Wave-Particle Duality
Particle Behavior of Wave
In the previous section we have discussed interaction between atomic oscillators and
electromagnetic wave. Planck’s radical assumption was that the electromagnetic radiation
emitted or absorbed as quanta, with energy 𝐸 = ℎ.
In this lecture we will discuss particle properties of wave and wave properties of particles. We
also discuss formation of particle from wave and formation of wave from particle.
After Planck's assumption, several experiments was proposed to investigate particle properties
of wave. These are namely:
Photoelectric effect
Compton scattering
Raman scattering
After particle behavior of wave accepted, the question became whether this was true only for
light or whether material objects also exhibited wave-like behavior. This is proposed and
formulated by de Broglie. This hypothesis have been experimentally confirmed.
Pair-Production shows production of particle from a wave.
Photo electric effect
After studying the photoelectric effect, Einstein concluded that electromagnetic radiation
could be shown to behave as a beam of particles, with energy 𝐸 = ℎ.
Let us briefly summarize the photoelectric effect experiment:
When a light is shone on a metal surface, electrons are ejected from the surface of the metal.
The ejected electrons are termed photoelectrons.
This interaction can be explained as follows; Energy of incident light is absorbed by electrons
within the metal. According to the classical Maxwell wave theory of light, the more intense
the incident light the greater the energy with which the electrons should be ejected from the
metal. Classically, the average energy carried by an ejected (photoelectric) electron should
increase with the intensity of the incident light. Experiment shows this is wrong!
The energies of the emitted electrons to be independent of the intensity of the incident
radiation.
Einstein (1905) successfully resolved this paradox. He proposed that the incident light
consisted of individual quanta, called photons, that interacted with the electrons in the metal
like discrete particles, rather than as continuous waves.
For a given frequency, or 'color,' of the incident radiation, each photon carried the energy
𝐸 = ℎ𝜈.
In this model, Increasing the intensity of the light corresponded, to increasing the number of
incident photons per unit time (flux), while the energy of each photon remained the same (as
long as the frequency of the radiation was held constant).
In Einstein's model, increasing the frequency, rather than the intensity, of the incident
radiation would increase the average energy of the emitted electrons.
Both of these predictions were confirmed experimentally.
The photoelectric effect is perhaps the most direct and convincing evidence of the existence
of photons and the 'corpuscular' nature of light and electromagnetic radiation. That is, it
provides undeniable evidence of the quantization of the electromagnetic field and the
limitations of the classical field equations of Maxwell.
Mathematical Formulation and Experimental Procedure of Photoelectric effect
The photoelectric effect exhibits the following:
1)
There is a minimum frequency, 𝜈𝑐 , called the threshhold frequency (or cutoff
frequency) required for the effect to occur.
2)
The maximum kinetic energy of the photoelectrons does not depend on the intensity of
the light.
3)
The maximum kinetic energy of the photoelectrons increases as the frequency of the
light increases.
4)
There is no appreciable time delay between the illumination of the surface and the
emission of the photoelectrons.
1
2
𝑚𝑒 𝑣𝑚𝑎𝑥
= 𝐾𝑚𝑎𝑥 = 𝑒𝑉𝑠
2
The electrons is bounded to the metal surface with a potential energy. 𝑈  − 𝜑, where U =
potential energy of an electron, and -φ is the highest value of U. Energy conservation requires:
(𝑒𝑛𝑒𝑟𝑔𝑦 𝑜𝑓 𝑝ℎ𝑜𝑡𝑜𝑛) = (𝑤𝑜𝑟𝑘 𝑡𝑜 𝑒𝑗𝑒𝑐𝑡 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛) + (𝐾𝐸 𝑜𝑓 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛).
ℎ = 𝜑 + 𝐾
φ is called the photoelectric work function of the metal.
The particle-particle collision concept explains the immediate ejection of photoelectrons.
Since Kmax cannot be less than zero, the minimum frequency is explained: for
𝐾𝑚𝑎𝑥 = 0,
ℎ𝜈𝑐 = 𝜑
Where νc is “cutoff” frequency and the result shows no dependence on light intensity for Kmax
Without further details I will summarize the other observations that leads to understand
particle properties of wave.
Raman scattering and quantum hypothesis
The scattering of light, as a result of interaction with matter, can be elastic (Rayleigh) or
inelastic (Raman). In case of Rayleigh scattering, the scattered light is observed at the same
frequency as the incident light. This scattering is responsible for the blue color of the sky.
In case of Raman scattering, scattered light is detected at different frequencies compared with
incident light; photons interact with the molecules in such a way that its energy is either
increase or decrease so that the scattered photons are shifted in frequency.
Photons of source (preferable a laser) are absorbed by the sample and then reemitted.
Frequency of the reemitted photons is shifted up or down in comparison with original
monochromatic frequency, and these facts represent the basis of Raman effect. This shift
provides information about vibrational, rotational and other low frequency transitions in
molecules, because the shift of photons energy is directly related to the molecule spectra.
