Quantisation of Energy

1
Quantisation of Energy

• Black Body Radiation
• Bohrs Atom
• Franck-Hertz Experiment

2
Black Body Radiation

• Black Body
• An ideal radiator eg. Cavity radiator
• Radiation within cavity in thermal equilibrium
  with surface
• Spectrum observed from small hole in cavity is
  continuous

3
BBR Spectrum

• Independent of material, shape etc. of the cavity
• Depends only on temperature
• Quantified by spectral emittance (W/m2 Hz)
• Total emittance is s T4
• Radiation within cavity quantified by spectral
  energy density

4
Attempts at Fitting Spectral Energy Density

• Weins Formula
• Rayleighs Formula
• Plancks Formula

5
Weins Formula

• Used Maxwells equation and thermodynamics
• U(v, T) A v3 e-bn/T
• A and b are fitting constants
• O Fails at low frequencies

6
Rayleighs formula

• Assumed that the radiation within the cavity were
  standing waves
• U(v, T) 8p/c3 . v2 kT
•  
• Suffered uv catastrophy

7
Plancks Formula

• Proposed empirical law
• Note that he proposed the formula before having
  the explanation
• Treated walls as consisting of SHOs
• Each SHO exchanged energy with cavity depending
  on its frequency v
• The oscillators absorbed/emitted energies 0, hv,
  2hv, 3hv..
• Large number of oscillators resulted in the
  continuous nature of the spectrum

8
Conclusion

•     The oscillators emit and absorb energy in
  discrete quantities (quanta) nhv

9
Interesting exercise Show that for low (high)
frequencies Plancks equation yields
Rayliegh-Jeans (Weins) formula.
10
Bohrs Atom

•  If atom consists of positive center with
  electron in orbit..
• atom would radiate itself into nonexistence
• Radiation from atom could be due to electron
  vibration (acceleration).
• continuous spectrum expected
• Line spectrum observed

11

• Balmer proposed equation

for n 3,4,5.
12
Bohrs postulates

• Only certain discrete orbits (stationary states)
  are allowed for the electron
• Electron in a stationary state does not radiate
• Classical mechanics apply to electron in a
  stationary state (not between states)
• When an electron moves from one SS to another, a
  change in energy occurs involving the emission
  (or absorption) of a single photon of frequency v
  DE/h
• Permitted orbits (SS) are those in which angular
  momentum can take on only the discrete values
  nh/2p

13
Applying the postulates and applying coulombs
force as the centripetal force he obtained
14
Conclusion

• Energy changes in atom occur in discrete
  quantities.
• Angular momentum of electron within the atom is
  also quantised.

15
Franck-Hertz Experiment

• Electrons accelerated through mercury vapour
• Accelerating voltage increased gradually
• Measured resulting current

16
observation

• Current initial increases
• Sharp decrease at specific voltages
• Increases thereafter

17
Explanation

• Elastic collision between electrons and Hg atoms
• At specific voltages, inelastic collisions occur
• Electron gives all its energy to the electron of
  the atom
• Causes excitation of atom
• Having given up its energy the electron cannot
  overcome the slight negative pd at collector
• Spectral lines observed only after excitation

18
Conclusion

• Discrete energy levels do exist within the atom
• Atoms can accept energy corresponding only to
  transitions between these energy levels
• The spectral lines correspond to transition
  between the energy levels

19
Photon

• Photoelectric Effect
• Comptons Effect
• Read The Tiger and the Shark by B.R. Wheaton

20
Photoelectric Effect

•  Recall
• Light is an electromagnetic wave
• Maxwell
• Hertz demonstrated Maxwells waves experimentally
• Youngs interference
• Only waves can have interference pattern
• Einstein proposed that not only did the
  oscillators of the BB emit/absorb quanta of
  energy but the radiation itself exists in these
  quanta.

21
Classical Expectations of PE

• Kinetic energy of emitted electrons should depend
  on light intensity
• A time lag should exist between illumination and
  electron emission
• Electrons should be emitted at all frequencies

22
Observations

• A threshold frequency below which no electrons
  are emitted.
• The threshold frequency depends on the metal
  used.
• Instantaneous emission for v vT
• Maximum KE depends on v and not on intensity.
• Current increases with intensity

23
Einsteins Explanation

• Energy in light is not distributed in space.
• Rather light consists of discrete quanta of
  energy
• E hv
• These photons collide with the electrons of the
  metal.
• If the energy is great enough causes the emission
  of electron.

24
Conclusion

• Energy in light is localised in space.
• Energy of light depends on its frequency.
• Upon colliding a photon gives up all of its
  energy to the electron.

25
Comptons Effect

• X-ray scattering off graphite
• Classical expectation
• Incident and scattered X-rays have equal
  wavelengths.

26
Observation

• Scattered X-rays have longer wavelengths
• Change in wavelength is independent of incident
  wavelength
• Greater the angle of scattering results in
  greater increase in wavelength

27
Resolution

• X-rays consists of photons
• Photons collide with electrons
• Energy and momentum are conserved during the
  collision
• The recoiling electron absorbs some of the energy
  of the photon resulting in a longer wavelength
  X-ray

28
Conclusion

• X-rays (part of the EM spectrum) behave like
  particles having localised energy and momentum

29
Wave Particle Duality

• Wave particle duality observed for light
• Electromagnetic wave
• Youngs Exp.,Maxwell, Hertz
• Photon
• Photoelectric Effect
• Compton Effect
• De Broglie proposed wave nature of matter
• l h/p
• Confirmation
• Davisson Germer experiment
• Electron interference
• Particle
• Spot on florescence screen