Tag / Wave Mechanics

The Rise of Wave Mechanics by Erwin Schrödinger
by David D. Nolte

100 Years of Quantum Physics: Schrödinger’s Wave Mechanics (1926)

By the middle of 1925, a middle-aged Erwin Schrödinger was casting about, bogged down in mid career, looking for something significant to say about the rapidly accelerating field of quantum theory.  He was known for his breadth of knowledge, and for his belief in his own creative genius [1], but a grand synthesis had so far eluded him. 

Einsteinian Gas Theory

In the middle of the year, Schrödinger was deep analyzing two papers recently published by Einstein (Sept. 1924 and Feb. 1925) on the quantum properties of ideal gases where Einstein applied the new statistical theory of Bose to the counting of states in a volume of the gas [2].  One of the intriguing discoveries made by Einstein in those papers was a close analogy between the fluctuation of gas numbers and the interference of waves.  He stated:

‘I believe that this is more than a mere analogy; de Broglie has shown in a very important work how a (scalar) wave field can be coordinated with a material particle or a system of material particles’

referring to de Broglie’s thesis work of 1924 that associated a wave-like property to mass.  Einstein had been the first to attribute a wave-particle duality to the quantum phenomenon of black-body radiation, giving a lecture in Salzburg, Austria, in 1909 showing relationships between the particle-like and the wave-like properties of the radiation, and he found similar behavior in the properties of monatomic ideal gases, though his derivations were purely statistical.

Erwin Schrödinger photo
Fig. 1 Erwin Schrödinger (Source)

Schrödinger was suspicious of the “unnatural” way of counting states used by Einstein and Bose for the gas, and he sought a more “natural” way of explaining how the elements of phase space were filled.  It struck him that, just as Planck’s black-body radiation spectrum could be derived by assuming discrete standing-wave modes for the electromagnetic radiation, then perhaps the behavior of ideal gases could be obtained using a similar approach.  He and Einstein exchanged several letters about this idea as Schrödinger dug deeper into de Broglie’s theory.

The Zurich Seminar

At that time, Schrödinger was in the Chair of Theoretical Physics at the University of Zurich, holding the same chair that Einstein had held 15 years earlier.  Following Einstein, the chair had been occupied by Max von Laue and then by Peter Debye who moved to the ETH in Zurich.  Debye organized a joint seminar between the University and ETH that was a hot social gathering of physicists and physical chemists, discussing the latest developments in atomic and quantum science. 

In November of 2025, Debye, who probably knew about the Einstein-Schrödinger discussion on de Broglie, asked Schrödinger to give a seminar on de Broglie’s theory to the group.  A young Felix Bloch, who was a graduate student at that time, recalled hearing Debye say something like

“Schrödinger, you are not working right now on very important problems anyway. Why don’t you tell us sometime about that thesis of de Broglie, which seems to have attracted some attention?”[3]

Schrödinger gave the overview seminar in early December, showing how the Bohr-Sommerfeld quantization conditions could be explained as standing waves using de Broglie’s theory, but Bloch recalled Debye was unimpressed, saying that de Broglie’s way of talking was “childish” and that what was needed for a proper physics theory was a wave equation.

This exchange between Schrödinger and Debye was recalled only in later years, and there is debate about what exactly was said and what effect it had on Schrödinger.  From Schrödinger’s letters to friends, it is clear that he was already well into his investigations of wavelike properties of matter when Debye asked him to give the seminar.  Furthermore, he had already tried to construct wave packets using superpositions of phase waves propagating along Bohr-Sommerfeld elliptical orbits but had been led to ugly caustics when he tried to apply the packets to the hydrogen atom [4].  Therefore, although Debye was probably not the source of Schrödinger’s interest in de Broglie, it is possible that Debye’s quip about “childishness” may have spurred Schrödinger to find a wave equation subject to boundary conditions rather than working with packets following ray paths.

The Christmas Breakthrough

By this time, Christmas was approaching and Schrödinger arranged to take a vacation away from his family to the Swiss Alpine village of Arosa, and given his unconventional belief in the link between personal pleasure and genius, he did not go alone.  There is no record of what transpired, and no record of which mistress was with him on this particular trip, nor how she spent her time while he worked on his theory, but two days after Christmas he wrote a letter to the physicist Willy Wien saying

“At the moment I am struggling with a new atomic theory. If only I knew more mathematics! I am very optimistic about this thing, and expect that, if only I can . . . solve it, it will be very beautiful.. . . I hope that I can soon report in a little more detailed and understandable way about the matter. At present I must learn a little mathematics in order to completely solve the vibration problem …”[5]

He had uncovered his first wave equation.  When he returned to Zurich, he enlisted the help of his friend, the mathematician Hermann Weyl at the University in Zurich, and Schrödinger had his first eigenfrequencies for hydrogen.  But they were wrong!

Fig. 2 Schrödinger’s first wave equation was relativistic.

