# Wave Particle Duality of Light

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```					The Interaction of Light and Matter

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Wave-Particle Duality
   Today, we know (have shown) that elementary particles (e.g., protons, electrons,
neutrons), atoms, and molecules exhibit wave-like properties.
   This revolutionary understanding of the behavior of matter was
first proposed in 1924 by the French physicist Louis de Broglie
in his PhD thesis.
   Recall that, in 1905, Einstein used Planck’s idea that the energy
of an electromagnetic wave is quantized in the manner

whereby a quantum of energy is carried by a photon, to explain      Louis de Broglie,
the photoelectric effect.                                             1892-1987

   Recall that from the Theory of Special Relativity, the energy and momentum of a
photon are related by

as had been verified in 1922 by Compton through the Compton effect.
Wave-Particle Duality
   Purely from symmetry arguments, de Broglie proposed that all matter (from
elementary particles to people, planets, stars, and entire galaxies) also exhibit
wave-like properties such that their frequency and wavelength are given by

   In this view, the wave-particle duality applies to everything in the physical world:
- everything exhibits its wave properties in its propagation
- everything manifests its particle nature in its interactions
Wave-Particle Duality
   In 1927, Clinton J. Davisson and Lester H. Germer directed a beam of electrons
on a highly polished single crystal of nickel. The lines of atoms on the surface
form a sort of diffraction grating for the electron waves with a spacing equal to the
lattice spacing of the nickel crystal of 0.215 nm, about the size of an atom.

Clinton J Davisson (1881-1958; left)
and
   The electron beam they used had an energy of 54 eV,      Lester H. Germer(1896-1971; right)
corresponding to a de Broglie wavelength of 0.167 nm.
The first-order (m=1) maximum should therefore occur
at  = sin-1 (/d) = 51o, in agreement with their measurements.
Wave-Particle Duality
   Modern experiments such as that performed in 1989 by A. Tonomura and his
colleagues at the Hitachi Advanced Laboratory and Gakushuin University in
Tokyo (based on Clauss Jönsson’s experiment in 1961) is equivalent to the
double-slit experiment for light.
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   To produce an interference pattern, each electron must pass through both slits,
thus revealing the electron’s wave-like properties.
Wave-Particle Duality
   Similar experiments have been performed on other elementary particles as well as
atoms.
Wave-Particle Duality
   de Broglie reasoned that the quantization of orbital angular momentum in Bohr’s
model of the atom is simply a manifestation of the wave-like nature of the
electron. The circumference of an electron’s orbit must be equal to an integral
number of wavelengths for the electron to undergo constructive interference.
Otherwise, the electron will find itself out of phase and suffer destructive
interference.
   Based on this consideration, one can show that
the electron can only have angular momenta
given by                     .

Assignment
question
Probability Waves
   Quantum mechanics describe particles in terms of probability waves.
   Consider a particle that comprises the following probability wave, Ψ, a sine wave
with a precise wavelength  propagating along the x-direction

Probability wave Ψ :                                (x,t) = 0 ei(kx-t)
x     where k = 2 / 
=2ν
The momentum, p = h / , of a particle described by such a wave is known
precisely as the wavelength is known precisely.
   The probability of finding the particle at a given location x is given by

P(x) =  *                                                                   =
[0 ei(kx-t)] [0 e-i(kx-t)]
= |0|
which is a constant independent of x or t. Thus, the particle can be found with
equal probability at any point along the x-direction: its position is perfectly
uncertain; i.e., a sinusoidal wave has no beginning or end.
Probability Waves
   Consider now a particle that has a probability wave, Ψ, that is equal to the
addition of several sine waves with different wavelengths
Probability wave Ψ :

