# PHY197Fnotes

## Wavelike nature of quanta

### Quantum Theory

Last time, we learned how the Ultraviolet Catastrophe puzzle led to major advances in understanding the microscopic realm, via the new idea of the quantum. We discovered that energy comes not in continuously variable amounts, but instead comes in discrete lumps. We also learned that for EM radiation (photons) the energy of the photon is directly proportional to its frequency: $E=hf$. We did the example for a CBC Radio One photon and found that a single one has only about $7 \times 10^{-26}$ Joules of energy. By comparison, the average air molecule at room temperature and sea-level pressure has a kinetic energy of about $10^{-21}$J, which is over ten thousand times bigger. So quanta are very very small compared to human scales.

### Wavelike Nature of Quanta

All quanta a.k.a. subatomic particles show wavelike properties. How did physicists come up with this outlandish statement? There were two hallmark experiments that proved beyond reasonable doubt that subatomic particles actually have wavelike properties:-

• The Young Double Slit experiment (1803), which showed that photons interfere;
• The Davisson-Germer experiment (1927), which showed that electrons diffract.

Interference and diffraction are typical behaviours of waves, not of classical particles like bullets.

I sometimes call quanta wavicles for fun, because the word wavicle gets across the idea that a quantum (e.g. an electron) is at the same time particle-like and wave-like. Some physicists make a Really Big Deal about this fact that quanta have both wave-like and particle-like properties, referring to it as Wave/Particle Duality. I prefer to use an analogy to people like me with nonbinary gender. I have feminine aspects, and I have masculine aspects -- at the same time! That is just fine: it is entirely possible to have both feminine and masculine aspects in one person, like for your professor. You could make a big deal out of it and say that I have male/female duality, or you could just get over the fact that I am nonbinary and be comfortable with my existence in the universe! It is a bit like this with wave/particle duality: it is not hard to understand if you just stretch your imagination a bit further than it was used to stretching. But it is counterintuitive at first, if you are not used to it. Everyone who studies quantum mechanics has to get over wave/particle duality, and it can take awhile before it feels natural. Just like with non-normative genders. :D

### Interference

Interference occurs whenever two or more wave sources are present and close enough to interact. Interference patterns result from waves either being in phase i.e. adding together, or waves being out of phase, i.e. cancelling each other out. In the bathtub with your brother, you could achieve constructive interference of water waves at a particular place by ensuring that, at that place, the peaks of the waves from your end of the bathtub coincide with the peaks of the waves coming from your brother’s end. Destructive interference happens when your peaks coincide with your brother’s troughs. Interference can be continuous or transient, depending on the wave sources.

### ￼Young Double Slit Experiment

If photons were like bullets (the classical picture), then a beam of photons shone upon a double-slit setup would give two peaks on a detector behind it, located precisely at the two slit positions. But if you do the experiment on real photons, you find something entirely different. When Young did the Double Slit experiment on photons, he found alternating light and dark areas on his detector screen, which physicists call interference fringes. The same qualitative behaviour is observed for all particles you do this experiment with, whether they be photons or electrons or even neutrons. Quantum mechanics is wild, eh?

￼The really mindblowing thing is that if you do this experiment one photon at a time you still see an interference pattern in your detector. In a very real sense, each photon goes through both slits on its way to the detector. But if you boss the wee photon around, e.g. by covering one of the two slits so it cannot go through that one, the interference pattern disappears.

### Diffraction

Diffraction is the apparent bending of waves around small obstacles or the spreading out of waves past small openings. Clearly, classical particles (like bullets) do neither of these things. So if you were a classical physicist, you never would have expected to see diffraction from subatomic particles. What the discoverers of quantum physics found was that this is not at all how subatomic particles behave in the microscopic realm. If you look at small scales, you do see diffraction!

Shining a red laser beam (coherent light) onto a square aperture gives the following diffraction pattern

￼Shining white light on a circular aperture gives a different but related kind of pattern:-

￼There are lots and lots more fun experimental results that I could show you, but I do not want to clutter the stage too much intellectually.

### Davisson-Germer Experiment

Davisson and Germer did their famous experiment in the period when quantum mechanics was brand-spanking-new. What they did was to heat up a filament of metal known to be good at emitting electrons. They caught the electrons and accelerated them using an applied voltage (a concept closely related to electric field). The accelerated electrons were then allowed to strike the surface of some nickel metal - which has a regular, periodic, internal structure. What the classical physicists predicted was reflection of the photon bullets off the nickel surface. What they saw, however, was completely and absolutely different. They discovered that there was a whole range of angles for deflected electrons, not just a single one, and that there were peaks and troughs in the intensity of deflected electrons. To discover the dependence of intensity on angle, they rotated their detector apparatus around. The graph with peaks and troughs below is a representation of what they saw.

This experimental result was really confusing to Davisson and Germer at the time. They did not expect wavelike behaviour from subatomic particles, because of their intuition about bullets. But this experiment helped lead physicists to invent quantum mechanics, and so we celebrate it.