The theory of de Broglie was fundamentally a relativistic theory, motivated by mapping the behavior of matter onto the behavior of light.  Therefore, Schrödinger’s first attempt was also relativistic, equivalent to the Klein-Gordon equation.  But there was no clear understanding of electron spin at that time, even though it had been established as a fundamental property of the electron.  It was only several years later when Dirac correctly accounted for electron spin in a relativistic wave equation. 

Dirac relativistic quantum wave equation
Fig. 3 The full relativistic Dirac equation.

The Schrödinger Wave Equation

Convinced that he was onto something big, and unwilling to fail, despite his failure to obtain correct values for hydrogen, Schrödinger went back to first principles, to the classical theory of Hamiltonian mechanics, identifying Hamilton’s characteristic function with the phase of an electron wave and deriving a non-relativistic equation using variational principles subject to boundary conditions.  The eigenvalues of this new equation, when applied to hydrogen, matched the Bohr spectrum perfectly!

Fig. 4. Schrödinger’s second wave equation was non-relativistic and correctly matched the Bohr energy levels of hydrogen.

It had been only a few weeks since Schrödinger had given his previous seminar to the Zurich group, but in January he gave his update, probably given with some degree of satisfaction, having Debye in attendance, showing his now-famous wave equation and the agreement with experiment.  Schrödinger wrote up his theory and results and submitted his paper on January 27, 1926, to Annalen der Physik [6].

Hydrogen quantum orbitals
Fig. 5 Hydrogen quantum orbitals (Source)

Schrödinger had been known, but not as a forefront thinker, despite what he believed about himself.  Now he was a forefront thinker, vindicating his beliefs but not always on the right track.  He continued his unconventional lifestyle, marginalizing him socially, and he resisted Max Born’s and Niels Bohr’s probabilistic interpretations of the meaning of his own quantum wavefunction, marginalizing him professionally.  Yet his breakthrough gave him a platform, and his skeptical reactions to his colleague’s successes helped illuminate the nature of the new physics (“Schrödinger’s Cat” [7]) through the decades to follow.

Bibliography

A very large body of historical work exists on the discovery of the Schrödinger equation, partially fueled by the lack of first-person accounts on how he achieved it.  There has been a lot of speculation and a lot of sleuthing to uncover his path of discovery. Several accounts differ mainly in the timing of when he derived his equations, although all agree on the sequence: that the relativistic equation preceded the non-relativistic one. Here is a small sampling of the literature:

• Hanle, P. A. (1977). “The Coming of Age of Erwin Schrödinger: His Quantum Statistics of Ideal Gases.” Archive for History of Exact Sciences, 17(2), 165–192. DOI: 10.1007/BF00328532.

• Hanle, P. A. (1979). “The Schrödinger‐Einstein correspondence and the sources of wave mechanics.” American Journal of Physics, 47(7), 644–648. DOI: 10.1119/1.11587.

• Mehra, Jagdish. “Erwin Schrödinger and the Rise of Wave Mechanics. II. The Creation of Wave Mechanics.” Foundations of Physics, vol. 17, no. 12, 1987, pp. 1141-1188.

• Renn, J. (2013). “Schrödinger and the Genesis of Wave Mechanics.” In W. L. Reiter & J. Yngvason (Eds.), Erwin Schrödinger – 50 Years After (pp. 9–36). Zurich: European Mathematical Society. DOI: 10.4171/121-1/2.

• Wessels, L. (1979). “Schrödinger’s Route to Wave Mechanics.” Studies in History and Philosophy of Science Part A, 10(4), 311–340.

Notes

[1] He led an unconventional lifestyle (some would say emotionally predatory) based on his belief in the personal origins of genius.  Although this behavior presented significant social barriers to his career, he refused to abandon it.

[2] A. Enstein, ‘Quantentheorie des einatomigen idealen Gases’, Preuss. Ak. Wiss. Sitzb. (1924) pp. 261 – 267, and (1925), pp. 3-14.

[3] Quoted in Mehra, pg. 1150

[4] Mehra, pg. 1147

[5] Wessels, pg. 328

[6] Schrödinger, Erwin. “Quantisierung als Eigenwertproblem (Erste Mitteilung).” Annalen Der Physik, vol. 384, no. 4, 1926, pp. 361-376.

[7] Schrödinger, E. (1935). “Die gegenwärtige Situation in der Quantenmechanik.” Naturwissenschaften 23: 807–812, 823–828, 844–849.  Schrödinger, E. (1980). “The Present Situation in Quantum Mechanics: A Translation of Schrödinger’s ‘Cat Paradox’ Paper.” Proceedings of the American Philosophical Society 124 (5): 323–338. (Translated by J. D. Trimmer).

Books by David Nolte at Oxford University Press
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by David D. Nolte

100 Years of Quantum Physics: de Broglie’s Wave (1924)

One hundred years ago this month, in Feb. 1924, a hereditary member of the French nobility, Louis Victor Pierre Raymond, the 7th Duc de Broglie, published a landmark paper in the Philosophical Magazine of London [1] that revolutionized the nascent quantum theory of the day.

