Welcome to the second video in this series on quantum mechanics. In the first video we
saw that classical physics failed to explain the relationship between temperature and electromagnetic
radiation, such as arises in an incandescent light bulb. The classical prediction is that
radiation intensity will always increase with frequency. Planck was able to derive a radiation
law that matches what is observed – intensity reaches a peak at a certain frequency and
then rapidly drops off at higher frequencies. But to achieve this Planck had to assume that
radiated energy comes in discrete chunks, or quanta. At frequency nu the quantum of
energy is Planck's constant h times nu. He called this assumption an act of desperation,
and although it produced the correct formula, he was highly skeptical of it.
The assumption worked because at a temperature T there is an average thermal energy kay tee
available to excite a radiation mode. Here k is called Boltzmann's constant. For low
frequencies the energy quantum is much less than kay tee. The mode can have an energy
of kay tee (which is the classical prediction) by containing some number of quanta. But at
high enough frequencies the energy quantum will be more than kay tee. These modes either
have to have no energy or more energy than, on average, is thermally available. So at
higher frequencies the probability becomes greater that a mode will have no energy at
all.
This is how the quantum hypothesis explains the drop off of radiation intensity with increasing
frequency.
Planck's skepticism about the quantum hypothesis was understandable. Radiation takes the form
of electromagnetic waves, and the energy of a wave is distributed throughout space. How
could this energy exist in discrete chunks. Discreteness is something we associate with
particles, not waves, and nobody had ever seen direct evidence of any type of radiation
particle.
But then it was pointed out that such evidence had indeed been observed in a phenomenon called
the photoelectric effect. A solid material contain electrons. These can be pulled loose
by supplying electrical energy, as is done in a spark plug. Energy supplied in the form
of light should also be able to remove electrons. But if a material is illuminated with low-frequency
light, represented in red here, no electrons are ejected, even if the light is very intense.
On the other hand, if the illumination is high-frequency light, represented here in
blue, electrons are emitted, even if the light is very weak.
It was Einstein who in 1905 explained this by considering Planck's quanta as particles
moving at the speed of light, what we now call photons. If W is the energy needed to
remove an electron, only a photon with an energy greater than W can eject an electron.
The total light energy, which represents the number of photons striking the material, will
determine the number of electrons ejected, but not whether electrons are ejected. And
the kinetic energy of the ejected electrons will simply be the energy left over from this
photon-electron process.
A photon of low-frequency light does not have enough energy to free an electron. So, even
if there are many such photons, that is, the light is very bright, no electrons will be
ejected from the material. Thus Einstein explained how Planck's quanta manifest themselves as,
quote, real particles of light.
A more prosaic example is how photons explain sunburn and sunscreen. If you look at the
spectrum of solar radiation, you find that the great majority of the energy is in the
visible and infrared regions. Only a tiny fraction is in the high-frequency, short-wavelength
ultra-violet region. Yet sunscreen protects us from sunburn by blocking only this small
amount of UV radiation.
Let's assume the chemical reaction responsible for sunburn requires a certain energy W and
light is composed of discrete photons. If a molecule absorbs a photon of energy less
than W there will be no chemical reaction and no sunburn. If a molecule absorbs a photon
of energy greater than W then there will be a chemical reaction leading to sunburn. For
sunburn the required energy corresponds to photons with frequencies in the ultra-violet
region of the spectrum.
So by blocking only a tiny fraction of the sunlight's energy our skin is protected against
sunburn. The great majority of the sunlight's energy still reaches our skin, but no single
photon corresponding to these frequencies and wavelengths has enough energy to cause
sunburn inducing chemical changes.
We've seen that there is strong evidence that light consists of discrete particles we call
photons having energies given by Planck's formula. But it had long been firmly established
that light is a wave phenomenon. If you put a small pinhole in an opaque screen, place
another screen to its right and shine a beam of light from the left, on the right screen
the light intensity will display a so-called diffraction pattern that is the unmistakable
signature of a wave. If you open a second pinhole so that another diffraction pattern
appears on the screen, a very rapid variation of intensity, called interference fringes
will appear. These observations are a slam dunk for the wave theory of light.
Let's look at this experimentally. First we take an image immediately after the pinhole
screen. This shows light emerging from the two pinholes, one large and one small. Next
we take an image some distance away from the pinholes. Here we see that light from the
pinholes has spread out into ringed diffraction patterns. Notice how the rings for the large
hole are more closely spaced than are the small-hole rings. This features are precisely
explained by the wave theory of light.
Moving even farther away from the pinholes the ring patterns grow in size and overlap
more. Looking at this in more detail we see closely spaced vertical lines superimposed
on the diffraction patterns. These are interference fringes. There is no reasonable way to explain
these observations without invoking the concept of waves.
Now, we think of a particle as a little thing at a single point in space, possibly moving
in some direction at some velocity. We think of a wave as something spread out in space
oscillating with some frequency. If light is so precisely described as a wave, but also
displays unmistakable particle properties, then what is it exactly? What is a photon?
How do we picture it? The answer is: not very well. We can draw a little wiggle indicating
a wavelike oscillation with an arrow indicating a particle-like thing moving in some direction.
But the fact is we simply don't have any day-to-day experience with something that has the properties
of both a wave and a particle. So we don't have a frame of reference in which to understand
such a thing in any intuitive sense.
We've seen images that clearly show the wave nature of light. We've also presented evidence
that argues for the particle nature of light, but we haven't actually seen light particles
with our own eyes. So, reasonably we might still be skeptical about that. Especially
since the implications are a bit mind bending. Fair enough.
