Download The Wave-Particle Duality for Light So is Light a Wave or a Particle

The Wave-Particle Duality for Light
Matter as both particles and waves
In 1924, French doctoral student Louis de Broglie proposed
that particles, such as electrons, could act as waves just as
waves of light could sometimes act as particles. Every
particle is endowed with wave characteristics as it travels.
Light behaves like a wave in traveling
from a source to where it is detected.
This behavior is evident from wave
interference measurements and the
prediction of the speed of light from
the electromagnetic wave equation of
Maxwell.
wavelength =
Planck's constant
;
momentum
The de Broglie wavelength of a car (mass = 1500 kg) traveling
at 30 m/s (67 mph) is only 1.5x10-38 m, which is much smaller
that any atom or nucleus.
So is Light a Wave or a Particle?
Seemingly contradictory, the waveparticle duality implies that light has
both particle-like and wave-like
properties. This duality addresses the
inability of classical concepts like
"particle" and "wave" to fully describe
the nature of light in all situations.
de Broglie also proposed
that the stable Bohr
orbits of the electrons in
an atom are those where
the electron wave closes
back on itself.
stable
PHYS 1010Q
h h
=
p mv
What is the de Broglie wavelength of an electron (mass =
9.1x10-31 kg) traveling at 1,000 km/s? (The kinetic energy of
this electron is about 3 eV.)
Light behaves like a particle when it is
being emitted, absorbed, or scattered
by an atom. This behavior is key to
understanding the Planck radiation law,
the photoelectric effect, and the
spectra of atoms.
13.1
λ=
© D.S. Hamilton
unstable
13.2
PHYS 1010Q
© D.S. Hamilton
Wave interference
The electron wave must fit evenly into the circumference of
the circular orbit, ie C = nλ.
Show that this requirement of C = nλ is equivalent to Bohr’s
quantization rule for the angular momentum, L = mvr = nh/2π.
The radius of the n=1 Bohr orbit is 0.0531 nm. What is the
wavelength and speed of an electron in this orbit?
13.3
PHYS 1010Q
© D.S. Hamilton
An interference pattern for light (left)
and electrons (right) passing by a sharp
"knife-edge". Interference is conclusive
evidence for wave-like behavior.
An electron microscope makes practical
use of the wave nature of electrons. The
de Broglie wavelength of high-speed
electrons is typically thousands of times
shorter than the wavelength of visible
light, so the electrons are able to
distinguish details not apparent with light.
13.4
PHYS 1010Q
© D.S. Hamilton
What is waving??
In Schrodinger's wave equation, the thing that waves is a
mathematical entity called the wave function Ψ (psi = "sigh").
The wave function represents the possibilities that can occur
for a system.
For an electron of mass m confined in a one-dimensional
box of length L, find expressions for the de Broglie
wavelength, momentum p, velocity v, and kinetic energy KE.
This figure show the probability
distribution Ψ2 of an electron
cloud of the lowest energy state
in the hydrogen atom. We can not
tell exactly where the electron
will be found in the atom at a
given moment, but only the
likelihood (probability) of finding
it there.
The calculation of the wave function is completed by solving
the Schrödinger wave equation for Ψ(x).
Compare the momentum p of an electron in the n=1
quantum ground state of a one-dimensional box of length L to
the momentum when the electron is in the n=3 excited state.
For a particle confined to a
rigid box of length L, the
solutions Ψ(x) to the
Schrödinger wave equation
are those standing waves
that just fit inside the box.
The probability distribution
for finding the particles is
then proportional to Ψ2.
13.5
PHYS 1010Q
© D.S. Hamilton
Uncertainty Principle
13.6
PHYS 1010Q
© D.S. Hamilton
Uncertainty Principle continued
There are pairs of quantities in physical systems that are
linked in such a way that we cannot know them both
simultaneously with infinite accuracy. For waves, it is well
known that the longer we have to measure the frequency f of
the wave, the smaller the uncertainty in the value of f.
Mathematically, this can be stated as Δf·Δt ≥ 1.
By adding several waves of different
wavelengths λ, you can produce an
interference pattern which localizes
the wave. But that process involves a
spread in momentum values. There is
an inherent increase in the uncertainty
Δp when the wave becomes more
particle-like and Δx decreases. The
more certain we are about its position,
the more uncertain we become about
its momentum.
Show that short acoustic "click" of duration Δt=0.1 msec
contains a band of frequencies having a width Δf=10 kHz.
Quantum uncertainties stem from the wave nature of matter.
Show that Δf·Δt ≥ 1 implies that ΔE·Δt ≥ h.
Because matter has wave properties, there is another pair of
such variables, the position of a particle and its momentum
along the same direction that have an uncertainty relation, ie,
Δx·Δpx ≥ h.
Suppose you have an instrument which can measure the
speed of an object to a precision of 1%. What is the minimum
uncertainty in the position of a 0.145 kg-baseball moving at
40 m/s (90 mi/h)? For an electron with a typical velocity of
106 m/s in an atom?
A wave with a definite wavelength λ
implies that the momentum is also
precisely known since p=h/λ. But the
wavefunction Ψ and the probability Ψ2
of finding the particle is now spread
out over some region of space. The
more certain we are about its
momentum, the more uncertain we
become about its position.
13.7
PHYS 1010Q
© D.S. Hamilton
13.8
PHYS 1010Q
© D.S. Hamilton
Complementarity Principle
The wave particle duality is an underlying principle of the
Universe. The complete description of an electron or a photon
requires both its wave and particle aspects. If two concepts
are complementary, an experiment that clearly illustrates one
concept will obscure the other. For example, an experiment
that illustrates the particle properties of light will not show
any of the wave properties of light. Complementarity is not a
compromise with the truth being somewhere in between a
particle picture and a wave picture. Rather it is a statement
of the dual nature of the quantum reality.
Opposites are seen to complement each other in
this view of the world. Neils Bohr was knighted
for his work in quantum physics and selected
the yin-yang symbol for his coat of arms.
Determinism
Newton's laws showed us how to calculate of the future
motion of any particle. Is then the motion of the entire
Universe completely predetermined? And does this depend on
our ability to calculate the future?
Quantum mechanics suggests the future is a statistical issue.
We can calculate the probabilities for certain events, but we
do not know exactly where or when they will actually occur.
13.9
PHYS 1010Q
© D.S. Hamilton