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Modern Physics Lecture III. The Quantum Hypothesis. In this lecture we examine the evidence for “light quanta” and the implications of their existence Waves as Particles The photoelectric effect Compton scattering Particles as Waves Electron diffraction The Double Slit Revisited.

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Modern physics lecture iii

Modern PhysicsLecture III

The quantum hypothesis
The Quantum Hypothesis

  • In this lecture we examine the evidence for “light quanta” and the implications of their existence

    • Waves as Particles

      • The photoelectric effect

      • Compton scattering

    • Particles as Waves

      • Electron diffraction

    • The Double Slit Revisited

Modern physics lecture iii

Photoelectric effect

  • When light strikes the cathode, electrons are emitted

  • Electrons moving between the two plates constitute a current

Photoelectric effect
Photoelectric Effect

  • Properties of the photoelectric effect

    • Electrons are only emitted above a certain “cut-off” frequency

    • This frequency is different for different materials

    • It is called the “work function”

    • Below the “work function” no electrons are emitted no matter how intense the light is

    • The maximum energy of the ejected electron is Kmax=eDVs

Photoelectric effect1
Photoelectric Effect

  • Properties

    • No photoelectrons are emitted if the frequency falls below some cut-off frequency fc

    • The maximum energy of the photons is independent of the light intensity

    • The maximum kinetic energy of the photoelectrons increases with increasing frequency

    • Photoelectrons are emitted almost instantaneously from the surface

Photoelectric effect2
Photoelectric Effect

  • Explanation

    • Einstein extended Planck’s explanation for blackbody radiation to suggest that in fact the quanta of energy used in blackbody radiation are in fact localised “particle like” energy packets

    • Each having an energy given by hf

    • Emitted electrons will have an energy given by

      • Where f is known as the “work function” of the material

Photoelectric effect3
Photoelectric Effect

  • Quantum interpretation

    • If the energy of a photon is less than the work function f, the photon cannot give enough energy to the electron to leave the surface

    • Kmax does not depend on light intensity, because doubling the number of photons would only double the number of electrons and not double their energy

    • Kmax increases with frequency because energy and frequency are related

    • If light is particle-like, then all of the energy will be delivered instantaneously thus liberating an electron with no time delay between the light hitting the surface and the electron escaping

Inverse photoelectric effect production of x rays
Inverse photoelectric effect - Production of X-rays

  • Photons are absorbed in whole - electrons can transfer part of their energy.

  • Brehmsstrahlung - electrons decelerate in electromagnetic field of nuclei: Ef = Ei - h , wide distribution

  • Maximal frequency - minimal wavelength (observed empirically first) eV = h max = h c / min

  • Also discrete spectrum (atomic levels)

Roentgen lamp
Roentgen Lamp

Tungsten - wolfram

Compton scattering
Compton Scattering

  • If light is like a particle does it have momentum?

  • In Compton scattering x-rays impart momentum to matter, scattering electrons like billiard balls

  • Thus photons also have momentum. The momentum of a photon is given by

Recoiling electron




Photon, l0


Photon, l’

Photons and electromagnetic waves
Photons and Electromagnetic Waves

  • How can light be considered a photon (particle) when we know it is a wave

  • Light has a dual nature: it exhibits both wave and particle characteristics

    • There is a smooth transition of these properties across the electromagnetic spectrum

    • At low frequencies (radio waves) photons have a vanishingly small energy and the wave properties dominate

    • At high frequencies (x-rays, g-rays) it is the particle properties that dominate


Louis de broglie 1892 1987
Louis de Broglie1892 - 1987

Wave properties of matter
Wave Properties of Matter

  • In 1923 Louis de Broglie postulated that perhaps matter exhibits the same “duality” that light exhibits

  • Perhaps all matter has both characteristics as well

  • Previously we saw that, for photons,

  • Which says that the wavelength of light is related to its momentum

  • Making the same comparison for matter we find…

De broglie wavelength of electrons
de Broglie Wavelength of Electrons

  • We now calculate the wavelength of a charged particle accelerated through potential V

  • Assume that the particles have mass m and charge q

  • Equate kinetic energy of the particles with the electrostatic energy

    K = m v 2/2 = q V

    momentum p = m v

    We can express kinetic energy in terms of momentum

    K = p 2/(2 m) = q V

    Reorganise to get

    p = (2 m q V )1/2

    de Broglie’s hypothesis gives

    l= h / p

    Substitute for pto get

Does matter really have a wavelength
Does Matter Really Have a Wavelength

  • The wavelength of matter waves is very small. This is why we do not see them in our every day experience

  • To see diffraction a grating a very small slit width is required (eg the space between two atoms in a crystal)

  • This is exactly how electron diffraction was first found!

    • G. P. Thompson of Scotland and Davisson and Germer from the USA used the close spacing between atoms in a crystal lattice to diffract electron waves thus proving that matter can also exhibit diffraction and interference

C j davisson and l g germer
C.J.Davisson and L.G.Germer



de Broglie

Electron interference
Electron interference

a, b, c – computer simulation

d - experiment

Electron waves
Electron Waves

  • Electrons with 20ev energy, have a wavelength of about 0.27 nm

  • This is around the same size as the average spacing of atoms in a crystal lattice

  • These atoms will therefore form a diffraction grating for electron “waves”

  • Several pictures are shown left (see the web links on the course home page)

Conflicting hypotheses

Wave Properties

Reflection, Refraction

A property of both particles and waves

Interference and Diffraction

Young’s double slits

Waves Only


Waves Only

Particle properties

Consists of discreet particles

(atoms or molecules)


A well defined trajectory

Does not diffract or interfere

1 particle + 1 particle = 2 particles

Conflicting Hypotheses

Conclusion: Our initial hypothesis is incorrect. We need to form a new hypothesis for the differences between matter and light. One could be that matter has a rest mass!

From this lecture
from this lecture…

  • The solution to the blackbody spectrum leads to the concept of photons, and to a solution for the photoelectric effect

  • The maximum excess energy of a photoelectron is

  • The particle nature of light is also shown by Compton scattering of electrons by photons

  • Scattering shows that photons have momentum given by

  • This implies that matter also has wavelike properties given by the de Broglie formula

  • The de Broglie wavelength leads to phenomena such as electron diffraction. A common tool in modern crystallography