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Introduction

Quantum Mechanics is one of the 20th century's great achievements. It is however very strange. It is so strange that Richard Feynman famously stated that no one understands it.

At the turn of the 20th Century the world was conceptually a much simpler place. The geometry of Euclid had existed for nearly 2000 years. The physics of Newton, revolutionary in it's day, had proved flawless in it's predictions for 200 years. Netwonian Physics had some spectacular successes - Neptune had been discovered as a result of a search for an unseen planet causing minute discrepancies in the predicted orbit of Uranus. There was no obvious reason to doubt its correctness.

Newtonian physics also had a certain intuitiveness. Things in the Newtonian world react to forces. People feel forces. They apply forces to objects every day of their lives. They are aware when forces are applied to them. They are aware that rigid body can transmit forces. They are aware that if a substance is compressed, the material pushes back against whatever is compressing it. They are aware of friction. They are generally aware of abstract principles like Galileo's Principle of Relativity - anyone who has travelled in a car, boat or aeroplane knows it impossible to know how fast you are going without reference to some external point (looking out the window). All motion is relative.

A few, rare individuals had guessed that other options existed and seriously considered them. Georg Riemann climbed the Alps with a theodilite to check that the angles of a triangle did indeed add to 180^o. Fifty years later, Einstein would use Riemann's mathematics to construct the General Theory of Relativity.

The Atom

The theory of atoms dates back to the ancient Greeks. Leucippus and Democritus (460 BC - 371 BC) proposed that matter is made up of small, eternal, indivisible particles called atomos; atoms of the same type being identical in all respects.

John Dalton (1766-1844) and others proposed that atoms could be used to explain the weights and proportions of compounds produced and consumed in chemical reactions. By 1860, the atomic weight of the 60 elements then known had been established. In 1869, Mendeleyev and Meyer produced the chemical periodic table, since been updated by Ramsay, Mosley and Seaburg.

Boyle, Dalton and others had used the concept of atoms to explain the properties of gases. Ludwig Boltzmann (1844 - 1906) developed a statistical description of the motion of atoms/molecules in gases.

The existence of atoms was still controversial by the beginning of the 20th Century. Bitter debates, frustration and rejection ultimately lead to Boltzmann's suicide (1906). In 1905, Einstein published a paper on Brownian motion. Small grains of pollen "judder" about under the microscope as the result of atoms crashing into them at high speed. Within a matter of years of publication, the existence of atoms became universally accepted within the scientific community.

But what do atoms look like?

Version 1 - The classic atom

The picture of the atom taught at school resembles that of the solar system. A heavy nucleus sits at the centre of the atom playing the part of the Sun. Smaller lighter electrons play the part of the planets orbiting the nucleus. The force of gravity is replaced by electrical attraction – each electron carries a single negative charge and is attracted towards the positively charged nucleus.

The "classic" picture of an atom slowly emerged from a series of experiments.

• Milikan's experiment showed that charge was quantised - charge is lost or gained as additional electrons are lost or gained.
• Thomson demonstrated the existence of negatively charged particles (electrons) using Cathode Ray Tubes.
• Rutherford demonstrated that most of the mass of an atom resided in a small positively charged core by firing alpha particles at thin gold foil. Most of the particles passed through the gold foil as if it consisted of empty space (because it mostly is). A few struck the nucleus, recoiling back at sharp angles and allowing for the calculation of the size of the nucleus.
• Chadwick discovers the neutron (required to explain atomic weights)

Is the classical picture correct?

The model is fine for teaching purposes in junior high school. Classical Electromagnetic theory however suggests that the electron should rapidly spiral down and crash onto the nucleus, radiating energy as it goes. That's not what happens - atoms are stable. The model is wrong.

Version 2 - The electron cloud

The second picture of the atom is based on a niave interpretation of the equations of QM discovered by Schrodinger. Atoms are portrayed as a nuclei surrounded by electron clouds. (Electrons swarm around the nuclei like flies around a cowpat ).

Is the electron cloud picture correct?

