Use of eigenvalues of operators in quantum mechanics

Question

My very basic understanding of Quantum Mechanics and its history is that first, some physical quantities were thought to be continuous but experiments showed that they only took discrete values.

My question is when and how did people introduce the notion of Hermitian operators and postulated that the observed discrete values actually were the eigenvalues of some Hermitian operator?

Things that would be appreciated in the answer:

• What were the first quantities and corresponding operators to be introduced?
• Was the motivation for the eigenvalues formalism mostly experimental or were there some theoretical arguments as well?

Answer 1

Comments to answer: the question is hard to answer because, there is more than one question and more than a way to interpret it.

If the question is when discrete values and eigenvalues were introduced as part of the formalism of quantum mechanics? The answer is from the very first paper of Planck on quantum theory from 1900. Why? because the idea of eigenvalues was already being used for the development of classical theory of waves. Note that since the Pythagoreans or even before that, we know that the natural frequencies of an ordinary string are discrete. So the discreteness of quantum phenomena was quickly understood as a wave-like behaviour since the very beginning.

If the question is when the notion of Hermitian-like matrices was introduced to quantum mechanics then it is better to ask about the history of operators directly. Check:

• Origin of operators in quantum mechanics
• How did quantum mechanics operators come into being?

The most important contribution for the notion of operators are the three papers of Werner Heisenberg (1925-1926) on matrix mechanics. The period from 1900 to 1925 is now known as old quantum theory. Heisenberg's 1925 reinterpretation paper led to modern quantum mechanics. He introduced the idea that we needed a non-commutative algebra, but it was with the help of Pascual Jordan and Max Born that he was able to formalize it using matrices. Note that the dimension of these matrices was not necessary finite. In 1927, Jordan seems to have introduced even the symbol $\dagger$ of Hermitian conjugate, see: Who introduced the "dagger"symbol as conjugate transpose in quantum mechanics? And Born and Norbert Wiener have already a section called "calculus of operators" in their paper A New Formulation of the Laws of Quantization of Periodic and Aperiodic Phenomena in 1926.

In parallel, Schrödinger's introduced at the same time as Heisenberg his wave equation as an eigenvalue problem, but he was borrowing the mathematical tools from analytical mechanics and from the classical theory of waves.

First operators to be quantized

You asked what were the first operators introduced? Position and momentum, because from classical mechanics these are the important variables for any dynamics. Heisenberg and Schrödinger chose these two "operators" in their original papers. They also defined the energy or Hamiltonian operator (as a function of $x$ and $p$).

Last question

As for

Was the motivation for the eigenvalues formalism mostly experimental or were there some theoretical arguments as well?

If the question is: does the use of eivenvalues was motivated experimentally? the answer is yes, but again there was no surprise on how to address that for the community.

If the question is does the Hermitian algebra was motivated by experiments? The answer is no so clear. Again Schrödinger's approach did not need that (of course he was already naming things operators in the same manner as you call $\nabla^2$ a differential operator). I would argue that this falls more in the realm of "is math invented or discovered", everybody was already using some underline theory of operators without proper rigorous understanding. Heisenberg was not looking back at experiments, but more for another way of deriving previous results. Note that the later task of showing that this was all the same was undertaken later by Paul Dirac, John von Neummann and others. Von Neumann introduced more formally the concept of self-adjoint operators in 1932 (borrowing also from the works of Hilbert and Hermann Weyl on integral operators who were working on that since the beginning of the 20th century).

Answer 2

The story is long and quite complicated. Here is a very short and necessarily incomplete sketch.

It begins with experimental observations about atomic spectra. The main fact is that each atom has some fixed discrete sequence of frequencies of light that it emits or absorbs. This was known from the early 19th century (Fraunhofer) and constitutes the basis of spectral analysis. In 1885 Balmer discovered a simple empiric formula for the spectrum of hydrogen atom. Then Planck discovered the discreteness of black body radiation and relation between energy and frequency. This enabled Einstein to explain photoelectric effect. Both Planck and Einstein were explaining experimental facts about radiation.

Next important step was made by Bohr, who proposed a model of hydrogen atom which explained Balmer's formula. This was the basis of the so-called Old quantum theory. It explained many experimental facts but was based on some ad hoc "quantization rules".

Linear operators came to the theory in the work of Heisenberg (matrices) and Schrodinger (differential operators), followed very shortly shortly by the work of Born, Jordan and Dirac. From their work it became clear that the discrete values of energy (related to the frequency by Plank's formula) are actually eigenvalues of certain Hermitian operator. This final formulation is due to Dirac, and rigorous mathematical background was developed by von Neumann.

So the first quantized physical quantity were energy and action. And this was a result of complicated theoretical development whose goal was to explain experimental facts, mainly about emission and absorption of radiation. Chronologically, the first hint that measured values of physical quantities are eigenvalues of certain operators comes from the work of Heisenberg. But Heisenberg did not know the relevant mathematics and had to invent matrices himself. The connection with known mathematical theories was established by Born, Jordan and Dirac. And necessary further development of these mathematical theories is due to von Neumann.

Recommended literature: For the first period (from Fraunhofer to Bohr): S. Sternberg, History of 19th century spectroscopy. Appendix F to his book Group theory and physics Cambridge UP, 1994.

For the second period (from Bohr to Dirac) B. L. van der Waerden, Sources of quantum mechanics. It includes English translations of the main original articles with short comments by van der Waerden.

Also A. Sommerfeld, Atombau und Spektrallinien (1921) is a very good source for the transition period between old and modern quantum mechanics. Later editions of this book were expanded to include the work of Heisenberg and Schrodinger.