Prior to de Broglie’s theory of quantum matter waves, quantum physics had been mired in ad hoc phenomenological prescriptions like Bohr’s theory of the hydrogen atom and Sommerfeld’s theory of adiabatic invariants.  After de Broglie, Erwin Schrödinger would turn the concept of matter waves into the theory of wave mechanics that we still practice today.

Fig. 1 The 1924 paper by de Broglie in the Philosophical Magazine.

The story of how de Broglie came to his seminal idea had an odd twist, based on an initial misconception that helped him get the right answer ahead of everyone else, for which he was rewarded with the Nobel Prize in Physics.

de Broglie’s Early Days

When Louis de Broglie was a student, his older brother Maurice (the 6th Duc de Broglie) was already a practicing physicist making important discoveries in x-ray physics.  Although Louis initially studied history in preparation for a career in law, and he graduated from the Sorbonne with a degree in history, his brother’s profession drew him like a magnet.  He also read Poincaré at this critical juncture in his career, and he was hooked.  He enrolled in the  Faculty of Sciences for his advanced degree, but World War I side-tracked him into the signal corps, where he was assigned to the wireless station on top of the Eiffel Tower.  He may have participated in the famous interception of a coded German transmission in 1918 that helped turn the tide of the war.

Beginning in 1919, Louis began assisting his brother in the well-equiped private laboratory that Maurice had outfitted in the de Broglie ancestral home.  At that time Maurice was performing x-ray spectroscopy of the inner quantum states of atoms, and he was struck by the duality of x-ray properties that made them behave like particles under some conditions and like waves in others.

Fig. 2 Maurice de Broglie in his private laboratory (Figure credit).
Fig. 3 Louis de Broglie (Figure credit)

Through his close work with his brother, Louis also came to subscribe to the wave-particle duality of x-rays and chose the topic for his PhD thesis—and hence the twist that launched de Broglie backwards towards his epic theory.

de Broglie’s Massive Photons

Today, we say that photons have energy and momentum although they are massless.  The momentum is a simple consequence of Einstein’s special relativity

And if m = 0, then

and momentum requires energy but not necessarily mass. 

But de Broglie started out backwards.  He was so convinced of the particle-like nature of the x-ray photons, that he first considered what would happen if the photons actually did have mass.  He constructed a massive photon and compared its proper frequency with a Lorentz-boosted frequency observed in a laboratory.  The frequency he set for the photon was like an internal clock, set by its rest-mass energy and by Bohr’s quantization condition

He then boosted it into the lab frame by time dilation

But the energy would be transformed according to

with a corresponding frequency

which is in direct contradiction with Bohr’s quantization condition.  What is the resolution of this seeming paradox?

de Broglie’s Matter Wave

de Broglie realized that his “massive photon” must satisfy a condition relating the observed lab frequency to the transformed frequency, such that

This only made sense if his “massive photon” could be represented as a wave with a frequency

that propagated with a phase velocity given by c/β.  (Note that β < 1 so that the phase velocity is greater than the speed of light, which is allowed as long as it does not transmit any energy.)

To a modern reader, this all sounds alien, but only because this work in early 1924 represented his first pass at his theory.  As he worked on this thesis through 1924, finally defending it in November of that year, he refined his arguments, recognizing that when he combined his frequency with his phase velocity,

it yielded the wavelength for a matter wave to be

where p was the relativistic mechanical momentum of a massive particle. 

Using this wavelength, he explained Bohr’s quantization condition as a simple standing wave of the matter wave.  In the light of this derivation, de Broglie wrote

We are then inclined to admit that any moving body may be accompanied by a wave and that it is impossible to disjoin motion of body and propagation of wave.

pg. 450, Philosophical Magazine of London (1924)

Here was the strongest statement yet of the wave-particle duality of quantum particles. de Broglie went even further and connected the ideas of waves and rays through the Hamilton-Jacobi formalism, an approach that Dirac would extend several years later, establishing the formal connection between Hamiltonian physics and wave mechanics.  Furthermore, de Broglie conceived of a “pilot wave” interpretation that removed some of Einstein’s discomfort with the random character of quantum measurement that ultimately led Einstein to battle Bohr in their famous debates, culminating in the iconic EPR paper that has become a cornerstone for modern quantum information science.  After the wave-like nature of particles was confirmed in the Davisson-Germer experiments, de Broglie received the Nobel Prize in Physics in 1929.

Fig. 4 A standing matter wave is a stationary state of constructive interference. This wavefunction is in the L = 5 quantum manifold of the hydrogen atom.

Louis de Broglie was clearly ahead of his times.  His success was partly due to his isolation from the dogma of the day.  He was able to think without the constraints of preconceived ideas.  But as soon as he became a regular participant in the theoretical discussions of his day, and bowed under the pressure from Copenhagen, his creativity essentially ceased. The subsequent development of quantum mechanics would be dominated by Heisenberg, Born, Pauli, Bohr and Schrödinger, beginning at the 1927 Solvay Congress held in Brussels. 

Fig. 5 The 1927 Solvay Congress.

By David D. Nolte, Feb. 14, 2024

[1] L. de Broglie, “A tentative theory of light quanta,” Philosophical Magazine 47, 446-458 (1924).

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