The Quantum World

Successes of the Newtonian World View

Newton's Principia (1687) is the greatest scientific treatise ever written. It contains laws and methods that, for over three hundred years, have provided the basis for a detailed and precise description of ballistics, planetary motion, ocean tides, the dynamics of fluids, the motion of rigid bodies, the stability of bridges, houses, skyscrapers and the human skeleton to name but a few topics.

In short, Newton's laws can describe a vast array of everyday things; principally, those things which are on a scale that we can directly perceive. The fundamental principle is: Given all the forces (causes) acting on a body its motion (effects) could, in principle, be predicted with arbitrary accuracy. All effects had causes. The Newtonian world was deterministic. Our ability to predict was limited only by our practical inability to measure all the forces with infinite precision.

Towards the end of the 19th century, however, it became apparent that all was not well with the Newtonian world view.

Failures of the Newtonian World View

Atoms can't exist!

Perhaps the most catastrophic failure of the Newtonian world view was its failure to explain the stability of matter. The laws of Newton together with the laws of electromagnetism established by James Clerk Maxwell (1831-1879) predict that all accelerating electrons emit electromagnetic radiation. This, after all, is how radio waves are generated.

The problem arises because of the observation that a continuously accelerating electron will have its energy gradually reduced to zero. Consequently,  electrons in atoms must radiate away all their energy since the electrons orbit the nucleus and are, therefore, accelerating. To their dismay, the 19th century physicists' were forced to conclude that the orbiting electrons must spiral into the nucleus in less than one hundred millionth of a second, thereby destroying the atom. But, as far as we know, atoms have been around for several billion years!

The ultraviolet catastrophe

Another spectacular failure was the prediction (Lord Rayleigh, 1842-1919) that a hot oven with a tiny hole through which electromagnetic energy could escape would emit an infinite amount of energy. This was called the ultraviolet catastrophe. These failings, and others, precipitated a deep crisis in physics in the waning years of the 19th century.

Inspired desperation

In 1900 the physicist Max Plancktried to derive a formula that could describe the experimentally observed spectral distribution of radiation from a hot oven. The spectrum was that of a perfect absorber and a perfect emitter of radiation; a so-called black-body. Planck's idea was to model the radiation in an oven as a collection of harmonic waves. He tried the Rayleigh approach, but failed. In an act of inspired desperation Planck introduced what he considered a truly horrible idea, namely: that the energy in a bounded system, like an oven, can change only in discrete amounts. He called these discrete changes in energy quanta (plural of quantum).  Thus was born the quantum hypothesis.

Planck found that he could get a formula that correctly described the black-body spectrum if he assumed the energy of an harmonic wave (or harmonic oscillator) to be given by

where n is the frequency of the harmonic wave. The constant h (= 6.63 x 10-34 joule-second) was new to science. It is called Planck's constant in his honor.

Planck regarded his quantum hypothesis as an expedient mathematical trick which subsequent work would surely prove unnecessary. But he was mistaken. This was no mere trick; in fact,  it represented a complete break with the Newtonian world view, known also as classical physics.

The photoelectric effect

In an act of genius, Albert Einstein took the quantum hypothesis seriously: he interpreted Planck's quantum, literally, as  particles of light that he called photons.  In 1905, Albert Einstein, in one of three truly revolutionary papers published in the German scientific journal  Annalen der Physik, used his photon hypothesis to explain the photoelectric effect. In this effect light falls onto a metal and is found to liberate electrons from the metal's surface only when light above a definite frequency is used. This is true even for extremely low light intensities. Moreover, the number of liberated electrons is directly proportional to the intensity of the incident light. In Maxwell's wave theory the energy of light does not depend on its frequency, so it was very difficult to understand why the frequency of light had anything to do with the liberation of electrons from a metal's surface.

Einstein assumed that a definite amount of energy W is needed to liberate an electron from a metal's surface. Therefore, one had only to shine photons of sufficient energy (and therefore frequency) to liberate the electrons. Moreover, the energy of the liberated electrons Ee is  nothing more than the difference between the energy of the incident photons and the binding energy W of the electron to the surface:

For metals this binding energy W is called the work function. The intensity of light is just given by the number of photons in the beam; the more photons the bigger the intensity of the light and therefore the greater the number of liberated electrons. But only photons whose energy exceeds the work function W are able to strip electrons from the metal. Einstein won the 1921 Nobel Prize in Physics for this amazingly simple but utterly revolutionary explanation of what until 1905 was a must puzzling phenomenon.

Einstein's photon hypothesis is a bold declaration that light has particle-like properties. But we know that light diffracts and thus exhibits wave-like properties. In classical physics, particles and waves are contradictory notions: a thing cannot be both a wave and a particle. Yet, light, electrons, atoms, indeed it is thought everything can, under the right circumstances, exhibit one or the other aspect of these contradictory notions.

The resolution of this paradox required a complete deconstruction, and subsequent reconstruction, of the way we understand the world. Indeed, even what is meant by an ``understanding" has changed. The new understanding was so radical that even Einstein, one of the instigators of the revolution, could not bring himself to fully accept  it.

