed the scattering of alphaparticles from such a nucleus.

Rutherfords assumptions: The nucleus and the alpha particle can beconsidered as point-like charged particles. The nucleus is much heavier than the alphaparticle and thus can be considered to remainat rest in the problem. The electrostatic interaction is mediated by a1/r potential 1/r2 force) leading to ahyperbolic path of the alpha particle with thenucleus in the outer focal point. b is the impact parameter and the scatteringangle


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kinetic energy of particle remainsconstant because nucleus is assumed toremain 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 cannot be determinedexperimentally. Therefore we will determine the number of particles that pass by the nucleuscloser than an impact parameter 0f b and thus will be scattered by an angle of at least .


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Rutherford Scattering Equation

Consider scattering of alpha particles from a metal foil of thickness and an irradiatedarea 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 scatteringangle and alpha particle energy Ek can be used to determine charge of nucleus


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Estimate Size of Nucleus

approximate maximum kinetic energy of natural alpha particles:

Consider head on collision with impact parameter . All kinetic energy is transformedinto 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 thenucleus.

Electron Orbits

In an atom model in which the negatively charged electrons move around the smallpositively charged nucleus stable orbits are possible. Consider the simple example of an atom with a nucleus of charge of +e and one electronwith charge e on an orbit around it like in the hydrogen atom).

centrifugal force:

electrostatic force:

stability criterion:

wave properties of particles

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