From particles to waves, Part 3a: Eigen Spiel

“. . . 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, 2^nd 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.

From particles to waves, part 2b

 So far, I’ve discussed the meaning of periodic reflective boundary conditions for the 1-D wave equation and looked at the general solution, or what might be called the necessary generic form of the solution, to this equation as it applies to a small disturbance (y-direction small-amplitude motion) on a taut string having linear density σ and tension τ. This solution is written as

y(x,t) = f(x-ct) + g(x+ct)

or (see end of previous post)

y(x,t) = f(t-x/c) + g(t+x/c),

where c = (τ/σ)^1/2 is the speed of the disturbance, which is actually two disturbances when both terms in either of these equations are nonzero—the f-disturbance travelling to the right and the g-disturbance travelling to the left.

I’ll first look at how we get from the generic traveling wave solution to a traveling sine wave solution for the infinitely long string, then look at the form the generic solution takes for the semi-infinite string (where there’s a boundary at one end), and finally look at what happens when the traveling wave (the “incident wave”) on a semi-infinite string is a sine wave.  Part 3 (the next, and last, post on this subject) will be a return to the string fixed at both ends and a look at what happens to the generic form of the solution and the sine wave solution in that case. The beginning of Schrödinger’s wave mechanics in 1926 will also be discussed in Part 3. 

(The titles of Erwin's first four wave mechanics papers were "Quantization as an Eigenvalue Problem" Parts 1, 2, 3, 4. Since many people, myself included, have eigenproblems understanding the proper eigencontext of the prefix "eigen," I will include some eigencomments in my Part 3.)

The generic form of the 1-D wave equation solution above is derived without consideration of initial conditions or boundary conditions and has a unique speed associated with it, but no single frequency or wavelength like we might expect from a wave. But, as you probably realize, if it did have a frequency and wavelength in place of speed—if we just tried to substitute λν = c or ω/k = c into the generic solution—it wouldn’t be a generic solution anymore because f and g would have to be pure sine waves having their own frequencies ν or ω, and wave numbers k,  with f and g necessarily expressible, for instance, as Acos(kx ± ωt + φ), where A, k, ω, and φ are constants, although we can have a superposition of sine waves of different amplitudes and frequencies. 

Dudley Towne says in Section 1.7 of Wave Phenomena, where he’s still considering waves on a string of infinite length:

 

Contrary to the impression which may be created by the fact that waves of sinusoidal form are the most frequently cited examples of waves, it is to be noted that neither wavelength nor periodicity is an essential characteristic of a wave. An initial waveform of any desired shape determines an allowable solution to the wave equation.

 

After making this cautionary statement, Towne gives several reasons why sinusoidal waves are fundamental in the study of wave phenomena: 1) a sine wave is “one of the simplest analytic functions which is bounded on an infinite interval,” 2) it represents a pure tone in sound and a “spectral color” of light, and 3) it “achieves an overwhelming importance through Fourier’s theorem.” Dudley also mentions dispersion: “In some contexts involving the phenomenon of dispersion, the wave equation is not satisfied except for waves of sinusoidal form, and then only when a wave propagation velocity appropriate to the given frequency is substituted.”

 

Sinusoidal Progressive Wave on an Infinite String

For the 1-D wave equation, how do we get to a traveling sine wave solution, y(x,t) = Acos(kx±ωt + φ), from the generic traveling wave solution, y(x,t) = f(x-ct) + g(x+ct)? The same way we always get from the general to the particular when dealing with differential equations: initial conditions and boundary conditions. For the infinite string, there are no boundaries, so we just have initial conditions. And for partial differential equations like the 1-D wave equation, it’s really initial functions we have to deal with, a subject I’ll mention again further down in this post (see the paragraph below that starts with “Now”).

Right now, I just want to particularize the generic solution to the case of a traveling sine wave. That means we start with an initial sinusoidal disturbance on our infinitely long string, courtesy of Towne’s Section 1.7, “Description of a Progressive Sinusoidal Wave,” where he says

 

An initial waveform which is sinusoidal is described by the function

y(x, 0) = Re {ye^ikx}

where k and y are given constants. If the complex amplitude y is written in the form

y = y_me^iφ,

the real amplitude y_m determines the maximum displacement of the string from the x-axis and the phase φ determines the position of the curve with respect to translation parallel to the x-axis. … When the quantity (kx) increases by 2π, the function y(x,0) goes through one cycle of its values. The corresponding increment in x is, of course, the wavelength, λ. Thus k(x  + λ) = kx + 2π, or kλ = 2π, or k = 2π/λ. The parameter k is referred to as the wave number and may be thought of as the number of waves contained in a distance of 2π meters.

