Contents [Back to Course Contents]

A. Advanced Wave Behaviour of Light

2. Diffraction Through an Opening

3. Other Interference and Diffraction Effects

i. Thin film interference

ii.

iii. Poisson’s Spot

B. The Particle Nature of Light

C. Particle-Wave Duality of Light

D. Particle-Wave Duality of Matter

2. Matter Waves

E. Applications of Particle-Wave Duality of Light

2. The Gravitational Red Shift

F. Significance of Particle-Wave Duality of Light and Matter

1. The Heisenberg Uncertainty Principle

G. Application of Particle Wave Duality:

Bohr’s Explanation of the Hydrogen Spectrum

1. Derivation of the Frequencies of the Hydrogen Lines

As you saw in the previous section, waves radiating from two point sources vibrating with the same frequency produce a distinctive pattern.

In the “pond” in the above diagram, the waves coming through the two openings produce a double-source interference pattern. If you were to travel across the water along a horizontal path near the top of the diagram, you would experience a succession of wavy – calm – wavy – calm – wavy – calm regions.

If light were a wave, we would expect a similar result, except we would experience bright – dark – bright – dark – bright – dark regions.

The light has to be monochromatic (all one colour) because for standing waves, the frequencies of the waves interfering have to be the same.

When we do the experiment with light, an interference pattern is seen.

Note that this diagram is very much out-of-scale. The slits in the card have to be very close together (less than 0.1 mm apart). The screen must be far away. The point is, though, that there are a series of light and dark bands, called “interference fringes”. If one of the two slits is covered, some bands disappear. Reopening the second slit causes dark lines to appear where there was initially light. At those locations the second light (i.e. the light through the second slit) cancelled out the light through the first slit. Two lights are producing darkness—a classic wave phenomenon.

When waves pass through a small opening they spread out and form nodal lines. A supplementary section below details the quantitative nature of this diffraction. The important thing is that light does this, too. You can see it yourself by bringing your index finger and thumb together and looking between them at a light source. Just before they touch, you should notice parallel black lines. These are the nodal lines.

When light passes through a single slit and shines on a screen or photographic plate, it produces this diffraction pattern:

Note the wide central maximum and the decreasing intensity of the bright areas as you move away from the centre.

**i.
Thin film interference**

The colours in an oil slick are an interference phenomenon, as are the colours in the soap film in the diagram below.

Here’s how this thin film interference works. Light is reflected from the front and back surfaces of the film or, in the case of an oil slick on water, light is reflected from the top of the oil and the top of the water.

In the above diagram, light ray “a” splits several times. Part of the ray reflects (the dashed line) and part is transmitted through the oil. Splitting occurs again at the bottom surface of the oil. Part bounces back up, refracting as it heads out into the air and travelling into our eye. Part of the light that hit the water refracts and heads down into the water. Light ray “b” does the same thing. This time the part that reflected from the top surface of the oil happens to line up with the light from ray “a” that came from the bottom surface of the oil. The result is that two light rays are entering the eye, one that travelled through the oil (down and back) and one that did not. Suppose the oil is approximately a quarter wavelength wide. Then the extra distance from that ray “a” travelled to the eye was about a half a wavelength. The two light waves entering the eye would be out of phase, and cancel. If the scene was being illuminated with monochromatic light, the eye, looking at that spot on the oil slick, would see a dark region. At other places where the oil film was thicker, (or at other angles), the light would not cancel. When you look at an oil slick on water under monochromatic light you see light and dark bands.

Sunlight contains light of various colours, which corresponds to various wavelengths. At a place where, say, green cancels out but red and blue does not, the oil slick will appear violet. At other regions where blue cancels out, the oil slick may appear orange. The colour swirls in an oil slick or soap bubble depend on the angle of the viewer and the varying thickness of the film. They provide excellent support for a wave description for light.

**ii.
Newton’s Rings**

Isaac Newton noticed that concentric circles appeared when he looked down at a lens resting on a plate of glass.

