PMU199S notes

Quantum weirdness

Seeing Colours of Light

White light hitting a prism made of clear material will refract, resulting in a rainbow of colours.

A prism splitting white light into its ROYGBIV colour spectrum.

Have you ever stopped to wonder why we see colours? Atoms and molecules in different objects interact with (reflect, absorb, transmit) photons differently, depending on wavelength. Red things reflect only red to your eye; they absorb green and blue. Similarly, something green absorbs red and blue.

How colour works. A red apple absorbs blue and green while reflecting red.

Why do we say hot objects have a colour? For instance, why do we refer to something as a red-hot poker? Because it emits its peak power in the red part of the spectrum. Generally, any object emitting thermally at a particular temperature will have a spectrum of light emitted and it will peak at different wavelengths for different temperatures. What you find experimentally is that hotter things emit at bluer wavelengths. Our sun's emission peaks at a green wavelength of about 500 nanometres.

Blackbody radiation thermal spectrum peaks at different wavelengths depending on temperature. The specific equation is known as the Wien displacement law.

Blackbody Radiation

A blackbody is an idealization that describes a perfect absorber. Because heat cannot build up in a body in thermal equilibrium, a blackbody is also a perfect emitter. Blackbody radiation is electromagnetic radiation in thermal equilibrium with a blackbody. Why are blackbodies interesting? Well, because thinking about them led to the development of a whole new branch of physics called Quantum Mechanics (QM). QM was crucial to the development of every electronic gadget in use today like your smartphone. These devices would literally not exist if humans had not grown to understand how quantum mechanics works.

Two branches of classical (non-quantum) physics were well understood in the later part of the 1800s: electromagnetism, and thermodynamics, the study of how heat can be used to do mechanical work. What totally puzzled physicists of the day is that that if you compute the power spectrum for electromagnetic radiation from a blackbody, Maxwell's equations and thermodynamics give a completely nonsensical answer. They predict what is called the ultraviolet catastrophe: that the power emitted by a blackbody should rise for shorter wavelengths and reach all the way to infinite power at zero wavelength. This is a problem because it completely and utterly fails to explain experimental results: hot blackbodies don't emit infinite amounts of zero-wavelength radiation that could kill us! Physicists at the time thought this little problem of the ultraviolet catastrophe was just a niggle: a wrinkle that would surely iron itself out in time. They could not have been more wrong. Solving it would turn out to require entirely new thinking known as quantum mechanics.

What got people so confused about blackbody radiation? Consider a box of photons, in thermal equilibrium. What are the possible modes for photons in this box? In other words, what are the allowed ways for electromagnetic waves to vibrate inside the box? The answer is determined by the physical requirement that the electromagnetic fields must vanish at the walls. This is to ensure that there is no possibility that photons could leak out of the box. What is the answer that you get? You find that there is a fundamental mode, which vibrates at the lowest possible frequency, and then overtones, which occur in integer multiples of the fundamental frequency. The integer that labels them is called the quantum number $n$.

The different harmonics for vibration - the fundamental and the overtones.

Are we now in a position to explain where the ultraviolet catastrophe comes from? Not quite. We need one more crucial piece of information from thermal physics, which tells us the average energy contributed by any given electromagnetic mode. The problem is that this average energy per mode ${\frac{1}{2}}k_B T$ is exactly the same regardless of which mode you are talking about. This is dangerous because classical electromagnetic fields can have arbitrary frequency and hence arbitrary energy. You get

$$ E_{\rm total} = (\#{\rm{modes}})\times E_{\rm single\ mode} = (1+1+1+\ldots) \times {\frac{1}{2}}k_B T = \infty \,! $$

Oh dear. We have just done something extremely naughty. How can the total energy of a finite system be infinite? That just does not make sense physically. The problem was our assumption that we could use Maxwell's classical theory of electromagnetic phenomena to tell us the correct answer. We assumed that EM waves could have absolutely arbitrary energy or frequency. But quantum mechanics says that the physics of electromagnetic phenomena is described naturally in terms of discrete lumps, not continuous stuff.

This essential lumpiness of the quantum world compared to the smooth classical world is like the difference between water molecules and liquid water. At large distances compared to the nanoscale of water molecules, water looks continuous. It looks like a fluid that can flow arbitrarily. But if you look more closely with a very powerful microscope you would be able to see the individual water molecules. Life at short distances is grainy, not continuous.

Energy comes in lumps called quanta. (This is simply the plural of quantum.) The energy of a photon of any given frequency f has energy

$$ E=h\,f \,. $$

This equation applies only to photons, not to massive subatomic particles. The constant of proportionality $h$ is known as Planck's constant and has the approximate value in SI units of $6.63\times 10^{-34}$Js. Quantum mechanics rules the micro world. Quantum effects on human-sized scales are small -- because $h$ is so tiny. So -- how does the big flat classical world, apparently continuous to us, emerge out of the grainy world of quantum physics?

Quantum numbers (like the mode number $n$ above) are non-negative integers. At small $n$, if you add 1 to the total, that gives rise to a substantial percentage change. But at large $n$, if you add 1 to the total, the result looks almost like a continuous change. The quantum change between the $n$th and the $n{+}1$th is so small that a large object like a human being does not notice it. This is how the classical limit emerges in any quantum theory context -- at large mode numbers $n$. A discrete distribution seen with bad eyeglasses will look continuous to the viewer.