What if we repeat the pinhole experiment but with light so weak that at any given time
there should no more than a single photon in the system? What will we see? Doing this
successfully requires extremely sensitive equipment, but it is possible. Here's we're
going to look at frames from a truly beautiful video titled from photons to waves that shows
the results of this experiment. The link is given below.
After a short time we see a seemingly random pattern of dots. These are the images of individual
photons. A while later more photons have arrived and there is a hint of some pattern emerging.
As time goes on the pattern becomes clearer while the contributions of individual photons
are no longer so apparent. We finally have the fringe pattern that characterizes wave
interference, yet we've seen that it's actually built up from a great many point events. The
light energy does indeed arrive at the image in discrete photon packets at specific places
at specific times. Yet somehow the photons quote “know” how to arrange their arrivals
so that after some time the average effect of a great many photons is to create a wave
interference pattern.
So we have a wave picture in which interference patterns are the result of a wave passing
through two pinholes or slits, creating two diffraction patterns that spread out and overlap.
But we also have a very convincing particle picture where light energy is concentrated
in photons which occupy a particular place in space at a particular time. We typically
think of a particle as being at one point at time t1 and at another point at time t2
and having followed some specific path between the points. But we've seen that photons arrive
at the image screen in accordance with an interference pattern. Do single photons interfere
with themselves in the presence of two slits? If so, then how can a single particle pass
through two slits at the same time? If the photon only goes through one slit then how
does the existence of the other slit influence how the photon arrives at the image screen?
This is one of those “nobody understands it” moments. But the problem is in our heads.
There is a precise mathematical description of this whole process. Light energy manifests
itself as discrete photons each with a certain energy and arriving at a certain place at
a certain time. This occurs at random and the classical wave intensity is just the probability
that you will find a photon at some place and time. What paths do the photons follow?
That's not part of the theory, and in fact it doesn't even appear to be a meaningful
question. Asking it is simply trying to force our limited concepts of particles and waves
onto a realm of Nature where they don't seem to be as valid as they are in our macroscopic
world.
Let's continue to try and push back against the idea of particles of light. We might argue
that real particles carry momentum and you can exert force by scattering them off of
other particles. If light is composed of real particles then you should be able to push
an object around by shining light on it. Well it turns out you can.
In classical theory light has two components, an electric field and a magnetic field, shown
here as red and blue arrows. The length of an arrow signifies the strength of the field
at that point. The fields are perpendicular to each other and to the directio n in which
the light is moving. Suppose the light passes by a charged particle.
Classical theory predicts the phenomenon of radiation pressure. This slide is a little
technical, but in the end the important thing is the final result. The steps are just for
completeness and there's no need to follow them. Radiation pressure works like this.
Suppose at a particular time the charge q sees an electric field E and a magnetic field
B which are moving in the direction S. The magnetic field strength is the electric field
strength divided by the speed of light. Suppose the charge is moving with a velocity v. The
electric field pulls on the charge with force q times E. The energy the field transfers
to the charge which we ca ll the work it does, is this force times the distance the charge
moves. The distance is velocity v times elapsed time, so the field does work E q v t. The
magnetic field will also exert a force on the charge. A force exerted for a given time
transfers a momentum of force times time. The force exerted by a magnetic field on a
moving charge is the magnetic field B times the charge q times the velocity v. For an
elapsed time t this transfers momentum B q v t to the charge. The magnetic force is peculiar
in that it pushes in a direction perpendicular to both the magnetic field and the charge's
velocity. This is in the direction S. The momentum expression is the same as the work
expression except the E field is replaced by the B field. But the B field equals the
E field over the speed of light, so we have the following result. If light transfers some
energy to a charged particle, it also transfers a momentum equal to the energy over the speed
of light. The transferred momentum is in the direction that the light is traveling.
The reality of radiation pressure was spectacularly demonstrated by a resent Japanese space mission
to Venus which deployed a so-called solar sail to use radiation pressure from the sun
to accelerate and navigate the spacecraft.
One more detail before we summarize all this. If a light wave propagates at the speed of
light, c, for one second, it travels a distance c. We call the distance between wave peaks
the wavelength lambda. If the light frequency is nu, then in one second there are nu wave
cycles. So nu wavelengths must add up to the distance c. Therefore the wavelength is the
speed of light divided by the frequency, c over nu.
The take-away from this discussion is that if light consists of photons of energy E equals
h nu, then the existence of radiation pressure requires that those photons must have momentum
p equals E over c, which is h nu o ver c, which is h over the wavelength lambda. So
photons do indeed carry momentum, just like we'd expect for quote real particles. Later
we'll see that this relationship between momentum and wavelength will provide a critical insight
when we try to understand the behavior of electrons.
The proverbial final nail the in coffin of skepticism toward the photon concept was provided
by observations of so-called Compton scattering, discovered by Arthur Compton. He found that
if an electron at rest is illuminated by high-frequency light, the electron scatters off in some direction
carrying kinetic energy and momentum while the light is scattered off in another direction
and with a longer wavelength. Using the photon concept, the longer wavelength implies the
scattered photon has lost energy and momentum. These are precisely the energy and momentum
seen to be transferred to the electron. Thus a photon can be considered to literally bounce
off an electron analogous to a collision between two billiard balls.
There is rock solid evidence that light is neither a classical wave nor a classical particle,
but has both wavelike and particle-like properties simultaneously. This wave-particle duality
represents, in Richard Feynman's words, the wonderfully different behavior of things on
very small scales. A behavior that is unlike anything we have direct experience with, yet,
which countless experiments tell us is simply how Nature is. As compelling as this new picture
may be, it is also quite unsettling.
In the next video we'll see that wave-particle duality undermines classical views of determinism
and causality, and of our ability to in principle determine physical quantities with arbitrary
precision.