The explanation is the same as for thin film interference. Light reflects off top and bottom surfaces of the “air film”, cancelling at some widths and reinforcing at others.

**iii.
Poisson’s Spot**

If two light waves originate from the same place, travel equal distances, and converge to one spot, there should be constructive interference. Whatever arrives at the spot from one wave (crest or trough) should match what’s arriving from the other wave. What about light passing a small circular object (such as a dime)? There will be a circular shadow, of course. If you look closely you will see diffraction fringes around the edge of the shadow. And if you look carefully in the centre of the shadow, which is equidistant from the edges, there should be a bright dot (as if the dime had a tiny hole in the centre.)

By now you have seen that light shows definite wave-like characteristics. Two lights can cancel each other out to give darkness, definite wave behaviour. (It’s hard to imagine machine gun bullets doing that!)

Also, you have seen a mechanism to explain light: electromagnetic waves.

You have seen that the wavelength of
visible light corresponds to the light’s colour. Widening the range of
wavelengths showed you that light ranges from large wavelength radio waves
(e.g. 3 m) to very small wavelength X-rays (e.g. 10^{-12} m).

A wave model for light looks very solid. Would you be surprised to learn that there are experiments that show that light can’t possibly be a wave?

When light hits the film inside the camera a chemical reaction occurs. To simplify the discussion, let’s look at a simple reaction in the original black and white photography:

light + silver bromide à silver + bromine

The unexposed film is covered in silver bromide, a white solid. Light supplies enough energy to break the silver bromide bond, leaving a speck of black silver and bromine. During development the bromine is washed away leaving the negative: black where light hit. Taking a negative of the negative gives the positive print.

Imagine taking a photograph of white wall with a small aperture (opening). Suppose you need a five second exposure to take a picture of the wall. (Shorter times leave the wall underexposed, too dark.) The camera’s small aperture does not allow much light in at a time. Wave after wave of light washes over the film. Eventually enough energy has arrived that to supply the bond energy of silver bromide and the film turns black.

Visible light has a frequency of about 10^{14}
Hz, so the energy contained in an individual light wave is tiny. If you exposed
the film for, say, 1/1000^{th} of a second, not enough energy would
have arrived at the film to break the silver bromide bonds. If you developed
the film after that small an exposure, you would not expect to see any grains
of silver.

But you do! Here and there you find grains
of silver, places where, apparently, enough energy had arrived to break a
silver bromide molecule. Remember that waves have their energy spread over a
region. (Think of a wave washing up on a

When light hits metals it can knock electrons right out of the atoms. This is the photoelectric effect. If light were a wave, you would expect that if the frequency of light is increased, more electrons would be ejected from the metal per second. (Frequency is waves per second.) Also, you might expect that strong light would give the electrons more kinetic energy than week light.

That does not happen! It turns out that the kinetic energy the electrons receive from the light is proportional to the frequency (colour) of the light, not the intensity. Strong light does not give the electrons any more energy than weak light. It does, however, knock off more electrons per second, what we expected, incorrectly, an increase in frequency might do.

Einstein explained the photoelectric effect by saying that light comes in particles of energy hf, where h was a constant (now called Planck’s constant). The kinetic energy of an ejected electron equalled the energy of one light particle minus the binding energy of electron in the atom (called the work function.)

The key thing is that light comes in particles, photons, each with an energy that depends on the frequency. High frequency photons have higher energy. Thus blue light has higher energy than red light. That’s why photographic dark rooms, where film is being developed, have red bulbs in them rather than blue lights. (Some steps in the development must be done in complete darkness.)

Energy of photon = hf, where h = 6.626 x 10^{-34} J-s

(Of course, frequency is a wave term, so it seems strange that we are saying that it has frequency. That’s OK, though, because we have already assigned wavelength and frequency values to the different colours of light and types of electromagnetic radiation, so we might as well use them!)

**Sample
question**

How many photons per second does a 60 W red light bulb emit?

Solution

Light bulbs are very inefficient. Perhaps only 10% of the electrical energy reaching the bulb comes out as red light. The rest is heat. But let’s calculate the number of photons as if all the electrical energy going to the light was converted into red photons.

