Egan, Greg - Foundations 4 - Quantum Mechanics

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Foundations

by Greg Egan

4: Quantum Mechanics

The first three articles in this series dealt with special and general relativity, the two great

twentieth-century theories of the geometry of spacetime and its relationship with matter

and energy. This article will describe the ideas behind a second, simultaneous revolution

in physics, one that has had even more profound philosophical and technological

consequences: quantum mechanics.

The Birth of Quantum Mechanics

In the second half of the nineteenth century, the Newtonian description of the dynamics

of material objects was supplemented by an equally successful theory encompassing all

of electrostatics, magnetism and optics. The physicist James Clerk Maxwell brought

together a number of disparate laws that had been found to govern quite specific

phenomena — such as the force between two motionless electric charges — into a unified

description of an electromagnetic field. Light, and most other forms of radiation,

were seen to consist of oscillations in this field, or electromagnetic waves. This

confirmation of the wave-like nature of light made sense of many long-standing

observations, including the phenomenon of interference: if you allow light of a single

wavelength to travel through two adjacent narrow slits in a barrier and then recombine on

a screen, it produces patterns of dark and light stripes. Since the difference in the time it

takes for light waves from the two slits to reach the screen varies from place to place, the

waves shift in and out of phase with each other, resulting in varying degrees of

constructive interference (where the contributions to the field from both slits point in the

same direction), and destructive interference (where they point in opposite directions).

Newtonian dynamics and Maxwellian electrodynamics cut a wide swath through

the scientific problems of the day. However, by the end of the nineteenth century a

number of serious discrepancies had been found between experimental results and

predictions based on these two theories. Newtonian physics was soon to be superseded

by special relativity, but the most glaring problems had nothing to do with the motion of

objects at high velocities, so the explanation had to lie in another direction entirely.

One of the biggest puzzles involved the spectrum of radiation emitted by hot

objects: thermal radiation. This is visible to the naked eye when, for example, the

tungsten wire in a light bulb becomes white hot. There's an idealised class of objects for

which this effect is particularly easy to analyse: if an object is a perfect absorber and

emitter of electromagnetic waves across the entire spectrum, its thermal radiation should

depend solely on its temperature, rather than any idiosyncratic properties of the stuff from

which it's made. Physicists call this a black body, since it should appear black to the

naked eye at room temperature. The cavity of a furnace containing nothing but the

thermal radiation from its heated walls, with a tiny hole through which radiation can

escape to be observed, serves as a good approximation to a black body, both theoretically

and experimentally, so black body thermal radiation is also known as cavity radiation.

Maxwell's theory suggested that the electromagnetic field inside a cavity should

be treated as something akin to the three-dimensional equivalent of a piano string being

bashed at random, simultaneously vibrating with every possible harmonic. A piano

string has evenly spaced harmonics, say 500 Hz, 1000 Hz, 1500 Hz, and so on, which

occur when an exact number of half-wavelengths fit the length of the string; the fact that

the ends of the string are fixed prevents other frequencies being produced. An

electromagnetic field in a three-dimensional cavity is subject to similar boundary

Egan: "Foundations 4"/p.2

conditions, but unlike a piano string the field's vibrations are free to point in different

directions. For example, the field in a cubical cavity might vibrate in such a way that 5, 7

and 4 half-wavelengths span the cavity's width, breadth and height respectively, because

of the way the waves are oriented with respect to the walls. But waves of exactly the

same frequency, oriented differently, would fit just as well with 4, 5 and 7 half-

wavelengths spanning the same three dimensions.

This makes the situation more complicated than it is for a piano string, but it's still

not too hard to count the modes available to the field: the number of distinct ways in

which it can vibrate. Figure 2 isn't a drawing of a furnace cavity; rather, each point here

represents a different mode, with the x, y and z coordinates of the point giving the

number of half-wavelengths that fit across the width, breadth, and height of the cavity.

The more tightly packed the waves are, the shorter their wavelength and the greater their

frequency. The exact frequency of any mode is proportional to its distance from the

centre of the diagram — that's just a matter of Pythagoras's theorem, and the relationship

between frequency and wavelength. So the number of points between the two spherical

shells counts the number of modes in the frequency range ∆F. For small values of ∆F,

this is proportional to the surface area of the inner sphere, which is proportional to F2.