x

The position of such a particle can be determined with a greater certainty because
P(x) =  * is large only for a narrow range of locations.
   On the other hand, because Ψ is now a combination of waves of various
wavelengths, the particle’s momentum, p = h / , is less certain.
   This is nature’s intrinsic tradeoff: the uncertainty in a particle’s position, Δx, and
the uncertainty in its momentum, Δp, is inversely related. As one decreases, the
other must increase. This fundamental inability of a particle to simultaneously
have a well-defined position and a well-defined momentum is a direct result of the
wave-particle duality of nature.
Probability Waves and the Two-Slit
Experiment
 Let us see how we can use the idea of particles as
probability waves to explain the two-slit experiment
for electrons.
   To produce a sharp interference pattern, we need to
precisely control the wavelength, and therefore
momentum, of the electron beam. Otherwise, no
interference pattern is seen; i.e., if the incident wave
has a range of wavelengths, constructive or
destructive interference will occur at different
locations at different wavelengths and hence the
interference pattern smear out.
   If we precisely control the momentum and hence wavelength but not the position
of the electrons, they can pass through either slits and interfere with each other.
Heisenberg’s Uncertainty
Principle
 In 1927, the German physicist Werner Heisenberg presented the
theoretical framework for the inherent “fuzziness” of nature,
showing that the uncertainty in a particle’s position, Δx, and the
uncertainty in its momentum, Δt, is related by

   He also showed that the uncertainty of energy measurement, ΔE,
and the time interval over which this measurement is taken, Δt, is   Werner Heisenberg,
1901-1976
related by

As the time available for an energy measurement increases, the
inherent uncertainty in the result decreases. Thus, making more
precise measurements require longer observing times.
Quantum Mechanical Tunneling
   Because of the inherent uncertainty between a particle’s momentum and position,
a particle can penetrate a barrier even though the particle’s energy is lower than
the barrier potential.
   If you know the energy of the particle precisely (and it is smaller than the barrier
potential), you do not know the position of the particle precisely and so it can be
on either side of the barrier (provided the barrier width is sufficiently small).
   So, a (small) fraction of particles on one side of a barrier can tunnel through to the
other side, an effect called quantum mechanical tunneling.

Probability wave Ψ :

x
Quantum Mechanical Tunneling
   A corresponding effect is seen in a classical wave such as a water of light wave.
When such a wave enters a medium through which it cannot propagate, its
amplitude decays with distance in this medium: the wave becomes evanescent
   If the barrier width is sufficiently small, the amplitude of the wave may not decay
away completely before reaching the other side of the barrier, where the wave can
once again propagate.
Quantum Mechanical Tunneling and Stellar Nuclear
Fusion
 Quantum mechanical tunneling plays an
essential role in stellar nuclear fusion.
   The first step in converting hydrogen to
helium is the fusion of two protons to
produce deuterium (proton + neutron nucleus)
along with a positron and a γ-ray photon.
   For two protons to overcome their Coulomb
repulsion and approach each other
sufficiently close for their strong nuclear
force to bind them together, their required
kinetic energy corresponds to a gas
temperature of ~1010 K.
Quantum Mechanical Tunneling and Stellar Nuclear
Fusion
 The central temperature of the Sun is only
1.57 × 107 K. Even taking into consideration
the fact that a significant number of particles
have speeds well in excess of the average
speed (i.e., particle speeds are distributed
according to the Maxwell-Boltzmann
distribution), not enough protons can
overcome their Coulomb repulsion to
produce the Sun’s observed luminosity.
   Quantum mechanical tunneling helps to
overcome the Coulomb repulsion between
protons for nuclear fusion to proceed.
Heisenberg’s Uncertainty Principle and the Atom
   Heisenberg’s uncertainty principle implies that electrons cannot have well defined
orbits as prescribed in Bohr’s model of the atom. Rather, the electron orbits must
be imagined as fuzzy clouds of probability, with the clouds being mode “dense” in
regions where the electron is more likely to be found.
   One of the predictions of such a model is that spectral lines cannot be infinitely
sharp but must have a certain width (natural linewidth), as is indeed observed.
Bohr’s model                             Quantum mechanical model
for the hydrogen atom                          for the hydrogen atom
Schrödinger’s
Equation
 Motivated to find a proper wave equation for the electron, in
1926 the Austrian physicist Erwin Schrödinger formulated the
wave equation

now known as Schrödinger’s equation that describes how the
Erwin Schrödinger,
quantum state of a physical system changes in time.                         1887-1961
   The Schrödinger equation can be solved analytically for the hydrogen atom,
giving exactly the same set of allowed energies as those obtained by Bohr. In
addition to the principal quantum number n, Schrödinger found that two other
quantum numbers, and , are required for a complete description of the
electron orbitals such that the angular momentum

