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Wave-Particle
Duality
Dr. Rohit Narula
Electromagnetic
waves or photons?
Wave-Particle Duality
PYL100: Electromagnetic Waves and Quantum
Mechanics (Fall 2016)
Dr. Rohit Narula1
1 Department
of Physics
The Indian Institute of Technology, Delhi
September 15, 2016
Wave-particle
duality
The principle of
spectral
decomposition
Matter or de
Broglie waves
Outline
Wave-Particle
Duality
Dr. Rohit Narula
Electromagnetic
waves or photons?
Wave-particle
duality
Electromagnetic waves or photons?
The principle of
spectral
decomposition
Matter or de
Broglie waves
Wave-particle duality
The principle of spectral decomposition
Matter or de Broglie waves
Is light wave- or particle-like?
Wave-Particle
Duality
Dr. Rohit Narula
Electromagnetic
waves or photons?
Wave-particle
duality
The principle of
spectral
decomposition
Matter or de
Broglie waves
?
Refection of light from a mirror
Wave-Particle
Duality
Dr. Rohit Narula
Electromagnetic
waves or photons?
Wave-particle
duality
The principle of
spectral
decomposition
Matter or de
Broglie waves
Figure: [Image from http://images.tutorvista.com]
I
On observing that light could bounce back upon
reflection from a mirror (See Fig 1), Sir Isaac Newton
deemed light to consist of a beam of particles.
Interference of light
Wave-Particle
Duality
Dr. Rohit Narula
Electromagnetic
waves or photons?
Wave-particle
duality
The principle of
spectral
decomposition
Matter or de
Broglie waves
Figure: [Image from Quantum Mechanics by Claude
Cohen-Tannoudji]
I
I
I
(b) The intensity pattern for each of the slits open
separately.
(c) The observed intensity pattern when both slits are
open simultaneuosly I 6= I1 + I2 .
The pattern on the screen suggests a wave-like
interpretation.
Diffraction of light
Wave-Particle
Duality
Dr. Rohit Narula
Electromagnetic
waves or photons?
Wave-particle
duality
The principle of
spectral
decomposition
Matter or de
Broglie waves
I
Wave-like.
Black-body radiation
Wave-Particle
Duality
Dr. Rohit Narula
Electromagnetic
waves or photons?
Wave-particle
duality
The principle of
spectral
decomposition
Figure: Spectral radiance as a function of wavelength.
I
I
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In 1900, the study of black body radiation (the EM
radiation within or surrounding a body in
thermodynamic equilibrium with its environment, or
emitted by an opaque, non-reflective body)
led Planck to suggest the hypothesis of the
quantization of energy
i.e., for an EM wave of frequency ν, the only possible
energies are integral multiples of h: the Planck’s
constant.
Matter or de
Broglie waves
Wave-Particle
Duality
The photoelectric effect
Dr. Rohit Narula
Electromagnetic
waves or photons?
Wave-particle
duality
The principle of
spectral
decomposition
Matter or de
Broglie waves
Figure: The photoelectric effect. [Image from Wikipedia.]
I
In 1905 Einstein explained the photoelectric (the
emission of electrons from a material when light is
shone on it).
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KEmax = hν − φ,
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quantization of energy of light.
(1)
The Compton effect
Wave-Particle
Duality
Dr. Rohit Narula
Electromagnetic
waves or photons?
Wave-particle
duality
The principle of
spectral
decomposition
Matter or de
Broglie waves
Figure: The Compton effect. [Image from Wikipedia.]
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I
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I
Twenty years later, the photon was finally shown to
exist via the Compton effect
that describes the inelastic scattering of a photon by a
charged particle, usually an electron.
The change in wavelength on scattering is given by:
h
(2)
λ0 − λ =
(1 − cos θ)
me c
Particle-like.
Wave-Particle
Duality
Conclusions from the experiments suggesting
particle-like behaviour?
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the interaction of an EM wave with matter occurs by
means of elementary, indivisible processes in which
radiation appears to be composed of particles, the
photons.
The particle parameters such as the energy E and the
momentum p of a photon, and the wave parameters
such as the angular frequency ω = 2πν and the wave
vector k = 2π/λ are linked by the fundamental relation:
E = hν = ~ω
p = ~k
I
Dr. Rohit Narula
(Planck-Einstein relations)
(3)
where ~ = h/2π is given in terms of Planck’s constant
h = 6.62 × 10−34 J s.
During each elementary process, energy and total
momentum must be conserved!
Electromagnetic
waves or photons?
Wave-particle
duality
The principle of
spectral
decomposition
Matter or de
Broglie waves
Wave-Particle
Duality
Returning to Young’s double-slit experiment
Dr. Rohit Narula
Electromagnetic
waves or photons?
Wave-particle
duality
The principle of
spectral
decomposition
Figure: [Image from Quantum Mechanics by Claude
Cohen-Tannoudji]
I
Matter or de
Broglie waves
Should we abandon the wave theory?
