Lecture 4' Particle properties of waves

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Lecture 4. Particle properties of waves
v/c
Relativistic mechanics, El.-Mag. (1905)
Relativistic quantum mechanics (1927-)
Classical physics
Quantum mechanics (1920s-)
h/s
S the actionmomentum?distance, units g?cm2/s

• Outline
• Light waves vs. particles
• Photons
• Photoelectric effect (demonstration of the
  energy of photons)

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Historical development
Newton (Opticks, 1704) light as a stream of
particles (corpuscles). Descartes (1637),
Huygens, Young, Fresnel (1821), Maxwell by
mid-19th century, the wave nature of light was
firmly established (interference and diffraction,
transverse nature of e.-m. waves). Physics of
the 19th century mostly investigation of light
waves physics of the 20th century interaction
of light with matter. One of the challenges
understanding the blackbody spectrum of thermal
radiation (to be considered later in the course).
Planck (1900) suggested a solution based a
revolutionary new idea emission and absorption
of electromagnetic radiation by matter has
quantum nature the energy of a quantum of e.-m.
radiation emitted or absorbed by a harmonic
oscillator with the frequency f is given by the
famous Plancks formula
where h is the Plancks constant
- at odds with the classical tradition, where
energy was always associated with amplitude, not
frequency
Also, in terms of the angular frequency
where
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Historical development (contd)
This progress leads to the concept of photons as
quanta of the electromagnetic field. However,
Planck thought that the quantum nature of light
reveals itself only in the processes of
interaction with matter. Otherwise, he thought,
classical Maxwells equations adequately
describe all e.-m. processes.
Einstein (1905) put forward more radical ideas.
He is purported to have said, I know that beer
comes in pint bottles (in referring to the
quantum features of blackbody radiation). What I
want to know is whether all beer comes in pint
bottles (that is, whether the quantum
character of the electromagnetic field has its
origins in the atoms or as an intrinsic property
of the field). H.J. Kimble, Strong Interactions
of Single Atoms and Photons in Cavity QED,
Physica Scripta T76, 127 (1998) (posted on our
Webpage).
Even when light propagates in free space, it can
be thought of as beam of quantum particles (not
old classical particles, as Newton thought).
Although surely the correct description of the
electromagnetic field is a quantum one, just as
surely the vast majority of optical phenomena are
equally well described by a semiclassical theory,
with atoms quantized but with a classical field.
... The first experimental example of a
manifestly quantum or nonclassical field was
provided in 1977 with observation of photon
anti-bunching for the fluorescent light from a
single atom (PRL 39, 691 (1977)). H.J. Kimble,
Physica Scripta T76, 127 (1998).
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Waves
Wave equation in one dimension for any quantity
??
Solution a plane wave traveling in the negative
(positive) direction x with velocity v
v the phase velocity
Harmonic plane wave traveling in the positive
direction x
?
angular frequency
wave number
A0
Electromagnetic waves (transverse in free space)
t
-A0
T
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Photons
According to the quantum theory of radiation,
photons are massless bosons of spin 1 (in units
h). They move with the speed of light
Quantum character of this equation is illustrated
by the fact that the energy is associated with
the frequency of oscillations rather than their
amplitude.
Light a shorthand notation for any e.-m.
radiation (? from 0 to ?).
The phase is a Lorentz-invariant
quantity, the (scalar) product of two 4-vectors
Particle properties of light
Wave properties of light
- both the time-like and space-like components
of these 4-vectors should transform under L.Tr.
in a similar way
Thus, if Plancks idea Eh? is correct, than we
must conclude that
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Some numbers
Visible light ? 0.4 0.8 micrometers
violet
red
for comparison, the momentum of an electron with
K3.1eV
more than two orders of magnitude greater than
for a photon with this energy!
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Photoelectric Effect
Historical Note The photoelectric effect was
accidentally discovered by Heinrich Hertz in 1887
during the course of the experiment that
discovered radio waves. Hertz died (at age 36)
before the first Nobel Prize was awarded.
Observation when a negatively charged body was
illuminated with UV light, its charge was
diminished. J.J. Thomson and P. Lenard
determined the ration e/m for the particles
emitted by the body under illumination the same
as for electrons. The effect remained
unexplained until 1905 when Albert Einstein
postulated the existence of quanta of light --
photons -- which, when absorbed by an electron
near the surface of a material, could give the
electron enough energy to escape from the
material. Robert Milliken carried out a careful
set of experiments, extending over ten years,
that verified the predictions of Einsteins
photon theory of light. Einstein was awarded the
1921 Nobel Prize in physics "For his services to
Theoretical Physics, and especially for his
discovery of the law of the photoelectric
effect." Milliken received the Prize in 1923 for
his work on the elementary charge of electricity
(the oil drop experiment) and on the
photoelectric effect.
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Photoelectric Effect (contd)
Parameters intensity (S) and frequency (f) of
light, applied voltage (V), measured photocurrent
(I)