Wavelength
of red light: assume wavelength = 6.6 x 10^{-7} m

You have seen that light comes in particles of energy hf. How, then, are interference patterns caused. The explanation for the dark bands in diffraction and interference patterns were easily explained by considering light to be a wave phenomenon. But if light is a particle, we now need a new explanation.

To investigate how light, even if
particulate, creates (what appears to be) an interference pattern,

Sample p

A 1.0 W green light bulb (wavelength 5.4 x
10^{-7} m) sits at the end of a 5.0 m long box. On average, how many
photons are in the box at a given time?

Solution:

Of course there would be fewer photons than
this in the box because over half of them would hit the walls behind, beside,
above, or below the light bulb.

What do you expect to happen? Suppose the one photon goes through the top slit. It should land near the top of the screen. A photon going through the bottom slit lands near the bottom of the screen. If we put photographic film at the screen and leave the light on for a long time, we expect two piles of photons.

That’s not what happens! When the light is left on for a long time, and the film developed, the film shows a typical double-slit interference pattern! Dark regions appear where photons never land. If the bottom slit is blocked and the experiment repeated we indeed get most of the photons near the top (forming a diffraction pattern). But with both slits open the photons refuse to land where they otherwise would if only one slit were open. The photons “knew” that both slits were open. How? They each, individually, must have gone through both slits at the same time!

As strange as this sounds, the answer is simple: the photon was a wave when it passed the slit. But when it hit the end it landed on one specific spot, breaking one silver bromide molecule and making a dot on the film. Many individual dots, built up one-at-a-time over long exposure, add up to a standard interference pattern.

Light is a wave when travelling but a particle when interacting with matter. This is called the particle-wave duality of light.

Light has wavelength and frequency, its
wave nature. Light comes in particles of energy E_{photon}=hf, where h
= Planck’s constant. This energy is, in a sense, the photon’s kinetic energy.
Particles that have kinetic energy have momentum. Can a photon have momentum
even though it has no mass?

The answer is yes, a photon has momentum. And the formula comes from the energy the same way a massive particle’s momentum comes from its kinetic energy. The momentum is the derivative of the kinetic energy with respect to v:

Energy of a
massive particle = ½mv^{2}

Momentum of a massive particle = mv

(If you don’t know calculus, don’t worry. You don’t have to know how to take derivatives, or even know what a derivative is, but just accept that calculus provides a way to get the momentum from the kinetic energy.)

Let’s use the same rule for light:

Energy of a photon: = hf = hc/λ. The derivative with respect to v (or c, in this case) would be h/λ. (Weak in calculus? Just accept it! Memorize the formula.) Thus using the same mathematical rule for light as for particles with mass gives a formula for the momentum of a photon.

Energy of photon: E = hf =hc/λ

Momentum of photon: p = h/λ

Experiment shows that light does have momentum. Photons can push objects. The momentum of the photons streaming out from the sun push a comets tail away from its head. Future spacecraft might unfurl giant reflective sails to use the momentum of sunlight to push them out to Jupiter.

You have seen that light is a wave when travelling and a particle when interacting with matter. At first, it seems that light is very special, having this dual nature.

But it turns out that all matter has a dual nature. You seem particulate. You have a mass. But you also have a frequency and wavelength.

Recall that a photon’s energy is E = hf = hc/λ and its momentum is p = h/λ.

Rearrange the momentum equation for photons: λ=h/p. If light is not special, then perhaps this formula applies to massive objects too:

wavelength of any particle = h/p = h/mv

So simple! This is the de Broglie equation. Lambda (λ) is the de Broglie wavelength for a particle with mass. At first the idea that a bullet, a car, and you have a wavelength seems ridiculous. The questions that matters is “Does the equation work? Does it give us anything useful?”

Try it. Use the de Broglie equation to calculate the wavelength of an electron.

Sample p

What is the wavelength of an electron
travelling at 90% the speed of light. (Mass of an electron is 9.1 x 10^{-31}
kg and the speed of light, c, is of course 3.0 x 10^{8} m/s.)