Basic Concepts

Physical States

Modern physics assumes that all systems in nature, from atoms to the entire universe, can be described by specifying the states in which they can exist. States are usually represented by the symbols For every state in which a system can exist there is an associated complex number a(*) called an amplitude.  The absolute value |a| of this amplitude gives the probability to find the system in the corresponding state; that is, the number |a| is the probability to find the system in the state |A>. A system can exist, in some sense, simultaneously in many states; a particle, like an electron, can, in some sense, be both here and there. This is called the superposition of states. Consider a system that has two possible states (a two-state system). A general state of this system can be expressed as where a and b are complex numbers. The states |A> and |B> are called base states. A radioactive nucleus can be modeled as a system that is in a superposition of two states: decayed and undecayed. The state of this nucleus would be written thus The mathematical theory that describes how to manipulate states and amplitudes is called quantum mechanics. Quantum mechanics is in a very curious position. Here is a theory that is a spectacularly successful description of Nature at its most fundamental level. Indeed, it is the most successful theory ever devised. Yet we still do not have a wholly satisfactory understanding of what the theory really means. In particular, what is the interpretation of the superposition of states? That concept is used routinely to explain all manner of things, from the behavior of stars to the dynamics of the DNA molecule. But there is no consensus about what it really means. What does it mean to say that a nucleus is both decayed and undecayed?

One of the founders of the new mechanics, Erwin Schrodinger, was so perplexed by what he helped to invent that he devised an example of superposition so strange that it puts the notion into extremely sharp focus. The example is called Schrodinger's cat.

A cat is in a box with a lid that is shut. Within the box is a radioactive atom that has a 50-50 chance of decaying in an hour. If the atom decays this triggers a mechanism that breaks a vial of poison gas which kills the cat. The cat has two states: alive or dead. Schrodinger argued that if we take seriously the idea of the superposition of states then we must write, for the cat's state, that is, the cat apparently is in a superposed state of life and death! Then we open the box.

According to the measurement hypothesis (discussed next) when we open the box, we are performing a measurement of the cat's state; this is said to cause the cat's superposed state to collapse into one base state or the other. The cat is found either pushing up the daisies, or purring for its milk. Schrodinger found this so totally absurd that (like Einstein) he could not bring himself to embrace fully the new mechanics he helped create.

This is such a strange notion, a cat that is both alive and dead, that many people have tried to devise different interpretations of the superposition of states that avoids such weirdness. There are many interpretations, but little agreement amongst scientists about which one is the most satisfactory.

Many Worlds: An Embarrassment of Riches

One interpretation of superposed states, proposed in 1957 by Hugh Everett (then at Princeton University), is to suppose that each base state physically exists, but in its own parallel universe. A superposed state corresponds to a superposed set of physically existing parallel universes. Everett would claim that the cat is, in fact, both dead and alive; but it is dead in one parallel universe and alive in another! Before one opens the box the parallel universes coexist (presumably sharing the same spacetime). However, when you open the box this disturbs the delicate superposition and causes the universes to part company; there are now two copies of you. One copy sees a cat happy to get out of the box; the other you is greeted with a depressing spectacle! When we measure (that is, interact with) a system it will be found to be in one of its base states with a probability given by the absolute value of the amplitude for that state. So if we interact with the nucleus in such a way as to discern its state we shall find it either undecayed or decayed, just as when we open the box we find the cat either alive or dead.

But it seems that we have no way to predict which outcome we shall find. We can predict only the probabilities of outcomes. We are unable to say anything about what causes a particular outcome to be realized. In this sense the world is non-deterministic. This is another decisive break with the Newtonian world view.

Quantum states can manifest properties in the sense that a property can be realized when one interacts with the quantum object. For example, the electron in a hydrogen atom manifest the properties: position and momentum. But try as we might, we cannot measure both of these properties (or attributes) simultaneously with arbitrary precision. If we try to measure the momentum very precisely we find that the position attribute becomes less well-known and vice versa. The physicist Werner Heisenberg (1901-1976) produced one version of the new mechanics, called quantum mechanics, in 1925. The other version was created by Erwin Schrodinger. Heisenberg discovered the following uncertainty principle This states that if you know the position of a particle with an accuracy Dx then you cannot know the momentum of the particle to better than Dp = h/Dx. There is an irreducible level of uncertainty in our knowledge of the world.

However, the interpretation of the uncertainty principle is controversial: it could mean that a particle does not have a well-defined position and momentum, or it could mean that it might have well-defined values for these properties, but we simply cannot determine their values with infinite precision. There is no agreement about which if either interpretation is correct.

The orthodox interpretation, called the Copenhagen interpretation---devised by one of the great founders of quantum theory Niels Bohr, adheres to the first interpretation. According to Niels Bohr, quantum systems do not necessarily have definite properties. Properties are revealed, made real, only when we interact with a quantum system. Therefore, an electron cannot be said to have a position or a momentum, or any other property for that matter, until we have interacted with it and made the property real.

