Quantum Theory 1

1
Quantum Theory 1
2
Topics

• Discovery of the Electron
• Millikans Experiment
• Blackbody Radiation
• An Act of Desperation
• Summary

3
The Discovery of the Electron

• 1838 M. Faraday
• Discovery of arc light in dilute gas
• 1857 - H. Geissler
• Discovery of fluorescence in dilute gas
• 1880s W. Crookes
• Development of cathode ray tube
• 1895 J. Perrin
• Cathode rays found to be negatively charged

4
The Discovery of the Electron

• 1896 P. Zeeman
• First evidence of atomic particles with a
  specific charge-to-mass ratio
• 1897 J.J. Thomson
• Measurement of charge-to-mass ratio of cathode
  ray particles.
• In effect, this was the discovery of the electron
  and the dawn of modern physics

5
The Discovery of the Electron
C cathode A,B slits D,E deflection plates
6
The Discovery of the Electron
C cathode A,B slits D,E deflection plates
Thomson found that whatever gas was used he
always got the same q/m ratio of about 0.7 x 1011
C/Kg. The cathode particles were called
corpuscles by Thomson and later electrons by
Lorentz
7
Millikans Experiment
Charged oil drops
Suspended oil drops
Millikan began experiments in 1909 He measured e
to be e 1.601 x 10-19 C
8
Blackbody Radiation
In 1879, Josef Stefan found an empirical
formula to describe the power radiated by an
ideal black body
s 5.6703 x 10-8 W/m2K4
Josef Stefan 1835 1893
Five years later, Ludwig Boltzmann was able
to derive this formula from thermodynamics
9
Blackbody Radiation
Spectral distribution found to depend only on
absolute temperature T
10
Blackbody Radiation
In the late 19th century, the race was on
to derive, from first
principles, the black body spectrum
The task was to compute the energy density of
radiation within a cavity
11
Rayleigh-Jeans Law
In 1900, Lord Rayleigh showed that the
energy density of radiation in a cavity must have
the form
In 1905, Sir James Jeans showed that a 8pk,
where k is Boltzmanns constant
Lord Rayleigh 1842 1919
12
Rayleigh-Jeans Law
Unfortunately, this law predicted an absurdity
the total electromagnetic energy in a cavity is
infinite!
Lord Rayleigh 1842 1919
This result, called the ultraviolet catastrophe,
revealed a serious flaw in classical physics
13
An Act Of Desperation
14
An Act of Desperation

• The October Revolution
• In early October 1900, Max Planck found that the
  formula
• could reproduce the experimental results

15
An Act of Desperation
16
An Act of Desperation

• Planck was to make a presentation at the German
  Physical Society meeting on October 19, 1900.
• But since he had arrived at his formula through
  inspired guesswork, he was rather anxious to
  derive his formula in a more respectable way!

17
An Act of Desperation

• Plancks Model
• A cavity is modeled as a system of N oscillators
  (presumably, atoms) that can emit and absorb
  electromagnetic radiation
• The radiation can be distributed amongst the
  oscillators in a large number of ways W
• Ludwig Boltzmann had earlier introduced the
  formula S k ln W, for the total entropy of a
  system

18
An Act of Desperation

• Plancks Model
• Planck computed the entropy per oscillator using
  a model in which the radiation is absorbed and
  emitted in discrete amounts
• He regarded this assumption as nothing more than
  a mathematical trick, an act of desperation.
  But, try as he might, he could not make headway
  without it

19
An Act of Desperation

• Plancks Model
• For a given l, Planck assumed that the energy in
  the cavity is in the form of M indistinguishable
  quanta of energy e distributed over N
  indistinguishable oscillators
• The total energy of the radiation is Me. So the
  average energy per oscillator is E M e / N
• Planck derived S, the entropy per oscillator,
  i.e., S / N

20
An Act of Desperation

• Plancks Model
• By computing the number of ways W to distribute M
  indistinguishable quanta amongst N
  indistinguishable oscillators he showed that the
  entropy per oscillator has the form
• He then had to determine the value of e

21
An Act of Desperation

• The Reluctant Revolutionary
• To determine e, Planck used the thermodynamic
  relation 1/T ?S/?E, between absolute
  temperature T, entropy S and average energy E, to
  derive the entropy per oscillator, starting with
  his formula for the blackbody spectral density
• When he compared his two expressions for the
  entropy he found they agreed only if he assumed
  each quantum had energy

22
An Act of Desperation
And thus did this reluctant revolutionary start,
in an act of desperation, the quantum
revolution
The constant, h, which does not appear in
classical physics is called Plancks constant
23
Extra Credit Problem

• Using Plancks model, derive Plancks formula for
  the entropy per oscillator
• Then use 1/T ?S/?E to show that the average
  energy per oscillator is given by
• E e /(ee/kT - 1)
• Hint
• for large N, ln N! N ln N N
• and E e M / N

due Feb. 22
24
Extra Credit Problem

• Plancks Model
• The number of ways of distributing M
  indistinguishable quanta amongst N oscillators is

25
Summary

• Discovery of Electron
• J.J. Thomson measured e/m ratio in 1897
• Millikan measured e in 1909
• Ultraviolet Catastrophe
• Rigorous application of the laws of
  thermodynamics, Newton and Maxwell by Lord
  Rayleigh led to an absurd prediction a hot oven
  should emit an infinite amount of energy!

26
Summary

• The Birth of Quantum Physics
• In 1900, Planck derived the correct formula for
  the blackbody spectral density under the
  assumption that electromagnetic energy changes in
  discrete amounts given by e hc/l, an assumption
  he regarded as an act of desperation
• He spent many years, thereafter, trying to evade
  this assumption but failed!