1 Beginnings

Quantum physics started out as a rather des­perate mea­sure to avoid some of the spec­tac­ular fail­ures of what we now call “clas­sical physics.” The story begins with the dis­covery by Max Planck, in 1900, of the law that per­fectly describes the radi­a­tion spec­trum of a glowing hot object. One of the things clas­sical physics had pre­dicted was that you would get blinded by ultra­vi­olet light if you looked at the burner of your stove.

At first it was just a fit to the data — “a for­tu­itous guess at an inter­po­la­tion for­mula,” as Planck him­self described his law. A few weeks later, how­ever, it was found to imply the quan­ti­za­tion of energy in the emis­sion of elec­tro­mag­netic radi­a­tion and thus to be irrec­on­cil­able with clas­sical physics. According to the clas­sical theory, a glowing hot object emits energy con­tin­u­ously. Planck’s for­mula implies that it emits energy in dis­crete quan­ti­ties pro­por­tional to the fre­quency f (in cycles per second) of the radiation:

E = hf,

where h = 6.626 069×10−34 Joule sec­onds is Planck’s con­stant. Often it is more con­ve­nient to use the reduced Planck con­stant ℏ (“h-​​bar”), which equals h divided by 2π (the ratio of a circle’s cir­cum­fer­ence to its radius). With this we can write

E = ℏω,

where the angular fre­quency ω (in angle per second) replaces f.

In 1911, Ernest Ruther­ford pro­posed a model of the atom based on exper­i­ments con­ducted by Hans Geiger and Ernest Marsden. Geiger and Marsden had directed a beam of alpha par­ti­cles (helium nuclei) at a thin gold foil. As expected, most of the alpha par­ti­cles were deflected by at most a few degrees. Yet a tiny frac­tion of the par­ti­cles were deflected through angles much larger than 90°. In Rutherford’s own words,[1]

It was almost as incred­ible as if you fired a 15-​​inch shell at a piece of tissue paper and it came back and hit you. On con­sid­er­a­tion, I real­ized that this scat­tering back­ward must be the result of a single col­li­sion, and when I made cal­cu­la­tions I saw that it was impos­sible to get any­thing of that order of mag­ni­tude unless you took a system in which the greater part of the mass of the atom was con­cen­trated in a minute nucleus.

The resulting model, which described the atom as a minia­ture solar system, with elec­trons orbiting the nucleus the way planets orbit a star, was how­ever short-​​lived. Clas­sical elec­tro­mag­netic theory pre­dicts that an orbiting elec­tron will radiate away its energy and spiral into the nucleus in less than a nanosecond.

In 1913, Niels Bohr pos­tu­lated that the angular momentum L of an orbiting atomic elec­tron was quan­tized: its pos­sible values are inte­gral mul­ti­ples of the reduced Planck constant:

L = nℏ,     n = 1,2,3.…

Bohr’s pos­tu­late not only explained the sta­bility of atoms but also accounted for the by then well-​​established fact that atoms absorb and emit elec­tro­mag­netic radi­a­tion only at spe­cific fre­quen­cies. It even enabled Bohr to cal­cu­late with remark­able accu­racy the spec­trum of atomic hydrogen — the par­tic­ular fre­quen­cies at which this absorbs and emits light, vis­ible as well as infrared and ultraviolet.

Yet apart from his quan­ti­za­tion pos­tu­late, Bohr’s rea­soning at the time remained com­pletely clas­sical. He assumed that the orbit of the hydrogen atom’s single elec­tron was a circle, and that the atom’s nucleus — a single proton — was at the center. He used clas­sical laws to cal­cu­late the orbiting electron’s energy E, expressed E as a func­tion of the clas­sical expres­sion for L, and only then replaced this expres­sion by L = nℏ.

In this way Bohr obtained a dis­crete sequence of values En. And since they were the only values the energy of the orbiting elec­tron was “allowed” to take, the energy that a hydrogen atom could emit or absorb had to be equal to the dif­fer­ence between two of these values. The atom could “jump” from a state of lower energy Em to a state of higher energy En, absorbing a photon of fre­quency (En − Em)/​h, and it could “jump” from a state of higher energy Em to a state of lower energy En, emit­ting a photon of fre­quency (Em − En)/​h. Or so the story went.

It took ten years, from 1913 to 1923, before someone finally found an expla­na­tion for the quan­ti­za­tion of angular momentum. Planck’s radi­a­tion for­mula had implied a rela­tion between a par­ticle prop­erty (E) and a wave prop­erty (f or ω) for the quanta of elec­tro­mag­netic radi­a­tion we now call pho­tons. Einstein’s expla­na­tion of the pho­to­elec­tric effect, pub­lished in 1905, estab­lished another such rela­tion, between the momentum p of a photon and its wavelength λ:

p = h/​λ.

If elec­tro­mag­netic waves have par­ticle prop­er­ties, Louis de Broglie rea­soned, why cannot elec­trons have wave prop­er­ties? Imagine that the elec­tron in a hydrogen atom is a standing wave on a circle rather than some sort of cor­puscle moving in a circle. (A standing wave does not travel. Its crests and troughs are sta­tionary; they stay put.)

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de Broglie's standing waves

Figure 2.1.1 Standing waves on a circle for n = 3, 4, 5, 6.

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Such a wave has to sat­isfy the condition

r = nλ,     n = 1,2,3.…

In (other) words, the cir­cum­fer­ence 2πr of the circle must be an inte­gral mul­tiple of the wave­length λ. With ℏ = h/​2π and de Broglie’s for­mula ph/​λ, this implies that

pr = nℏ.

But pr is just the angular momentum L of a clas­sical par­ticle moving in a circle of radius r. In this way de Broglie arrived at the quan­ti­za­tion con­di­tion Lnℏ, which Bohr had simply postulated.

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1. [↑] Cas­sidy, D., Holton, G., and Ruther­ford, J. (2002). Under­standing Physics, Springer, p. 632.