 

Towne chooses φ = 0, which makes the initial waveform a cosine function—but let’s keep the solution in exponential form for the moment. Next, we declare the initial waveform to be in motion in the positive x direction, making g(x+ct) in the generic solution of the wave equation identically equal to zero. The travelling wave solution is now

 

y(x,t) = f(x-ct) =  Re {y_me^ik(x-ct)}.

 

Towne says, “Note that if we focus on a particular particle of the string, say the particle [located at] x = x₁, the motion of this particle as a function of time is sinusoidal.” In a footnote, he says “in any progressive wave the curve which describes the history of a single particle is of the same geometric form as the wave profile of the string.” Then he continues on about the motion being sinusoidal, but I won’t quote him from this paragraph, except to say this is where he brings angular frequency into the argument: as the wave travels, the particles of the string undergo simple harmonic motion, with angular frequency ω = kc. Making this substitution in the above equation, we get

 

y(x,t) = Re {y_me^i(kx-ωt)} = y_mcos(kx-ωt).

 

Thus, starting from the generic form y(x,t) = f(x-ct) + g(x+ct), we’ve arrived at our desired expression for a traveling sinusoidal wave.

Now I’ll mention the use of “initial functions,” as discussed in Towne’s last section in Chapter One, “Initial Conditions Applied to the Case of a String of Infinite Length,” which, by the way, he says “may be omitted without loss of continuity.” (Dr. Rolleigh didn't omit this subject in his notes.)  I’ll omit the derivations but state the main idea. Since the wave equation is a 2^nd order partial differential equation, knowing the initial positions and initial velocities of all the particles on the string—the initial wave profile and the initial velocity profile—allows the functions f and g to be uniquely determined. As Towne says at the beginning of Chapter Three (Boundary Value Problems, where we’re about to go now), “the solution is uniquely determined for an infinite string by the requirements that y(x,t) and dy/dt reduce at t=0 to given functions.”

 

Sinusoidal Progressive Wave on Semi-Infinite String

The semi-infinite string extends from minus infinity on the x-axis to x=0, where it’s tied or fixed or anchored to a solid wall—the boundary. The 1-D wave equation must be satisfied within the interval – ∞ < x < 0. Towne chooses the form of the generic solution to be

y(x,t) = f(t-x/c) + g(t+x/c)                 (3-1)

“to simplify algebraic manipulations,” he says. I’ll just use two pages from the book to show the general waveform case and the sinusoidal waveform case. The boundary condition is y(0,t) = 0.

y(x,t) to vanish, regardless of the value of t. The fixed end is a node, and the spacing between successive nodes is a half-wavelength. Unlike the progressive wave, the sinusoidal waveform does not undergo translation parallel to the x-axis. This type of motion is referred to as a standing wave.

d) The amplitude of the simple harmonic motion of an individual particle depends on its location. The particles halfway between the nodes have the largest amplitude for their motion. These halfway positions are referred to as antinodes.

 

Before moving on to the string having boundaries at both ends, I’d like to note a couple of things about the semi-infinite string.

First, as you can surmise from the picture of the sine wave(s) above, there is no restriction on the frequency (or wavelength)  the standing wave can have. This is shown by the standing wave solution itself,

 

y(x,t) = 2y_msin(kx) sin(ωt),

 

where k and ω can take on a continuous range of values. (And now that we do have a sinusoidal wave, we necessarily have the relation ω/k = c = (τ/σ)^1/2.) This range of continuous values is in contrast to waves propagating in a “confined region,” such as on a string fixed at both ends, where the allowed frequencies are determined by the dimensions of the region, such as the length of the string.  This latter case has a discretely-infinite set of frequencies that are characteristic of the dimensions of the limited region--a periodicity in the frequency space and k-space. In the former case, there is a continuously-infinite number of possible sinusoidal standing wave frequencies that can be formed.

Secondly, the standing wave solution is a product of a function only of x and a function  only of t, something I’ll be returning to in Part 3—and something probably familiar to you from the usual “separation of variables” technique. I don’t like to invoke this rather abstruse technique unnecessarily, and neither does Dudley Towne, so I’ll follow his Wave Phenomena Chapter 15 discussion in Part 3.