So how is it that particles have wavelike properties? Does everything in the universe have a quantum wavelength? Actually, yes! :O

### De Broglie Wavelength

Louis de Broglie in 1924 surprised his fancy French PhD examiners by boldly proposing a formula on extremely scant evidence -- which later turned out to be right! ￼

$$\lambda_{\rm{de\ B}} = {\frac{h}{|{\vec{p}}|}} ￼$$

What this formula says is that quantum wavelength is inversely proportional to the magnitude of the momentum. This formula holds for all wavicles, whether they are massless or not.

Electron microscopes make very direct usage of this de Broglie wavelength concept. To probe smaller distance scales, we use a finer tooth comb in the form of a shorter-wavelength quantum. So to access smaller microscopic length scales in an electron microscope, we must use electrons that go faster, so that they have shorter quantum wavelength. Therefore, probing shorter distances requires bigger machinery, to allow acceleration of the electrons up to probing speed.

### Quantization of Energy

According to quantum mechanics (often abbreviated to QM), each quantum is not like a classical bullet but is rather a quantum wavicle. Wavicles can exhibit wave-like properties or particle-like ones, depending on how they are probed and with what. On large distance scales compared to their wavelength, wavicles behave to a good approximation like classical particles. But on distance scales similar to or shorter than their wavelength, wavicles are unavoidably wavelike.

Quantization of atomic energy levels is basically caused by having to fit an integer number of electron wavelengths around its orbit. This is needed in order to ensure that the electron wavefunction matches back up onto itself after a complete revolution around the nucleus. This quantization of energy levels has many sequelae, the best illustration of which is spectral lines of light emitted from atoms.

This is a picture of possible electron wave shapes in the hydrogen atom. The colours are false, and are chosen to group together wavefunctions with the same underlying symmetry pattern.￼

### Waves of What, Exactly?

You may be wondering at this point just what is doing the waving, if things like electrons have wavelike properties. The answer is the electron wavefunction. A wavefunction $\Psi(t,{\vec{x}})$ is a mathematical object that physicists use in order to embody the mathematical and physical behaviour of the subatomic particle's wavelike properties - such as its dependence on time $t$ and position ${\vec{x}}$. Using the wavefunction, physicists can calculate quantum mechanical things of interest. For example, the absolute square of the wavefunction $\left|\Psi(t,{\vec{x}})\right|^2$ gives the probability of finding the particle in question at time $t$ and position ${\vec{x}}$,

$${\mathcal{P}} \sim \left|\Psi(t,{\vec{x}})\right|^2 \,.$$

The sum of probabilities for all possible outcomes must add up to 100%.

Quanta with different spins and masses have different wavefunctions, and obey different wave equations.

Erwin Schrodinger was the first guy to derive the wave equation for non-relativistic wavefunctions of massive quanta. Here is the Schrödinger equation, recorded here only for artistic reasons:- $$i\hbar {\frac{\partial}{\partial t}}\Psi(t,{\vec{x}}) = \left( - {\frac{\hbar^2}{2m}}{\vec{\nabla}}^2 + V(t,{\vec{x}})\right) \,. \Psi(t,{\vec{x}})$$ ￼If you add in relativity to quantum mechanics, you get Quantum Field Theory. QFT handles both particles and antiparticles in the same breath. It is a very powerful apparatus, which we teach only in graduate school.

### Heisenberg Uncertainty Principle

The really amazing thing about quantum mechanics is that what actually ends up happening is purely up to random chance! Evolution of the wavefunction into the future is precisely computable, if you know the initial data exactly. But what you get is only the probability of various possible outcomes happening. Which one eventuates is entirely random and cannot be controlled by humans, as far as we know.

This essential randomness of quantum mechanics kills classical determinism. It totally ruins your day if you thought that the universe should run like a clockwork machine. Einstein really hated this indeterminacy of quantum mechanics, and was unrepentant about it even on his deathbed. He said God does not play dice with the universe!. Unluckily for him, history proved him wrong.

The wavelike nature of quanta, and the observed randomness of quantum mechanics, introduce an intrinsic fuzziness to physical observables like momentum or position. The Heisenberg Uncertainty Principle is a mathematical statement of this fuzziness. It governs only particular pairs of physical observables, so it does not apply across the board (contrary to popular opinion). After certain reasonable assumptions about the physical states being measured, you can express the Heisenberg Uncertainty relation as an equation which says that the uncertainty about momentum (in a particular direction) times the uncertainty about position (in the same direction) has to be bigger than Planck’s constant

$$\Delta x \,\Delta p \geq h \,.$$

Energy and time are also governed by an uncertainty relation

$$\Delta E \,\Delta t \geq h \,.$$

The precise meaning of the uncertainty of quantum mechanics is still hotly debated by physicists, some of whom really like the news media spotlight.