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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
    (fades away).
   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
    multi-electron atoms is therefore much more complicated. To learn more, you
    may enroll in my course on the Interstellar Medium.

				
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