Abstract
Louis Victor Pierre Raymond, Prince de Broglie, was an unlikely candidate to toss a bombshell into the field of physics, at a time when many classical concepts were withering and the new quantum concepts were as yet inchoate and murky.
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Endnotes
The second relation was introduced in a previous chapter. We will use wavelength or wave number indifferently, whichever is convenient. The two equations—the first introduced by Einstein and the second by de Broglie—are known as the Einstein-de Broglie equations.
[de Broglie 1924:450]; qu [French & Taylor 1978:61].
The momentum is 8000 kg x 1 m/s = 8000 kg•m/s. The wavelength is given by h/mv = (6.626 x 10-34 J•s)/(8000 kg-m/s) = 8 x 10-38 m.
(8 x 103/10-5) x (1/10-2) = 8 x 1010
At that rate of speed, it would take a month to go from my head to my toes. A useful number to remember is that there are close to 31 million seconds in a year (remember that as “10 million 7t”). Hence, one month contains -2.5 Ms. From these inputs, you can estimate my height, at least within a factor of 2 or 3.
You can derive these figures by yourself.
Such an electron could “put a girdle round the earth” in 10 s, a task for which Ariel required 40 minutes (A Midsummer Night’s Dream, Act ii, Scene 1). We deduce that Ariel’s top speed was a mere 17 km/s, twice that of an ICBM. Since Shakespeare did not disclose the sprite’s mass, we can determine neither his (its?) momentum nor the de Broglie wavelength. These must remain matters for speculation, along with “what songs the Sirens sang and what name Ulysses took when he went among women.” [Browne 1658].
Robert Browning, “By the Fireside.”
Von Laue, German scientist (1879–1960), and Nobel laureate, 1914. He is noted for his investigations of quantum effects in crystals and for his upright stance in the face of Nazi pressure. Einstein respected von Laue, not only as a professional colleague, but as a man of honor.
It might have been expected that the rays would be known as Röntgen rays, after their discoverer, but the term X-rays is universal.
In any particular experimental arrangement, you determine the value of n by finding several angles where the diffraction is strong. The largest angle is that for n = 1.
They were father and son and jointly received the Nobel Prize in 1915 for their work. This remains a unique distinction.
The competitor was General Electric. The experiment we are about to describe took place after the patent matter had been decided.
In the original apparatus, the incident electron beam hit the face of the nickel target normally, and the collector was swung in an arc to find the angle dependency. Subsequent investigators use more elaborate equipment in which the target itself is rotated and tilted; this allows access to more sets of planes of atoms.
For practical reasons, the intensity can’t be measured for a scattering angle of 0°: to do so would involve placing the collector in line with the electron gun.
This is a common way to reduce a metal oxide: that is, to drive off the oxygen.
There are minor effects which explain the discrepancy between 0.167 and 0.165 nm. Two such are the difficulty of measuring the precise angle at which the scattering is most intense, and the fact that the electron beam, which dives into the crystal and reemerges, gains and loses a small amount of energy in doing so, so its wavelength is slightly changed.
The algebra needed to manipulate these expressions is elementary. Do it yourself!
Toujours de l’audace! as de Broglie might have said. Try it!
Emerson, “Monadnock.”
You might reread the appendix on standing waves (Chapter 5).
Having read the appendix on Fourier analysis, you will appreciate that a finite pulse in the form of a sine wave—a waveform that is zero until a particular time, then a sine wave for a while, then zero thereafter—is composed of an infinite set of sinusoids (along the lines of RECT in that appendix). The full set of sinusoids is needed to form the sinusoid shape when the pulse is on and to ensure that the pulse has zero amplitude when off. The only “pulse” that consists of exactly one sinusoid is an infinitely long sine wave.
We are dealing here with the elements of Fourier analysis, by which an arbitrary wave shape can be built up as the sum of a set of sinusoidal waves. It is an attractive subject, neither subtle nor abstruse, but too lengthy for us to pursue in any detail. However, you are already acquainted with the subject, having read the appendix on Fourier analysis.
We are discussing an electron for definiteness, but the discussion applies to any type of particle.
De Broglie’s papers on the waves are difficult to read, at least in part because of his vague language. He recognized this but felt that his insight was so important that he ought not to withhold publication until the exposition was polished. Perhaps a degree of reticence would have helped promote his ideas. A detailed history of de Broglie’s concepts can be found in [Cushing 1994].
Wave packets that obey linear governing equations in a dispersive medium and display reasonable physical behavior must spread out with time. It has been observed (originally by watching a tidal bore roll for miles up a channel without losing its shape) that waves governed by nonlinear equations need not disperse. Much work has been done to determine whether such packets (known as solitons) can play a role in de Broglie waves. The work is inconclusive but not promising.
Efforts have been made to measure the energy carried by pilot waves, with no success. So, the waves carry either zero energy or so little as to escape measurement (so far). In either case, it is a puzzle how the waves can influence the path of the electron cum particle.
[Stapp 1977:191].
He first described his wave as une onde fictive associée au mouvement du mobile; qu [Jammer 1989:248].
qu [Lochak 1987:1192].
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Grometstein, A.A. (1999). A Princely Postulate (1924). In: The Roots of Things. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-4877-5_12
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