where                      an is the orbital quantum number and n is the
principle quantum number. For historical reasons related to how spectral lines
were first designated, = 0, 1, 2, 3, 4, 5, etc. are referred to as s, p, d, g, f, h etc.
Schrödinger’s
Equation
 The spin magnetic quantum number,        , yields the projection of the orbital
angular momentum in a specified direction (z-axis). The z-component of the
angular momentum vector, Lz, can only have values              , with     equal to
any of the      integers between      and       inclusive.
   Thus, the angular momentum vector can point in           different directions.
   E.g., for n = 2, = 0, 1 and     = −1, 0, +1.
Quantum States of the Hydrogen
Atom
 Quantum numbers and energies
for the ground (n = 1) and first
excited state (n = 2) of the
hydrogen atom.
   In the absence of any preferred
direction in space (e.g., as
defined by an electric or
magnetic field), different
orbitals with the same principal
quantum number n and therefore
the same energy are said to be
degenerate.
The Zeeman Effect
   An electron in an atom will feel the effect
of a magnetic field: the magnitude of this
effect depends on the electron’s orbital
motion (i.e., magnitude and orientation of
the electron’s orbital angular momentum
through the magnetic quantum number lll
) and magnetic field strength B.
   Electron orbitals with the same n and
but different values      therefore have
(slightly) different energies. The splitting
of spectral lines in the presence of a
magnetic field is called the Zeeman
effect.
   In the example shown, the three
frequencies of the split line are given by
The Zeeman Effect
   The Zeeman effect provides the only direct measure of magnetic field strengths in
astrophysics. (There are several indirect methods to estimate magnetic field
strengths).

slit

spatial dimension
along slit
λ
Electron Spin
   More complicated splitting patterns of spectral lines by magnetic fields are
sometimes seen, usually involving even number of unequally spaced spectral
lines. This effect is called the anomalous Zeeman effect.
   In 1925, George Uhlenbeck, Samuel Goudsmit, and Ralph Kronig suggested a
fourth quantum number that describes the spin of the electron. The electron spin
is not a classical top-like rotation (although this is often drawn for visualization
purposes) but a purely quantum effect.
   Each electron orbital (or quantum state) is therefore associated with four quantum
numbers. How many electrons can occupy the same quantum state?
   In 1925, based on empirical knowledge of the properties of
atoms, the Austrian theoretical physicist Wolfgang Pauli realized
that no two electrons can share the same four quantum numbers, a
rule of nature now known as the Pauli exclusion principle.
   In 1930, the Indian astrophysicist Subrahmanyan Chandrasekhar
used the Pauli exclusion principle to show that degenerate
electron pressure supports white dwarfs against collapse due to     Wolfgang Pauli, 1869-
1955
their own gravity.
Electron Spin
   In 1928, the English physicist Paul Dirac combined Schrödinger’s equation with
Einstein’s theory of special relativity to produce a relativistic wave equation for
the electron.
   Dirac’s solution
- naturally included the spin of the electron
- naturally explained Pauli’s exclusion principle as being applicable to all
particles with spin of an odd integer times      (such as electrons, protons, and
neutrons) known collectively as fermions
- particles (such as photons) that have an integral spin do not obey Pauli’s
exclusion principle and are known as bosons
- predicted the existence of antiparticles (identical to their corresponding
particles except for their opposite electrical charges and magnetic moments)
   Dirac showed that the electron’s spin angular momentum S is a vector of constant
magnitude

with a projection along the z-axis of             , with ms = ±½.
The Complex Spectra of
Atoms
 In summary, an electron in an atom is described by four quantum numbers
-   principal quantum number, n
-   orbital quantum number,
-   magnetic quantum number,     =       …
-   spin quantum number, ms = ±½
   To further complicate matters, the nucleus also has a spin quantum number!
   In an atom/ion with a single electron, the only interaction is between the electron
and the nucleus. The spectrum of such an atom/ion is hydrogen-like.
   In a multielectron atom, electrons not only interact with the nucleus but also with
each other through their spins and orbits angular momenta; i.e., spin-spin
interactions, orbit-orbit interactions, and spin-orbit interactions. The spectrum of