This wavelength is comparable to the wavelengths of X-rays. When X-rays are shot at a crystal, a diffraction pattern is formed. (The explanation is similar to that of thin film interference. The rows of atoms in the crystal act like the top and bottom surfaces of the film.) When a beam of electrons is fired at a crystal, the reflected electron beam forms a similar interference pattern. Electrons are waves when travelling but interact with matter as if they were particles.

It’s hard to picture what a wavelength of an otherwise “solid” particle represents. What does your wavelength represent?

Of course you are not really solid. The
very outside edge of you consists of a row of atoms, and atoms themselves have
no “skin”. So your edge is, in reality, a little fuzzy. If you use the de
Broglie equation to get your wavelength when travelling 1 m/s, your wavelength
works out to about 10^{-35} m.

Recall that small wavelength waves go
through slits without much diffraction, just as do particles. Thus the
behaviour of small wavelength waves is indistinguishable from particle
behaviour. That’s why you think of yourself as a particle. Presumably, though,
if we packed you and a hundred of your friends into a cannon and shot everyone
at double slits about 10^{-35} m apart, you all should all land in
equally-spaced piles, forming an interference pattern! The p

It’s only large wavelength waves that
behave differently from particles. So radio waves (λ = 3.0 m) are
decidedly wave-length. (They diffract around corners.) Bullets (λ = 10^{-34}
m) are decidedly particle-like. Electrons and visible light have wavelengths
small enough to act like particles in some experiments, but large enough to act
like waves in other experiments.

You have seen that photons can hit molecules such silver bromide and break bonds. The photoelectric effect showed photons colliding with atoms and knocking off electrons. It is possible to arrange an experiment that shows an electron hitting a free electron:

The diagram shows an electron coming in from the left bouncing off a stationary electron. The electron receives a push down to the right, gaining momentum and energy. Where did this energy and momentum come from?

After the collision the photon will have different energy and momentum. The values change just enough so that the laws of conservation of energy and momentum hold for this collision, just as they should for any elastic collision.

Conservation of energy:

E_{total
before} = hf = E_{total after} = hf_{new} + ½mv^{2}

Conservation of momentum:

Remember that momentum is a vector law. The angles of deflection of photon and recoil of electron work out perfectly so that the conservation of momentum triangle works. Light particles behave just like normal particles in elastic collision.

Now look what the wave nature of light
implies. The energy of the photon after must have been less than the original.
Therefore, hf_{new} < hf. The photon’s frequency must have
decreased. From the momentum triangle we see that the initial momentum arrow is
longer than the photon’s final momentum, so its wavelength must have increased.
(This makes sense: from c=fλ, if f decreased, λ must increase.)

The effect is that the light reddened. When
the experiment is done, scattered light is redder than the light that went
straight through, hitting no electrons. You have seen this

A bullet fired lifted straight up from the surface of the Earth slows down. It loses kinetic energy. The loss in kinetic energy equals the gain in potential energy:

ΔE_{k}
= ΔE_{p} = -mgΔH

(In the above equation, we used capital H for height, to save confusion with from Planck’s constant, which we are about to use. The negative sign means when kinetic energy decreased.)

But photons have a mass equal to zero, so
perhaps there is no change in kinetic energy? Actually, there is. From Einstein’s
famous formula E = mc^{2}, you can replace m with E/c^{2}. You
get:

Therefore,

:

(This formula is an approximation because g
is not constant as you go up. Also, should you divide the energy at the bottom
or the energy at the top, or some average energy, by c^{2} to represent
the mass of the photon.)

The result, though, is the frequency at the top is less than the frequency at the bottom. Einstein predicted that the light leaving a strong gravitational field should be red-shifted. The spectral lines from massive stars should be shifted toward the red. Experiment shows this effect to be true. In fact, we could get an estimate for the strength of the gravitational field.

Recall these points:

- light travels as if it were a wave

- light interacts with matter as if it were a particle

Therefore, the nature of light that you detect depends on the experiment you use to examine it.