The electron cloud model is based on the equation of QM. It is adequate for computational applications, including most of Quantum Chemistry and Quantum Mechanics. In particular it is (often) possible to to calculate or estimate such things as atomic energy levels, the spectrum of emitted and absorbed radiation, strength of chemical bonds and the properties of materials.

Problems only arise if one looks closely.

Quantum Waves

Light is a wave. Or is it?

By the beginning of the 20th century, it was generally regarded that experimental evidence such as diffraction showed light is a wave.
Maxwell Equations (1875) predict waves of Electromagnetic radiation. It is suggested light is a form of electromagnetic radiation. The speed of light predicted from Maxwells Equations matches the measured speed of light.

But...

Max Plank use his "quantum" hypothesis to explain the spectrum of black body radiation.(1901).
Einstein (1905) produced his light quanta (particle) hypothesis to explain the photo-electric effect.

Electrons are particles. Or are they?

J.J. Thompson demonstates particle nature of electrons using cathode ray tubes (1897).

But...

De Broglie (1924) suggests that matter might also behave like a wave in his doctoral thesis, which would have been rejected had it not been for the intervention of Einstein.
Davisson, Germer, Thompson (son of J.J.Thompson) demonstrate electron diffraction in a crystal (1927) - i.e. experimental proof that electrons behave like waves.

Schizophrenia

Experimental evidence now indicates that all sub-atomic particles can be made to demonstrate both particle and wave like properties. How can the two behaviours be reconciled?

In fact, the "best" description of the motion of a particle like an electron is that of a wave moving through space. It can be diffracted. Whenever energy is transferred form the wave, the energy is transferred as a quanta. QM only describes the motion of the wave. The transfer of energy from the wave cannot be predicted precisely, the best we can do is calculate probabilities of an interaction.

The diagram at left shows the result of a double slit diffraction experiment. QM predicts a waveform with alternating bands of high and low intensity. Frames (a)-(d) show snapshots over time as individual particle "arrive" (transfer energy) to the screen. Initially it is difficult to see any pattern (a)-(b). As time goes on, the waveform reveals itself (d).

It is tempting to think of electrons "hiding" inside the wave, only emerging when interacting with some other part of the world. But, as we shall see, that would probably be a step too far.

Quantum waveforms are complex valued. and interfere destructively. Probability waves are always "positive" and cannot interact distructively. If Y(x,t) is the waveform describing the state of a particle, then the position probability distribution P(x,t) is given by the formula

Why? A very good question. If you can come up with a really good reason for this, you will have solved one of the truly great mysteries of the Universe.

Notation

Dirac bra-ket notation is sometimes used to descrive quantum waveforms. For example, the waveform describing a electron might be denoted |e^- >, the waveform describing a sperm whale might be denoted |whale>.

Heisenberg's Uncertainty Principle

Heisenberg's Uncertainty Principle (HUP) lies at the heart of Quantum Mechanics. Without HUP, QM would be inconsistent.

To see that this is so, ask the question: Why are we forced only to deal with probabilities? In the case of an electron moving through space, why can we not build a measuring device and just measure the position of the electron? There are obvious engineering problems, but for the moment let us suppose that it can be done. QM certainly does not preclude the construction of such a machine.

To completely avoid probabilities, it would be necessary to also measure the electron's momentum. Once both the position and momentum are known, then it possible in theory (at least in classical theory) to calculate the position of the particle precisely into the future.

HUP kicks in: It is not possible to obtain exact measurements of both position and momentum. It is therefore not possible to bypass the probabilistic description of nature.

More precisely, HUP looks like

where s_x is the standard deviation of the probability distribution of a particle's position (x), s_x is the standard deviation of the probability distribution of the particle's momentum (p), h is Plank's constant = 6.625 x 10^-34 J-s. HUP guarantees that the more "precise" the information available about a particle's position is, the greater the uncertainty in the particle's momentum. Even if the a particle's momentum had been previously be determined with a high degree of accuracy, any measurement of the position would destroy the validity of the first measurement of momentum. Similarly the more precise the information about a particle's momentum is, the greater the degree of uncertainty in the particle's position. It is simply not possible to make accurate measurements of both position and momentum at the same time.