• Observations
• For a given material of the cathode, the
  stopping voltage does not depend on the light
  intensity.
• The saturation current is proportional to the
  intensity of light at f const.
• Material-specific red boundary f0 exists no
  photocurrent at f lt f0.
• Practically instantaneous response no delay
  between the light pulse and the photocurrent
  pulse (many applications are based on this
  property)

I
stopping voltage
V0
V
intensity of light increases, f const
retarding
I
cesium
calcium
stopping voltage
V
f
V0(f2)
V0(f1)
intensity const, f increases
red boundary
f0
f2 gt f1
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An attempt to explain Ph.E. using the wave
approach
1.
Classical equation of motion of an electron in
the light wave
Observation Kmax (and, thus, the stopping
potential) does not depend on the intensity S,
and is proportional to ?.
The photocurrent for sodium can be observed at S
as low as 10-6 W/m2. Number of electrons in the
volume 1m?1m ? 1 monolayer 1019. Thus, for a
single electron,
2.
Observation almost instantaneous response
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Photon-based explanation of Ph. E.
Absorption of a photon by an electron in
metal (inelastic collision between these
particles)
after
before
However, weve concluded that a free electron
cannot absorb a photon!
the rest RF of an electron after the collision
after
before
Whats wrong? The electron is not free, it is
embedded in metal, and the chunk of metal is the
second body that participates in the collision
energy conservation
momentum conservation
(see Slide 6)
energy conservation
Thus, while the electron is still inside metal
momentum conservation
The photon energy is absorbed by an electron (the
energy absorbed by metal is negligibly small),
but the momentum exchange between electron and
metal is crucial for momentum conservation.
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Photon-based explanation of Ph. E. (contd)
In the experiment, the electron is observed
outside the metal. It takes some energy to escape
(consider an attraction between an electron and
the positive image charge induced on the metal
surface)
metal
q
q-
The escape energy the work function W
(material-specific)
Thus, for the electron outside metal
red boundary of Ph. E.
Plancks constant measurements
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Photon-based explanation of Ph. E. (contd)

• Observations
• For a given material of the cathode, the
  stopping voltage does not depend on the light
  intensity
• the energy of photons is determined by the
  light frequency, not intensity
• The saturation current is proportional to the
  intensity of light at f const
• the saturation current is proportional to
  the number of photons, thus to the light
  intensity
• Material-specific red boundary f0 exists no
  photocurrent at f lt f0
• at f lt f0 (hf lt W) the photon energy is
  insufficient to extract an electron from metal
• Practically instantaneous response no delay
  between the light pulse and the photocurrent
  pulse
• single act of e-ph collision

metal
vacuum
energy of a free electron in vacuum with Ke 0
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Problems
1. The work function of tungsten surface is
5.4eV. When the surface is illuminated by light
of wavelength 175nm, the maximum photoelectron
energy is 1.7eV. Find Plancks constant from
these data.

• 2. The threshold wavelength for emission of
  electrons from a given metal surface is 380nm.
• what will be the max kinetic energy of ejected
  electrons when ? is changed to 240nm?
• what is the maximum electron speed?
• the loss of electrons due to the photoelectric
  effect will cause an isolated sphere of this
  metal to acquire a positive charge. Find the
  largest electric potential (in Volts) that could
  be achieved by this sphere for ? 240nm.

(a)
(c)
(b)
electron energy in the (repulsive) electric field
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Inverse Photoelectric Effect (production of
X-rays)
X-rays
?-rays
UV
X-rays
Production
The upper cut-off of the spectrum corresponds
to the full conversion of the electron kinetic
energy into the photon energy.