I
I (x) 6= I1 (x) + I2 (x)
I
I
(4)
The wave theory provides a satisfying interpretation of
the fringes.
i.e., the fringes are due to the superposition of the
electric fields emanating from each of the slits.
E(x) = E1 (x) + E2 (x)
(5)
Returning to Young’s double-slit experiment
Wave-Particle
Duality
Dr. Rohit Narula
Electromagnetic
waves or photons?
Wave-particle
duality
The principle of
spectral
decomposition
Matter or de
Broglie waves
I
Since I (x) ∝| E(x) |2 we get that,
I (x) ∝| E(x) |2 =| E1 (x) + E2 (x) |2
=| E1 (x) |2 + | E2 (x) |2 +2 | E1 (x) · E2 (x) |
=| E1 (x) |2 + | E2 (x) |2 +2 | E1 (x) || E2 (x) | cos(δ)
(6)
I
where 2 | E1 (x) · E2 (x) |2 is the interference term that
depends on the phase difference δ between E1 (x) and
E2 (x).
Returning to Young’s double-slit experiment
Wave-Particle
Duality
Dr. Rohit Narula
Electromagnetic
waves or photons?
Wave-particle
duality
The principle of
spectral
decomposition
Matter or de
Broglie waves
I
What happens when the source S emits the photons
practically one by one?
I
If we replace the screen E with a photographic plate
and increase the exposure time, we find that the fringes
develop eventually,
I
thus rejecting the hypothesis that the fringes are due to
some interaction between the photons.
Returning to Young’s double-slit experiment
Wave-Particle
Duality
Dr. Rohit Narula
Electromagnetic
waves or photons?
Wave-particle
duality
The principle of
spectral
decomposition
Matter or de
Broglie waves
I
If we shorten the exposure time of the photographic
plate we find that instead of very faint smears (a weak
interference pattern) on the photographic plate,
I
we get well-defined, discrete points indicating that each
photon results in a localized impact
I
thus rejecting the hypothesis of a purely wavelike view.
Returning to Young’s double-slit experiment
Wave-Particle
Duality
Dr. Rohit Narula
Electromagnetic
waves or photons?
Wave-particle
duality
The principle of
spectral
decomposition
Matter or de
Broglie waves
I
As we increase the integration time of the photographic
plate from zero we find that the dots that correspond to
individual photons initially appear to be randomly
distributed
I
(hinting at the probabilistic nature of the process),
I
and slowly, as the integration time is increased, the
image begins taking the shape of the fringe pattern.
Returning to Young’s double-slit experiment
Wave-Particle
Duality
Dr. Rohit Narula
Electromagnetic
waves or photons?
Wave-particle
duality
I
I
I
I
Since we have eliminated the possibility of
photon-photon interactions (by having the source S
emit photons one by one only)
we are presented with the intriguing paradox that the
photon passing through one slit, seems to be aware of
the presence of the other slit.
By placing a photodetector, we immediately observe
that the fringe pattern washes out, and we simply
the sum of the intensities of the individual slits, i.e.,
I (x) = I1 (x) + I2 (x).
Thus when one performs a measurement on a
microscopic system, one disturbs it in a fundamental
way.
The principle of
spectral
decomposition
Matter or de
Broglie waves
Wave-Particle Duality for EM Waves
Wave-Particle
Duality
Dr. Rohit Narula
The particle and wave aspects of light are inseparable.
Light behaves simultaneously like a wave and a like a
flux of particles, the wave enabling us to calculate the
probability of the manifestation of a particle.
Electromagnetic
waves or photons?
I
Predictions about the behaviour of the photon can only
be probabilistic.
Matter or de
Broglie waves
I
The information about a photon at time t is given by
the wave E(r , t), which is a solution of Maxwell’s
euqations.
I
This wave characterizes the state of the photon at time
t. E(r , t) is interpreted as the probability amplitude of a
photon appearing, at time t, at the point r .
I
This means that the corresponding probability is
proportional to | E(r , t) |2
I
Wave-particle
duality
The principle of
spectral
decomposition
Wave-Particle Duality for EM Waves
I
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I
I
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Since Maxwell's equations are linear and homogenous,
we can use a superposition principle: if E1 and E2 are
separately the solutions of the Maxwell's, then
λ1 E1 + λ2 E2 is also a solution, where λ1 and λ2 are
constants.
It is this superposition principle which explains wave
phenomena in classical optics (interference, diffraction).
In quantum phyiscs, the interpretation of E(r , t) is of a
probability amplitude.
Quantum theory merely allows one to calculate the
probability of occurence of an event.
Experimental verification must thus be founded on
statistics, or conducting a large number of identical
trials.
The interference pattern so obtained in the Young's
double-slit experiment is a manifestation of these
probabilities.
Wave-Particle
Duality
Dr. Rohit Narula
Electromagnetic
waves or photons?
Wave-particle
duality
The principle of
spectral
decomposition
Matter or de
Broglie waves
Wave-Particle
Duality
The principle of spectral decomposition
Dr. Rohit Narula
Electromagnetic
waves or photons?
