apollodorus: I believe I am quite well prepared to relate the events you are asking me about, for just the other day I happened to be going into Athens from my home in Phalerum when an acquaintance of mine caught sight of me from behind and called after me, jokily,

‘Phalerian! You there, Apollodorus! Wait for me, will you?’

So I stopped and waited.

‘I have just been looking for you, Apollodorus. I wanted to get from you the story about that party of Agathon's with Socrates, Alcibiades and the rest, the time when they were all together at dinner, and to hear what they said in their speeches on the subject of love. Someone else was telling me, who had heard about it from Phoenix, son of Philippus, and he said that you knew about it too. Actually he could not give any clear account of it, so you must tell me. You are in the best position to report the words of your friend. But tell me this first’, he went on. ‘Were you at that party yourself or not?’

‘It certainly looks as if your informant was rather confused’, I replied, ‘if you think the party you are asking about occurred recently enough for me to be there’.

Quantum mechanics as we have discussed it here works remarkably well. When we can figure out how to calculate something using quantum mechanics, we know of no situation where it gives the wrong answer. At a pragmatic level, it therefore works. It does, however, have some aspects about it that are very different from the classical view of the world. Interpreting what quantum mechanics says about reality is a very tricky subject with strange consequences and truly bizarre proposals about how reality actually works. Here, we try to give a short introductory summary of some of those key ideas. It is necessarily brief and incomplete and is certainly not conclusive. Indeed, this field of the interpretation of quantum mechanics can be considered unresolved, even if protagonists of various interpretations might attempt to convince us otherwise. Fundamentally, we do not know how to resolve some of the more philosophical aspects because we have no experiment to discriminate between them, though some points previously believed to be only philosophical have been resolved by experiments, with remarkable consequences.

Hidden variables and Bell's inequalities

Is quantum mechanics truly random? Perhaps the solution to the apparent randomness of quantum mechanics, with states collapsing into eigenstates with only statistical weights, is that quantum mechanics as stated is incomplete, in the sense that classical statistical mechanics is incomplete.

Thus far, we have dealt primarily with energy, position, and linear momentum and have proposed operators for each of these. One other quantity that is important in classical mechanics, angular momentum, is particularly important also in quantum mechanics. Here, we introduce angular momentum, its operators, eigenvalues, and eigenfunctions. If this discussion seems somewhat abstract, the reader can be assured that the concepts of angular momentum will become very concrete in the discussion of the hydrogen atom.

One aspect of angular momentum that is different from the quantities and operators discussed previously is that its operators always have discrete eigenvalues. Whereas linear momentum is associated with eigenfunctions that are functions of position along a specific spatial direction, angular momentum is associated with eigenfunctions that are functions of angle or angles about a specific axis. The fact that the eigenvalues are discrete is associated with the fact that for a single-valued spatial function, once we have gone an angle 2π about a particular axis, we are back to where we started. The wavefunction is presumably continuous and single-valued and, hence, we must therefore have integral numbers of periods of oscillation with angle within this angular range; this requirement of integer numbers of periods leads to the discrete quantization of angular momentum.

Another surprising aspect of angular momentum operators is that the operators corresponding to angular momentum about different orthogonal axes (e.g., and) do not commute with one another (in contrast, e.g., to the linear momentum operators for the different orthogonal coordinate directions).

The “Δ” quoted is the absolute value of the uncertainty in the last two digits of the quoted numerical value corresponding to one standard deviation from the numerical value given. Hence, for example, the possible values of Planck's constant within one standard deviation of the best estimate shown lie between 6.626 068 24 and 6.626 069 28 J s.

The speed of light in vacuum has been chosen to have the exact value shown because the meter is now defined as the length of the path traveled by light in vacuum during the time interval of 1/299 792 458 of a second. The magnetic constant (also known as the permeability of free space) is chosen to have the value shown because it is an arbitrary constant that arises from the choice of the system of units and the electric constant (also known as the permittivity of free space) then follows from it and the (chosen) velocity of light because, by definition, so all three of these quantities have no uncertainty by definition. The Bohr magneton is µB = eħ/2me . The fine structure constant is α = e 2/4πε0 cħ.

These values are the CODATA Internationally recommended values as of 1998. Reference http://physics.nist.gov/cuu/Constants/index.html.

So far, we have introduced quantum mechanics through the example of the Schrödinger equation and the spatial and temporal wavefunctions that are solutions to it. This has allowed us to solve some simple but important problems and to introduce many quantum mechanical concepts by example. Quantum mechanics does, however, go considerably beyond the Schrödinger equation. For example, photons are not described by the kind of Schrödinger equation we have considered so far, though they are undoubtedly very much quantum mechanical.

To prepare for other aspects of quantum mechanics and to make the subject easier to deal with in more complex problems, we need to introduce a more general and extended mathematical formalism. This formalism is actually mostly linear algebra. Readers probably have encountered many of the basic concepts already in subjects such as matrix algebra, Fourier transforms, solutions of differential equations, possibly (though less likely) integral equations, or analysis of linear systems, in general. For this book, we assume that the reader is familiar with, at least, the matrix version of linear algebra – the other examples are not necessary prerequisites. The fact that the formalism is based on linear algebra is because of the basic observation that quantum mechanics is apparently absolutely linear in certain specific ways as we discussed previously.

Thus far, we have dealt with the state of the quantum mechanical system as the wavefunction Ψ(r, t) of a single particle.

Quantum mechanics and real life

Quantum mechanics, we might think, is a strange subject, one that does not matter for daily life. Only a few people, therefore, should need to worry about its difficult details. These few, we might imagine, run about in the small dark corners of science, at the edge of human knowledge. In this unusual group, we would expect to find only physicists making ever larger machines to look at ever smaller objects, chemists examining the last details of tiny atoms and molecules, and perhaps a few philosophers absently looking out the window as they wonder about free will. Surely, quantum mechanics therefore should not matter for our everyday experience. It could not be important for designing and making real things that make real money and change real lives. Of course, we would be wrong.

Quantum mechanics is everywhere. We do not have to look far to find it. We only have to open our eyes. Look at some object, say a flowerpot or a tennis ball. Why is the flowerpot a soothing terra-cotta orange color and the tennis ball a glaring fluorescent yellow? We could say each object contains some appropriately colored pigment or dye, based on a material with an intrinsic color, but we are not much further forward in understanding. (Our color technology would also be stuck in medieval times, when artists had to find all their pigments in the colors in natural objects, sometimes at great cost.)

In this section, we return to the harmonic oscillator and consider it in a mathematically more elegant way. This approach leads to the introduction of “raising” and “lowering” operators that take us from one harmonic oscillator state to another. The introduction of these operators allows us to rewrite the harmonic oscillator mathematics quite economically. We then show that the electromagnetic field for a given mode can also be described in a manner exactly analogous to a harmonic oscillator. In this case, we describe the states of this generalized harmonic oscillator in terms of the number of photons per mode, with that number corresponding exactly to the quantum number for the corresponding harmonic oscillator state. The raising and lowering operators are now physically interpreted as “creation” and “annihilation” operators for photons and are key operators for describing electromagnetic fields.

By this process, we can describe the electromagnetic field quantum mechanically, rather than our previous semiclassical use of quantum mechanical electron behavior but with classical electric and magnetic fields. We say we have “quantized” the electromagnetic field. This quantization is then the basis for all of quantum optics. This approach also prepares us for discussions in subsequent chapters of fermion operators and of the full description of stimulated and spontaneous emission.