Recall the Compton Effect: detecting the position of the electron changed the electron’s momentum. Before you saw the electron you knew that it was (assumed to be) stationary in your equipment, but you didn’t know where it was exactly. Then you saw its location (if your eye happened to intercept the reflected photon.) The act of detection, however, changed the electron’s momentum, so you now you have less knowledge of where the electron is.

Suppose you get clever. You decide to illuminate the electron with lower frequency light so the electron is not given as much momentum and energy from the photon. So you have have more knowledge of the electron’s momentum because it didn’t change as much. Recall, though, that large wavelength waves diffract around objects more than small wavelength waves. The “view” of the electron will be fuzzier. You won’t have quite as good an idea of the electron’s exact location. If you use high frequency (short wavelength) radiation to detect the electron more clearly, you will have changed the momentum of the electron more.

The moral of the above story is that the
more you do (in your experimental procedure) to reduce the uncertainty in an
object’s position, the more you will __increase__ your lack of knowledge of
the object’s momentum. This leads to the Heisenberg Uncertainty Principle,
which adds a little mathematics to the idea:

ΔxΔp>h

There will always be some uncertainty in quantity values, and the more you know about the x position, the less you know about the momentum. (The Δ symbol here means “uncertainty” or “range of possible values”, similar to percent error.)

It is important to realize that the Heisenberg Uncertainty Principle is not a statement about the limits of experimental technique. It does not deal with experimental error. It says that there is a basic uncertainty in reality. It may look to you as if this principle is rather trivial. In fact, the Heisenberg Uncertainty Principle has deep significance for the nature of matter, atomic structure, the Big Bang Theory for the creation of the universe, the energy of a vacuum, the feasibility of quantum computers, and other scientifically important questions.

Here’s another way to arrive at this deep philosophical concept.

Recall these point:

- electrons are moved when hit by photons

- you see where an object is when light reflects off it and enters your eye

Therefore, the very act of detecting the position of an electron causes it to move to a different position. Therefore it is impossible to determine the location of an electron without changing the location of an electron (and anything else, for that matter, although admittedly the change in position is small.)

In science, by definition, reality is how
it is detected to be. (If you say that all tests show a phenomenon to be A but
you say “I still think it’s not A, then you are making a statement of
religion/belief/philosophy, not science. In science, if it looks like a duck,
walks like a duck, quacks like a duck, and all other tests show duckiness, then
__it’s a duck__!) If by detecting the location of something you change the
location, then the location is, in principle, unknowable exactly. There is an
uncertainty in position.

Here’s another deep philosophical implication of what you have learned. If reality is how it’s detected to be, and the observer by detection changes reality, then the observer does not merely observe the universe, the observer creates the universe by the act of detecting it.

When you do an experiment (or merely
witness life), you are not __discovering__ what something IS, you are __determining__
its properties. The universe does not exist independent of you!

The fact that the observer plays a role in
determining what happens in the universe leads to many interesting situations
and paradoxes. Your teacher might mention Schrödinger’s Cat, a thought
experiment which suggests that a cat is neither alive nor dead __until__ an
observer checks to see. This booklet was designed to give you just what you
need to know, so I will leave this experiment for you to read in your text.

Just for fun, though, here are a couple of
limericks which dwell on the significance of the observer. The first deals with
the existence of a tree in the quadrangle, a little park surrounded by the
physics building (at the

There
was a young man who said, 'God,

It has always struck me as odd

That the sycamore tree

Simply ceases to be

When there's no one about in the quad.'

The story goes that the day after the limerick was posted on the bulletin board in the physics building, a new limerick appeared below it:

'Dear
Sir, Your astonishment's odd;

I am always about in the quad:

And that's why the tree

Will continue to be,

Since observed by,

Yours faithfully, God.'

You have seen that light has a dual nature: it behaves like a particle when it interacts with matter and a wave when it travels. You have seen that light is not special, in that all objects have this dual nature, objects without mass (like photons) and objects with mass, like electrons.