HUP applies to sub-atomic particles, people, cars, galaxies, sperm whales and bowls of petunias, but Planks constant is very small. At normal energy (amd momentum) levels, the uncertainty in position of an electron is (not surprisingly) roughly the size of an atom.

Absorption and Emission of Radiation

Quantum Mechanics predicts that the energy levels of an atom are quantised; the energy levels are characterised by quantum numbers.

Chemists use the misleading term "orbitals" to describe the waveforms associated with each energy level. They also, for historical reasons, refer to the orbitals by letter (s,p,d,f) rather than quantum number. The figure below shows the shape of the various "orbitals". Note however that the orbitals do not have sharp boundary; rather the probability of finding an electron gradually drops off with dstance from the nucleus.

Protons and neutrons in an atomic nuclei behave in similar way to electrons in an atom; models using "orbitals" can be used to predict the stability of atomic nuclei. The pattern of stability is not the same as that of the periodic table since the energy levels associated with the nuclear orbitals do not match their atomic equivalents, and therefore each nuclear "shell" contains a different number of orbitals than the equivalent atomic shell.

Atoms randomly make transitions from high energy states to low energy states. If an atom starts in a state with energy E₀ and ends up in a state with energy E₁, then the difference in energy E₀ - E₁ will be carried away as a photon. The result is that radiation (radio waves, micro-waves, infra-red, light, ultra-violet, x-rays) emitted by an atom have very precise frequencies, giving rise to characteristic spectra.

The process works in reverse. Atoms may randomly "absorb" passing photons, but only if the energy of the photon matches the difference between its existing energy state and a higher energy state. If the photon is absorbed, the atom goes into the higher energy state.

Tunnelling

In general, the probability distribution of finding a particles at any point in space is not zero, rather it tapers off slowly approaching zero as the distance from the distribution mean increases.That allows particles to "tunnel" through any barrier, even the event horizon of a black hole. However the probability of a macroscopic object like a car tunneling through a barrier like a brick wall is almost zero.

HUP applies to measurements of time and energy. Mass is energy. It follows that a vacuum can create particles out of nothing - provided that they do not live longer than that the time interval predicted by HUP.

Double slit Diffraction

Double slit diffraction is the example of quantum behaviour par excellence. Feynman claimed that if it was possible to understand diffraction, then it was possible to understand all of Quantum Mechanics. A double slit diffraction apparatus consists of (1) a particle source - typically a light or electron source, (2) a screen with 2 parallel slits through which particles can pass, and (3) a screen where the particles that have passed through the double slits arrive. The characteristic double slit pattern is shown in the diagram below and can be understood as a result of "waves" interfering with each other.

Which slit did the electron go through?

It is tempting to consider the possibility that the interference pattern results from particles going through one slit interfering with the motion of particles going through the other slit, however it is possible to set the apparatus up so that statistically only one particle passes through the double slit apparatus at a time; the standard double slit interference pattern is still seen. That means that the photon or electron "interferes" with itself!

Richard Feynman came up with an intuitive method of calculating the probabilities of any event (E.g. the particle in a double slit diffraction arriving at the point A in the diagram below): (1) Draw diagrams showing every possible interaction, (2) calculate the amplitude of the event given the interactions in the diagram, and (3) add the amplitudes for each diagram together. The existance of virtual particles significantly increases the complexity of calculations.

There are some excellent resource on special relativity on the web so it does not make sense to duplicate them. Try here

The main features of Special Relativity relevant to Quantum Mechanics are

• The speed of light is always constant (in a vacuum).
• Time passes at different rates for observers travelling at different velocities.
• Events which appear simultaneous in one frame of reference will not be simultaneous in a frame of reference moving relative to the first.
• Minkowski diagrams (space-time diagrams) can be used to represent the structure of space-time.

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Copyright (c) Shaun O'Kane 2003.