The birth of quantum mechanics

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The birth of quantum mechanics
Until nearly the close of the 19th century,
classical mechanics and classical electrodynamics
had been largely successful in describing
phenomena in the world. Material particles were
determinate objects that obeyed the laws of
classical mechanics. Electromagnetic waves were
traveling waves of electric and magnetic fields,
in which the waves were continuous and exhibited
phenomena of interference and refraction that
could be explained from their wavelength and
frequency.
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In 1890-1910, there were problems!

• There were situations where electromagnetic waves
  exhibited properties that should be associated
  with particles!
• Black body radiation
• Photoelectric effect
• Frank-Hertz experiment
• Spectra of emission and absorption by atoms
• There were situations where material particles
  exhibited properties that should be associated
  with waves!
• Electron diffraction

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We will study several of these paradoxes, and
arrive at the wave-particle duality that spawned
quantum mechanics

• Light behaves like waves much of the time, but
  like particles some of the time
• Material particles behave like particles much of
  the time, but like waves some of the time.
• A successful description of both light and matter
  must somehow weave together both kinds of
  properties!

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First lets review the statistical mechanics of
material particles.

• (section 1.12 in the text)
• Suppose we want to calculate the dependence of
  the density of air as a function of altitude on
  Earth.
• We can get it by using the ideal gas law,
• pVNRT
• and the principle of detailed balance

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• The ideal gas law relates pressure p (force per
  unit area)
• to the number of moles N, the volume V, and the
  temperature T. R is the universal gas constant.
• For describing a gas, it is more convenient to
  use the number density of particles nNaN/V of
  particles
• where k 1.4 x 10-23 J/oK 10-4 eV/oK
• is Boltzmanns constant.

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How does the atmosphere vary as a function of
altitude?

1. A molecule must have work done upon it to elevate
   by a distance dz
2. The density of air decreases with altitude n(z)

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Detailed Balance
We require that the force on each particle of gas
be in balance (otherwise it would rise or
fall). Consider a horizontal slice of the
atmosphere at altitude z face area A, thickness
dz, mass of each particle m Downward force due to
gravity pulling on each particle Upward force
due to the pressure difference between the top
and bottom of the slice
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Now require Fup Fdown in each slice of the
sky Now connect n(z) and p(z) through the ideal
gas law So the ideal gas law
becomes Solution The air density decreases
exponentially, with scale length
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This result is easily generalized to any
situation where material particles are
distributed in a volume of space where the
potential energy that varies over the
region

• This is the Boltzmann distribution function. It
  governs the distribution of particles in the
  presence of any interaction potential.

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Black Body Radiation
When a material body is heated, it emits
electromagnetic radiation with a broad
spectrum. Mystery 1. It is observed
experimentally that the total intensity (power
per unit area) radiated by a black body is
determined solely by its absolute temperature.
There is no way to explain this result by
treating light as a wave!
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Stephan-Boltzmann law
? 5.7 x 10-8 W/m2/oK4
Example A steel rod is heated red hot (T 700
oC 1,000 oK). The rod is 1 cm diameter and 1 m
long. How much power does it radiate as
blackbody radiation? P ? T4 A (5.7 x 10-8
W/m2/K4)(1,000 K)4 (? x 10-2m)(1 m) 1,800 W.
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Mystery 2. The spectrum of light from blackbody
radiation cannot be explained by assuming that
the light is composed of waves.
The spectrum of light is the pattern of intensity
as a function of wavelength
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The spectrum of black-body radiation can be
explained (up to a point!) if we consider the
radiation to be produced by oscillations of the
atoms in the material.

• The light emitted should be proportional to the
  number of modes in which the oscillations of a
  given wavelength can be excited

Consider the modes that can be excited within a
cubic cavity of dimension L. modes in x Nx
L/l modes in y Ny L/l modes in z Nz L/l
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The power radiated in wavelength interval dl is
proportional to the fraction of a wavelength dP
dl/l So the energy density ? (J/m3) is
This energy spectrum was derived by Rayleigh and
Jeans by assuming that blackbody radiation is
emitted from atomic oscillators as a wave
process, and that there must be detailed balance
between the standing waves that can be supported
inside the solid and the emitted radiation that
comes out.
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This Rayleigh-Jeans theory is not too bad in its
description of the spectrum of long-wavelength
light. Unfortunately, it leads to an ultraviolet
catastrophe The power radiated at short
wavelength (high energy) increases without bound!
High-frequency (low-?) cutoff requires
factor
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Plank realized that he could (empirically) obtain
the observed spectrum IF he assumed that
blackbody radiation behaved as if it were emitted
by oscillators that could only change energy by
integer multiples of some minimum energy step u
E muThen he would have from the Boltzmann
distribution
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The total energy of all the oscillators emitting
this particular energy Emu is then
Following the derivation in the book, we
calculate the average energy of an
oscillator
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Put this together with the Rayleigh-Jeans result
for the number of oscillators of wavelength ?
This spectrum matched experimental observation
only if the energy u were inversely proportional
to wavelength ?
h 6.6 x 10-34 J s 2,000(2?) eV Å/c
This only makes sense if light is emitted in
quantized packets it behaves like a particle
when it is emitted!
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Example We can estimate the temperature at the
surface of a star by determining the wavelength
corresponding to the maximum intensity in its
spectrum, and assume that the emission is a
blackbody spectrum. This wavelength is 550 nm
(red) for the Sun, 430 nm (blue) for the North
Star, and 290 nm (ultraviolet) for Sirius.
Calculate the surface temperatures.
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This is a transcendental equation. We must
obtain an approximate solution. The solution
will be near ? 5.
hc 2000 eV Å 200(2?) eV nm
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Sun ?pk 550 nm T 4,600 oK North Star ?pk
430 nm T 5,800 oK Sirius ?pk 290 nm T
8,700 oK