Because the walls of the cavity are assumed not to favour any particular

frequency, every possible mode of the electromagnetic field should have, on average, an

equal share of the total energy. The trouble is, the field has an infinite number of modes

— at ever higher frequencies, you just keep finding more of them. If the energy from the

furnace really was free to spread itself between them, giving them all an equal share, that

would be a never ending process, like gas escaping into an infinite vacuum. The average

Egan: "Foundations 4"/p.3

frequency of the radiation in the cavity would wander off towards the ultraviolet and

beyond, never stabilising at any fixed spectrum.

The reality is nothing like this, as Figure 3 shows. The observed spectrum

reaches a peak at a certain frequency, then tapers off. Clearly, something prevents the

energy of the field from being equally distributed amongst all possible modes. But what?

The analysis we've given so far assumes that energy can be spread as thinly as

you like; as more and more modes share the energy of the field, each one ends up,

individually, with a smaller amount. But what if energy couldn't be endlessly subdivided

like this? What if you eventually reached a minimum amount, a “particle” of energy, as

indivisible as some particles of matter presumably are? Instead of taking on any value

whatsoever, energy would only be found in exact multiples of this amount.

In 1900, Max Planck proposed that this was the case, and called the minimum

amount a quantum. Though it might have been simplest to decree a fixed amount of

energy as the size of one quantum, like the fixed mass of an electron, that wouldn't have

solved the cavity radiation problem: with an infinite number of modes available, the finite

number of quanta would still have been free to “escape” to ever higher frequencies. The

only way to prevent this was to propose that higher frequency modes required a greater

minimum energy than lower frequency modes, raising a series of ever higher hurdles to

counteract the tendency for the energy to spread. Planck found that making the energy of

one quantum proportional to the frequency of the electromagnetic wave, as in Equation

(1), would yield a spectrum precisely in agreement with observation, if the constant of

proportionality was chosen correctly. This value, now known as Planck's constant,

is referred to by the letter h, and has a value of 6.625 x 10-34 Joules per Hz.

Egan: "Foundations 4"/p.4

E = h F (1)

You might be wondering how Equation (1) dictates the nice tapered curve in

Figure 3. What's to stop all the energy in the furnace from going into a single, super-

high-frequency quantum, making the spectrum an isolated peak way off to the right of the

graph? The same thing that stops all the energy in the Earth's atmosphere from ending up

concentrated in a couple of atoms: it's just not very likely. Of all the possible ways a

certain total amount of energy can be distributed between billions of possible modes of

cavity radiation, the vast majority look like the curve in Figure 3.

Over the first three decades of the twentieth century, many other experiments

confirmed the quantisation of light, and led independently to the same value for Planck's

constant. One famous example is the photoelectric effect. When ultraviolet light is

shone on a metal plate in a vacuum tube it blasts electrons off the surface of the metal.

The energy of the individual electrons released this way (as opposed to the total energy

they possess en masse) turns out to be completely independent of the intensity of the

light shone on the plate, and can only be increased by using light of a greater frequency.

This makes sense if the electrons are absorbing individual quanta, rather than gaining

energy from the electromagnetic field as a whole. More intense light of a given frequency

contains more quanta of the same energy, and can blast more electrons off the plate —

but only raising the frequency of the light, and hence the energy of the quanta, can

increase the energy of each individual electron.

Quanta of light, which came to be known as photons, were shown again and

again to behave like localised, indivisible particles. But there was no denying the fact that

light also behaved like a wave, exhibiting interference effects. Neither aspect could be

ignored, but it was not at all clear how to synthesise the two into a coherent new

description of electromagnetism.

In parallel with these revelations about light, physicists were grappling with the

problem of the structure of atoms. Electrons had been discovered in 1897, and in 1911

Ernest Rutherford had found strong experimental evidence for the theory, first proposed

by Hantaro Nagaoka, that atoms consisted of electrons orbiting a positively charged

nucleus. The puzzle here was that charged particles moving in a circle emit

electromagnetic waves, so the electron should have radiated away all its energy and

plunged into the nucleus. Not even Planck's quantised photons could rule this out.