Actual explanation supposes an interaction of incident photon with the electric dipole of that
molecule. The photon (light quantum), excites the one of the electrons into a virtual state, that
is to say, the energy is not yet enough to excite into a full quantum state. Then almost
immediately another photon is released and the molecule is left in a lower state. During this
process energy can be deposited into the molecule, which leaves the molecule in a higher
vibrational state. The scattered photon has a smaller energy compared with incident one.
When the molecule is already in an excited vibrational state, the energy of scattered photon is
higher than incident photon. This is possible if and only if light has a particle property.
Figure: Red shift satellite in Raman effect
Figure: Blue shift satellite in Raman effect
Compton Scattering
Arthur H. Compton observed the scattering of x-rays from electrons in a carbon target and
found scattered x-rays with a longer wavelength than those incident upon the target.
The shift of the wavelength increased with scattering angle according to the Compton
formula:
Compton explained and modeled the data by assuming a particle (photon) nature for light and
applying conservation of energy and conservation of momentum to the collision between the
photon and the electron. The scattered photon has lower energy and therefore a longer
wavelength according to the Planck relationship.
Compton experiment gave clear and independent evidence of particle-like behavior.
Wave Behavior of Particle: De Broglie's Hypothesis
In his 1923, Louis de Broglie made a bold assertion. Considering Einstein's relationship of
wavelength to momentum p, de Broglie proposed that this relationship would determine the
wavelength λ of any matter, in the relationship:
𝜆=
ℎ
𝑝
This wavelength is called the de Broglie wavelength. This equation and energy of the photon
can be written as:
𝑝 = ℏ𝑘 𝑎𝑛𝑑 𝐸 = ℏ𝜔
Where 𝑘 =
2𝜋
𝜆
is angular wavenumber and ω is angular frequency.
Significance of the de Broglie Hypothesis
The de Broglie hypothesis showed that wave particle duality was not merely an aberrant
behavior of light, but rather was a fundamental principle exhibited by both radiation and
matter. As such, it becomes possible to use wave equations to describe material behavior, so
long as one properly applies the de Broglie wavelength. This would prove crucial to the
development of quantum mechanics.
Experimental Confirmation
Elementary particles (Electron diffraction)
In 1927, physicists Clinton Davisson and Lester Germer, of Bell Labs, performed an
experiment where they fired electrons of energy E at a crystalline nickel target. The
measurements shows that:
𝜆=
ℎ
√2𝑚𝐸
The resulting diffraction pattern matched the predictions of the de Broglie wavelength.
Neutral atoms
Experiments with diffration and reflection of neutral atoms confirm the application of the de
Broglie hypothesis to atoms, i.e. the existence of atomic waves which undergo diffraction,
interference and allow quantum reflection by the tails of the attractive potential.
This effect has been used to demonstrate atomic holography, and it may allow the
construction of an atom probe imaging system with nanometer resolution. The description of
these phenomena is based on the wave properties of neutral atoms, confirming the de Broglie
hypothesis.
Waves of molecules
Recent experiments even confirm the relations for molecules and even macromolecules,
which are normally considered too large to undergo quantum mechanical effects. In 1999, a
research team in Vienna demonstrated diffraction for molecules as large as fullerenes. The
researchers calculated a De Broglie wavelength of the most probable C 60 velocity as 2.5
picometer.
In general, the De Broglie hypothesis is expected to apply to any well isolated object.
Macroscopic Objects & Wavelength
Though de Broglie's hypothesis predicts wavelengths for matter of any size, there are realistic
limits on when it's useful. A baseball thrown at a pitcher has a de Broglie wavelength that is
smaller than the diameter of a proton ... by about 20 orders of magnitude. The wave aspects
of a macroscopic object are so tiny as to be unobservable in any useful sense.
Formation of particle from a wave: Pair Production
Pair production is the formation or materialization of two electrons, one negative and the
other positive (positron), from a pulse of electromagnetic energy traveling through matter,
usually in the vicinity of an atomic nucleus.
Pair production is a direct conversion of radiant energy to matter.
It is one of the principal ways in which high-energy gamma rays are absorbed in matter.
For pair production to occur, the electromagnetic energy, in a discrete quantity called a
photon, must be at least equivalent to the mass of two electrons.
The mass m of a single electron is equivalent to 0.51 million electron volts (MeV) of energy E
as calculated from the equation formulated by Albert Einstein, E = mc2, in which c is a
constant equal to the velocity of light. To produce two electrons, therefore, the photon energy
must be at least 1.02 MeV. Photon energy in excess of this amount, when pair production
occurs, is converted into motion of the electron-positron pair. If pair production occurs in a
track detector, such as a cloud chamber, to which a magnetic field is properly applied, the
electron and the positron curve away from the point of formation in opposite directions in arcs
of equal curvature. In this way pair production was first detected (1933). The positron that is
formed quickly disappears by reconversion into photons in the process of annihilation with
another electron in matter.
References
In addition to the references books given for this course the following references are
adviced.
http://physics.about.com/
http://abyss.uoregon.edu/~js/glossary/
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/
http://www.physlink.com/
http://www.wikipedia.org/