Wave-particle
duality
The principle of
spectral
decomposition
Matter or de
Broglie waves
I
Imagine a plane monochromatic wave, polarized in the
p direction, and moving in the z direction:
E(z, t) = E0 p̂e i(kz−ωt) .
I
After passing through an analyzer oriented along the
Ox direction it becomes,
E 0 (z, t) = E00 x̂e i(kz−ωt) ,
I
(7)
intensity I 0 is ∝ to | E00 |2 is given by Malus' law:
(8)
Wave-Particle
Duality
The principle of spectral decomposition
Dr. Rohit Narula
Electromagnetic
waves or photons?
Wave-particle
duality
The principle of
spectral
decomposition
Matter or de
Broglie waves
I
I
I
I
Suppose we release the photons, one by one.
In the above experiment, there are only two possible
results: the photon crosses the analyzer or is stopped.
In a sense, the possible results of measurement from the
analyzer are quantized, which we shall call eigen
results.
To each of these eigen results, corresponds an
eigenstate. The eigenstates are characterized by:
p̂ = x̂
p̂ = ŷ .
and,
(10)
The principle of spectral decomposition
Wave-Particle
Duality
Dr. Rohit Narula
Electromagnetic
waves or photons?
Wave-particle
duality
The principle of
spectral
decomposition
Matter or de
Broglie waves
I
If p̂ = x̂ then we know with certainty that the photon
will traverse the analyzer;
I
if p̂ = ŷ , it will, on the contrary, definitely be stopped.
I
Thus, if the particle is, before the measurement, in one
of the eigenstates, the result of this measurement is
certain: it can only be the associated eigen result.
Wave-Particle
Duality
The principle of spectral decomposition
Dr. Rohit Narula
Electromagnetic
waves or photons?
Wave-particle
duality
The principle of
spectral
decomposition
Matter or de
Broglie waves
I
When the state before the measurement is arbitrary,
only the probabilities of obtaining the different eigen
results can be predicted.
I
To find these probabilities, one decomposes the state
of the particles into a linear combination of the
various eigenstates. Thus,
p̂ = cos θx̂ + sin θŷ
(11)
Wave-Particle
Duality
The principle of spectral decomposition
Dr. Rohit Narula
Electromagnetic
waves or photons?
Wave-particle
duality
The principle of
spectral
decomposition
Matter or de
Broglie waves
I
I
I
The probability of obtaining a given eigen result is
then proportional to the square of the absolute value of
the coefficient of the corresponding eigenstate.
For Px̂ ,
| cos θ |2
Px̂ =
(12)
| cos θ |2 + | sin θ |2
For Pŷ ,
Pŷ =
| sin θ |2
| cos θ |2 + | sin θ |2
(13)
Wave-Particle
Duality
Matter or de Broglie waves
Dr. Rohit Narula
Electromagnetic
waves or photons?
Wave-particle
duality
The principle of
spectral
decomposition
Matter or de
Broglie waves
Just as light has both wave-like and particle-like properties,
electrons and other matter also have wave-like properties.
Thus, in analogy with EM waves where we have,
E = hν = ~ω
p = ~k
(14)
Wave-Particle
Duality
Matter or de Broglie waves
Dr. Rohit Narula
Electromagnetic
waves or photons?
Wave-particle
duality
The principle of
spectral
decomposition
Matter or de
Broglie waves
de Broglie hypothesized in Ph.D. thesis of 1924 that the
wavelength λdB associated with an electron is:
λdB =
I
I
2π
h
=
k
|p|
(15)
It is important to note that the very small value of
Planck’s constant h explains why
the wavelike nature of matter is very difficult to
demonstrate on a macroscopic scale.
de Broglie Waves and the Bohr model of the H
atom
Wave-Particle
Duality
Dr. Rohit Narula
Electromagnetic
waves or photons?
Wave-particle
duality
The principle of
spectral
decomposition
Matter or de
Broglie waves
Neil Bohr’s model of the hydrogen atom was successful in
explaining the Rydberg formula:
En =
−13.6
n2
where En is the energy level (in eV ) of the nth energy
transition of the hydrogen atom.
(16)
de Broglie Waves and the Bohr model of the H
atom
Wave-Particle
Duality
Dr. Rohit Narula
Electromagnetic
waves or photons?
Wave-particle
duality
The principle of
spectral
decomposition
I
The Bohr model employed the use of the condition that
the angular momentum L is an integer multiple of ~:
L = mvr =
I
nh
= n~
2π
(17)
de Broglie's result led to a reinterpretation of this
condition as a standing wave condition or a condition
for constructive interference of an electron in a
circular orbit, i.e.,
2πr = nλn ,
(18)
Matter or de
Broglie waves
de Broglie Waves and the Bohr model of the H
atom
Wave-Particle
Duality
Dr. Rohit Narula
Electromagnetic
waves or photons?
Wave-particle
duality
The principle of
spectral
decomposition
Exercise 1.
Show that the de Broglie hypothesis leads to the
quantization of the angular momentum L for a circular orbit
in Bohr’s model of the hydrogen atom.
Matter or de
Broglie waves