A terrific example of treating particles with mass as having a wave and particle nature at the same time is Bohr’s explanation for the lines in the hydrogen spectrum.

Bohr proposed that electrons did not orbit the nucleus in a full range of orbits, as do planets about the sun. He suggested that electrons have specific orbits, each with its own energy. The full derivation of an expression for the energy levels in the hydrogen atom will be in your physics textbook. As this book is supposed to be “just what you need to know” we’ll cover the idea. You might follow your book as you read these steps.

1. Remember that using CF = GF you can find the velocity of a satellite in orbit about a planet of mass m at radius of orbit r. The mass of the satellite cancels out. Similarly, using CF = EF (centripetal force = electrical force) you can find an expression for the velocity of the electron about the proton in the hydrogen atom (1 electron orbiting 1 proton). The mass of the electron does not cancel, because there is no m on the right side.

2. Just as you can write an expression for the total energy (kinetic plus potential) of an orbiting satellite, you can write an expression for the total energy of the electron.

Steps 1 and 2 are treating the electron similarly to a satellite about a planet. That is, you are using standard Newtonian mechanics, treating the electron as a standard particle.

3. Here’s Bohr’s spectacular intellectual leap. He says “suppose the electron is a standing wave”. Remember that linear standing waves (like standing waves in a slinky) must have an integer number of half-wavelengths. That is, there are only certain frequencies that work for a slinky of a given length. Imagine bending the slinky around in a circle. Now only an integer number of full wavelengths fitting around the circumference will make a sustained standing waves. (If it’s a half-wavelength around the circle, a crest and a trough will match up everywhere, cancelling the wave.) This means that for a given orbit, only a specific set of wavelengths will fit.

4. Setting the circumference equal to an integer number of wavelengths, and substituting for lambda the terms from the de Broglie equation gives and expression for the allowable radii, in terms of the mass, and speed of the electron. Substuting the expression for the velocity of the electron in for v gives a formula for the radius of orbit of the electron without a v in it.

, where n = 1, 2, 3, …

What this means: the electron can only have specific r values. It can only be at specific distances (where its circular standing wave can fit), not at any random distance.

5. Substituting the above expression for r in the formula for the total energy of the electron gives this expression for the total energy of the electron in orbit about the nucleus:

, where n = 1, 2, 3 …

__Significance__:
the electron can only have specific, discreet energies, the value if n is 1 or
the value if n is 2, and so on. It cannot have an energy between those values.
n is the energy level number, the *first
quantum number*.

When the electron falls from one level to a lower level, it gives off a photon of energy equal to the difference between the levels (the difference will be positive). Each line in the hydrogen spectrum corresponds to one of these transitions.

Using Bohr’s equation for the energies of each orbit allows us to predict the frequency (using E = hf) of the light that should be emitted for each possible pair of energy levels (such as falling from level 3 to level 2). These frequencies match the colours we actually do see in the hydrogen spectrum. Wonderful!

Energy is quantized. A quantum is the smallest step between energy levels. (A quantum leap, therefore is NOT, as the media and general public thinks, a large step. It’s the smallest possible step.)

Using both particle behaviour formulas (for
orbits) and wave formulas (frequencies of standing waves) __at the same time__
led to a successful description of a phenomenon of nature, the colours of the
lines in the hydrogen spectrum.

Electron is no longer thought of as a
particle occupying a specific location and moving along a specific path.
Rather, it is a standing wave, with a high p

1. be in two places at the same time (e.g. a single electron can pass through both openings in a double slit simultaneously)

2. get from one point to another without travelling the space in between (e.g. the electron can be at an antinode, then later at a different antinode, but not travel through the plane of the node.)

Within a year or so, Bohr realized that his
explanation, even though beautiful and elegant, was flawed. He could not apply
it to any other atom (with more than one electron.) Still, Bohr’s derivation is
worth studying to see that tremendous intuitive leap he made, and appreciate
how particle and wave mechanics can be linked to provide an accurate
description of the atomic world. This was the beginning of __quantum mechanics__.