In 1913, Neils Bohr proposed that the energy of the electrons themselves was

quantised, and the existence of a minimum allowed energy kept them from falling into the

nucleus. Bohr came up with a formula for the energy levels of the single electron in a

hydrogen atom, constructed in order to agree with the observed spectrum of light emitted

and absorbed by hydrogen. This spectrum consisted of a discrete set of sharply defined

Egan: "Foundations 4"/p.5

frequencies, which could now be interpreted as the frequencies of photons whose

energies matched the differences in energy between the allowed states of the electron.

An electron could only move to a higher energy level by absorbing a photon that provided

exactly the right amount of energy, and it could only drop back to a lower level by

emitting a photon that carried the energy away again. This was by far the most

successful model of atomic structure to date, but Bohr's formula was even more

mysterious than Planck's. Why were only certain energy levels available to the electron?

The first hint at an answer came from the suggestion by Louis de Broglie in 1924

that matter, as well as radiation, might behave like both a wave and a particle. This was

confirmed spectacularly a few years later, in experiments showing that electrons fired at a

crystal were reflected back most often in certain directions: those in which a wave that

scattered off the regularly spaced atoms of the crystal would undergo constructive

interference. Since then, interference effects have been demonstrated for all kinds of

particles, including entire atoms.

To examine de Broglie's idea more closely, we need to ask what the wavelength

and frequency of the “matter wave” associated with a particle should be. One reasonable

starting point is the relationship that worked so successfully for Planck with photons:

E=h F. Since F is the frequency of the wave (the number of oscillations per second), the

period of the wave, the time each oscillation takes, is:

T = 1/F

= h/E (2)

Since the wave for a photon is moving forward through space at the speed of light, c,

each cycle is spread out over one wavelength:

L = c T

= c h/E

Throughout these articles we've been using units where c=1, but it's worth leaving the c

in here for a moment, and stating the fact that the momentum, p, of a photon with energy

E is always p=E/c. (This must be true in order for the 4-momentum of the photon to be a

null vector, a spacetime vector with an overall length of zero, as discussed in the

previous article. The relationship is obvious when c=1, but it holds regardless of the

units used.) So the wavelength of light is related to each photon's momentum by:

L = h/p (3)

Equations (2) and (3) are the formulas de Broglie proposed for the period and wavelength

Egan: "Foundations 4"/p.6

of matter waves. Let's see what such a wave might look like on a spacetime diagram.

Figure 4 shows a travelling sine wave with period T and wavelength L. We don't

actually know that a matter wave will ever take the form of a sine wave, but we might as

well start with a simple possibility like this and see where it leads us. The third axis on

the diagram represents the “strength” of the wave, or amplitude, traditionally labelled ψ

(the Greek letter psi). Exactly what ψ means, physically, is something we've yet to

determine. The equation for ψ in terms of x and t, the wave function, is:

ψ(x,t) =sin(2π(x/L – t/T)) (4a)

= sin(2π(px – Et)/h) (4b)

It's not hard to see that the wave defined by Equation (4a) will go through a complete

cycle whenever x increases by one wavelength, L, or time increases by one period, T.

The expression 2π(x/L – t/T) is known as the phase of the wave: each individual peak

(or trough) in Figure 4 has a certain constant phase, and successive peaks (or troughs)

have a phase of 2π more than the last one. The minus sign here, rather than a plus sign,

guarantees that a peak of the wave will move in the positive x direction: to keep

2π(x/L – t/T) constant, x must increase as t increases.

If we define the propagation vector for the wave, k, as:

k=(1/L)∂x + (1/T)∂t

and we write x = x∂x + t∂t for the spacetime vector that points from the origin to any

Egan: "Foundations 4"/p.7

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FoundationsbyGregEgan4:QuantumMechanicsCopyright©GregEgan,1999.Allrightsreserved.Thefirstthreearticlesinthisseriesdealtwithspecialandgeneralrelativity,thetwogreattwentieth-centurytheoriesofthegeometryofspacetimeanditsrelationshipwithmatterandenergy.Thisarticlewilldescribetheideasbehindasecond,simult...

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