“. . . when Werner Heisenberg discovered ‘matrix’ mechanics in 1925, he didn’t know what a matrix was (Max Born had to tell him), and neither Heisenberg nor Born knew what to make of the appearance of matrices in the context of the atom. David Hilbert is reported to have told them to go look for a differential equation with the same eigenvalues, if that would make them happier. They did not follow Hilbert’s well-meant advice and thereby may have missed discovering the Schrödinger wave equation.”
— Manfred Schroeder, in the Forward to his book Number Theory in Science and Communication, 2nd edition, corrected printing, 1990.
Keeping Up with the Eigens
In Quantum Concepts in Physics (Cambridge
University Press, 2013), Malcolm Longair says (page 267), “In seeking a wave
equation to describe de Broglie’s matter waves, Schrödinger began by attempting
to find an appropriate relativistic wave equation. … These first attempts at
the derivation of the relativistic wave equation were never published, but the
argument can be traced in Schrödinger’s notebooks and a three-page memorandum
he wrote on the eigenvibrations of the hydrogen atom.”
Professor Longair then says on page 268
that “de Broglie’s waves were propagating waves whereas Schrödinger had
converted the problem into one of standing waves, like the vibrations of a
violin string under tension.”
However, we know from my previous post
that the production of standing waves on a string can be done without the
string being fixed at both ends. Traveling sine waves of any frequency can be
sent in from –∞ on a semi-infinite string and their reflection at the x= 0
end of the string, where the string is tied, will produce a standing wave.
This is where we come to the subject of
all things eigen. The standing waves on a semi-infinite string are an
example of what “eigenvibrations” are NOT, simply because they can have any
frequency. Eigenvibrations occur only at eigenfrequencies, and these are the
frequencies that are characteristic of the length of a string tied at
both ends and the given boundary conditions at both ends. Indeed, eigenfriends,
“characteristic” and “proper” are most often used in math and physics books as
the English translation of eigen.
Eigen as “Own”
But let’s see what A Brief Course in
German by Peter Hagboldt and F. W. Kauffmann, published in 1946, has to say
on the subject. In the back of the book is a section called Vocabulary, which
gives the English translation of various German words, including Wien
and Wiener, which, just in case you didn’t already know, translate
respectively as “Vienna” and “Viennese.” I only recently thought of looking up eigen in the book.
Besides “characteristic” and “proper,” the
physics and math books sometimes also translate eigen as “special.” But none of
those are what A Brief Course in German says. There, the Vocabulary
section says Eigen translates as “own.” That’s it, no foolin’ around
with “characteristic” or “proper” or “special” by Hagboldt and Kauffmann.
Eigen as “Self”
And “own” itself has a near-synonym in
English. Among the Math Stack Exchange answers to a question dated 11
February 2013
and titled “What exactly are eigen-things?” there was one answer I especially
liked, written by Alex Chaffee. He (or she) says “eigen means proper only
insofar as ‘proper’ means ‘for oneself,’ as in ‘proprietary’ or French propre.
Mostly eigen means self-oriented.”
This reference to propre
connects with what seems to be a mistranslation used in relativity, where we have something called “proper” time, which some writers on the subject say comes from the French word "propre." It does seem
that proper time really should be called “own time,” because “own time” is
indeed what you read on your own watch, which never moves relative to you and
thus never changes its rate of ticking relative to you. This mistranslation of
French may also be why eigen gets mistranslated as “proper” instead of
“own.”
Before moving on to discussing
eigenwaves (sorry about that) on a finite length of string, I’ll mention one
other reference that says eigen refers to self. Last year in the Arkansas
Democrat-Gazette, a columnist named Philip Martin wrote about his German
grandmother and mentioned that one of the German phrases she sometimes used was
eigenlob stinkt. A few readers of this blog, such as Tom Mellett, are no
doubt aware of how this phrase translates, but for those who aren’t, here is
the English translation: self-praise
stinks.
I encourage you to
think in terms of “own” and “self” when you see eigen in the future. To
help you with that, here’s the History section from the Wikipedia article on Eigenvalues and Eigenvectors (it helped me) . . .
Eigenvalues are often introduced in
the context of linear algebra or matrix theory. Historically, however, they
arose in the study of quadratic forms and differential equations.
In the 18th century, Leonhard Euler
studied the rotational motion of a rigid body, and discovered the importance of
the principal axes. Joseph-Louis Lagrange realized that the principal axes are
the eigenvectors of the inertia matrix.
In the early 19th century,
Augustin-Louis Cauchy saw how their work could be used to classify the quadric
surfaces, and generalized it to arbitrary dimensions. Cauchy also coined the
term racine caractéristique (characteristic root), for what is now
called eigenvalue; his term survives in characteristic equation.
Later, Joseph Fourier used the work of
Lagrange and Pierre-Simon Laplace to solve the heat equation by separation of
variables in his famous 1822 book Théorie analytique de la chaleur.
Charles-François Sturm developed Fourier's ideas further, and brought them to
the attention of Cauchy, who combined them with his own ideas and arrived at
the fact that real symmetric matrices have real eigenvalues. This was extended
by Charles Hermite in 1855 to what are now called Hermitian matrices.
Around the same time, Francesco
Brioschi proved that the eigenvalues of orthogonal matrices lie on the unit
circle, and Alfred Clebsch found the corresponding result for skew-symmetric
matrices. Finally, Karl Weierstrass clarified an important aspect in the
stability theory started by Laplace, by realizing that defective matrices can
cause instability.
In the meantime, Joseph Liouville
studied eigenvalue problems similar to those of Sturm; the discipline that grew
out of their work is now called Sturm–Liouville theory. Schwarz studied the
first eigenvalue of Laplace's equation on general domains towards the end of
the 19th century, while Poincaré studied Poisson's equation a few years later.
At the start of the 20th century,
David Hilbert studied the eigenvalues of integral operators by viewing the
operators as infinite matrices. He was the first to use the German word eigen,
which means "own", to denote eigenvalues and eigenvectors in 1904,
though he may have been following a related usage by Hermann von Helmholtz. For
some time, the standard term in English was "proper value", but the
more distinctive term "eigenvalue" is the standard today.
The first numerical algorithm for computing eigenvalues and eigenvectors appeared in 1929, when Richard von Mises published the power method. One of the most popular methods today, the QR algorithm, was proposed independently by John G. F. Francis and Vera Kublanovskaya in 1961.
[References aren't cited above, but they are in the Wikipedia article. Also, the applications section of the Wikipedia article on Eigenfunction looks at the 1D wave equation and the Schrö eqn.]
In some cases, there might be confusion over whether "proper" in an English translation of "eigen" means proper in the relativistic sense or in the sense of the eigenvalues, etc. Here's a page from Einstein's 1909 paper "On the present status of the radiation problem," in English (Princeton Einstein Papers Project). In the first paragraph, you'll find the words " at the proper frequency." In the German version , not surprisingly, you'll find "bei der Eigenfrequenz." This usage is the one from relativity. If it hadn't been, the English translation of Eigenfrequenz would be eigenfrequency. Maybe! Translations depend on the translators' knowledge and preferences.
On page 358 of the English translation of this paper, you'll find our well-known acquaintance, the wave equation (in 3-D), along with some functions in its superposition solution that have t - r/c and t + r/c in their arguments: the "retarded" and "advanced" potentials, respectively. This paper of Einstein's is important because he note's Planck's mistake of assuming equal (a priori) probabilities for Boltzmann's "complexions" and he finds (equation 36) a wave and a particle term in the mean square fluctuations of thermal (blackbody) radiation--early evidence for the wave-particle duality.
To be continued in Part 3b.