Quantum Blackbody Non-Catastrophe

Physically, the discreteness of quantum mechanics is sufficient to solve the ultraviolet catastrophe convincingly. It softens the behaviour of the system at high energy or equivalently at short wavelength, taming the classical infinity into something much more physically reasonable. The result is called the blackbody power spectrum and is in awesome agreement with experiments. Since we promised to avoid complicated equations or algebra, here is a graph of what this power spectrum looks like -- from the classical vs quantum theory. As you can see, classical theory does not even come close to explaining the experimental results. Go, quantum theory!

The classical Rayleigh-Jeans graph of blackbody power versus wavelength compared to experimental results.

The wavelength at which a blackbody at temperature T has its maximum power output is inversely proportional to the temperature. This can be worked out using quantum and thermal physics, and is known as the Wien Displacement Law (the hotter, the bluer).

A graph of power emitted by stars vs wavelength in the blue, yellow and red.

We now turn to describing another watershed experiment in the lead-up to Quantum Mechanics.

The Photoelectric Effect

Contrary to popular opinion, Albert Einstein did not win his Nobel Prize in Physics for discovering special or general relativity. He actually won it for explaining a huge experimental puzzle called the Photoelectric Effect. What is this phenomenon?

Suppose that you are interested in how light interacts with metals. Metals are a good choice for something relatively simple to study, because they have electrons that are free to move about the metal. Those electrons can then conduct electricity and heat as they move around. Suppose further that you build the following kind of apparatus for studying the interaction of light with metals: a gold leaf electroscope. When the metal in the stem and leaf are initially charged up, positively or negatively, the gold leaf separates from the zinc stem by electrostatic repulsion like with static electricity and hair in winter.

The setup for the photoelectric effect experiment. UV light is shone on negatively charged Zn plate, which is connected to a gold leaf. Electrons flow and the leaf repels from the plate. If sub-UV-wavelength light is used, nothing happens!

What did experimenters find when radiation was shone on the plate? It depended on whether you initially charged the electroscope positively or negatively. Nothing happened with initial positive charge, regardless of what wavelength of radiation you shone on it. But when the initial charge was negative, if you shone UV light (but not red) on the plate, the electroscope discharged and the gold leaf fell. What happened was that the UV light liberated electrons from the plate. But if you shone light of longer wavelength on the zinc plate then nothing happened at all. Nothing. No movement. No electrons liberated. Despite using arbitrary intensity of impinging light. So: why the hell not?

Well, metals are atoms, and atoms are nuclei plus electron clouds. EM radiation consists of photons -- quanta of the EM field. These photons have specific frequencies, and only photons with sufficient energy to actually knock electrons out of orbit are able to make current flow in the Photoelectric Effect. A single photon knocks a single electron out of orbit: it is a one-for-one trade.

Schematic diagram of a photon kicking an electron out of atomic orbit.

If you have a photon of longer wavelength, then it has lower frequency, because frequency times wavelength always has to equal the wave speed. So longer wavelength photons have less energy. You might think that flooding your atom with a hundred photons, each of which has only a hundredth of the energy required to kick an atomic electron out of orbit, might be enough. Not so. The probability of being able to line up those hundred photons all on top of one another so that they made the transition as a group can be calculated, and is vanishingly small. So it never happens in the real world.

So the quantum hypothesis -- the idea that every hunk of stuff is granular, not continuous -- is enough to explain the observed experimental results of the photoelectric effect. It is this insight that earned Einstein his Nobel Prize.

Energy levels of electrons in atoms are set by the quantum theory of electrodynamics, which is called (surprise!!) Quantum ElectroDynamics, or QED for short. Electrons that change energy levels must either absorb or emit a photon in ordert to make the transition. Allowed frequencies for photons are called spectral lines when draw on a spectrum diagram, and they depend on the atom.

A schematic diagram of the quantized energy levels for electrons orbiting the atomic nucleus of Hydrogen (for simplicity).

So what is the take-home message from the Photoelectric Effect? Quantum mechanics rules. Ultraviolet photons had sufficient energy to kick atomic electrons out of orbit, leading to an electric current. Lower frequency photons couldn't clear the energy hurdle, no matter how many photons you tried to use. So that is why a long-wavelength light beam did not lift up the gold foil in the experimental apparatus.

It is like trying to win the high jump world record by lining up a hundred short people, each of which can jump 1/100 of the required height. This just does not cut the mustard. You only win the gold Olympic medal if you jump the highest in a single jump.

Cartoon of photons kicking electrons out of Na metal.

Why cellphone and wi-fi radiation is not dangerous to humans

You may enjoy knowing that the PhotoElectric Effect explains why cellphone and wifi radiation are not dangerous to human tissue. Atoms in our body have ionization energies corresponding to the ultraviolet and higher-frequency parts of the electromagnetic spectrum. Longer wavelengths than UV cannot harm DNA. Just remember to wear sunscreen and stay out of tanning booths and X-ray machines.

To give you an idea of how confident I am in Einstein's Nobel-winning PhotoElectric Effect explanation: I would not hesitate to buy a house under power lines or stand next to the CBC's biggest radio transmitter. Quantum mechanics will keep me safe.

The different types of ionizing radiation are known as alpha, beta, and gamma radiation. Alphas are Helium-4 nuclei (two protons, two neutrons), while betas are electrons or positrons, and gammas are photons.

Here is a fabulous cartoon showing how to evaluate risk from different sources of ionizing radiation, such as bananas, your sleeping spouse, and the Chernobyl nuclear accident.