If you don't look at the Moon is still there?

Quantum Fields

We imagine the universe filled with rather mysterious entities called quantum fields. In our current understanding there is a quantum field for every type of fundamental quantum object. There are two kinds of quantum field: fermionic and bosonic. The fermionic quantum fields are: the electron, muon, tau, and their associated neutrinos, as well six quark fields called (somewhat unimaginatively) up, down, strange, charm, bottom and top. The bosonic fields are: the photon, the Z and the W, gluons, the Higgs and maybe the graviton. Perhaps one day we shall discover that these different quantum fields are not really different, but are rather different aspects of one unified entity. We shall take up this question later.

Indistinguishability

All quantum objects of a given type are absolutely identical. Given a set of electrons it is impossible to keep track of an individual electron because there is no way to distinguish one electron from another. The indistinguishability of quantum objects has profound consequences. One is that all atoms of a given element, sharing the same environment, are identical. The consequences of indistinguishability vary according to whether we are dealing with fermions (like electrons) or bosons (like photons).

Fermions

At most one fermion can occupy a given quantum state. Were this not true, structure could not exist in the universe. Since electrons are fermions, each one in an atom occupies a different state. This is the famous Pauli Exclusion Principle, which explains the periodic table of elements. Electrons tend to keep apart from each other because they cannot occupy the same space.  Therefore, even if electrons were electrically neutral there would still be an effective repulsive force between them arising from their tendency to stay away from each other.

Bosons

Bosons on the other hand, like photons, can crowd into the same quantum state. Indeed, there is no limit to the number of bosons that can occupy the same state. If matter were purely bosonic everything in the universe would, eventually, condense into one giant universal blob! A laser is an example of bosonic condensation in which a very large number of photons occupy the same state, thereby producing an intense coherent beam of light.

Wave Particle Duality.

The particle-like aspects of matter emerge when we compute the energy of the quantum fields. We find that (at our relatively low energies) the energy of these fields can be written as multiples of a basic quantum of energy. Each such quantum is interpreted as a particle, analogously to Einstein's photon interpretation of light. An electron is the quantum of the electron quantum field; a top quark is the quantum of the top quark quantum field. And so on.

This dual nature of quantum objects, their wave-like and particle-like attributes, is most strikingly evident in the fundamental formula, first proposed by prince Louis-Victor-Pierre-Raymond de Broglie in 1924 while a graduate student at the University of Paris,

where p is the momentum associated with the particle-like aspects of the quantum field and l is the wavelength associated with its wave-like aspects. De Broglie predicted that electrons (and indeed all matter) should exhibit wave-like aspects. When this was shown to be true of electrons, by Davisson and Germer---working at Bell Labs in New York, de Broglie was rewarded with the 1929 Nobel Prize in Physics.

But again, like so much of quantum theory, the interpretation of this formula is controversial. Is the electron a particle or is it a wave? Perhaps, the only sensible answer is: It is most probably neither. An electron is a thing for which no analog exists in the macroscopic world. At a fundamental level neither waves nor particles exist, it would seem; but whatever the true nature of these quantum objects they exhibit aspects of both waves and particles.

Basic Law

The basic law of evolution, that replaces Newton's law of motion, is the Schrodinger Equation. The law states how the state of a system changes with time. The quantity H is called the Hamiltonian. Much of modern physics is concerned with trying to discern its detailed form.

Stability of Matter

The stability of matter is now understood to be a consequence of the fact that quantum systems, for example the hydrogen atom, can exist only in definite states, each with a reasonably well-defined energy. Every atom has a lowest state, called the ground state, into which it can settle. Since there are no lower states into which the electron can descend the latter cannot spiral into the nucleus. The atom remains stable in its ground state, in principle, forever.

Hydrogen is the simplest atom, consisting of an electron bound to a single proton. The basic energy levels of the hydrogen atom can be written as

The negative energy signifies that the electron when bound in an atom has less energy than when the electron is free. When n = 1, the hydrogen atom is said to be in its ground state; that is, the state of lowest energy. If the atom absorbs a photon of just the right energy it can be excited to the level n = 2, or higher. The energy required to go from n = 1 to  n = 2, for example, is just An excited atom does not remain in such a state for long. Indeed, typically atoms de-excite in about one hundred millionth of a second, releasing their energy in the form of photons. The pattern of radiation from an atom is unique for each element. The atom of a given element can emit only photons of a definite set of energies (and therefore frequencies). This is why every gaseous element shines with a different blend of colors.

Sometimes the amount of energy given to an atom, in its ground state, is large enough to strip an electron from the atom. To free the electron from the hydrogen atom requires the injection of 13.6 eV (electron-Volts) of energy. The electron would be free of the atom, but would have zero kinetic energy! If the injected energy, however, would be more than 13.6 eV the extra energy would be imparted to the electron as kinetic energy. This is just like the freeing of electrons from a metal surface in the photoelectric effect. An atom that has had one or more electrons liberated is said to be ionised.