In reading about vibrations in fields and wave functions and wave packets, some words that start with ‘e’ occurred a lot: eigenvalue, eigenvector, eigenfunction, eigenvalue equation, eigenequation . I needed to refresh my understanding of the mathematical models for macroscopic and microscopic systems .
Everyday we are surrounded by things that vibrate when excited. Some of these we sense and some not. Some are pleasant like harmonic tones from musical instruments. Some are unpleasant like when we drive on a rough road. Many things vibrate with natural or characteristic frequencies. We see and hear due to such frequencies. Colors and tunes enrich our lives. Science and math help us understand how all this works and empower us to shape vibrations in practical ways. In particular, to analyze and make systems where vibrations are sinusoidal oscillations.
At the macroscopic level, classical mechanics allows us to model oscillating systems. One of the simplest being the simple harmonic oscillator, which exhibits regular sinusoidal motion with a frequency dependent on some properties of such a system. Classically, a model of a single mass (object) oscillating with one degree of freedom (up-down).
When we get to more interesting systems, ones with more than one degree of freedom and multiple masses, their analysis rapidly gets harder — their models are more complex. More powerful mathematical techniques are required. Many of these systems still involve sinusoidal motion. What we discover is that we can determine certain properties of these systems because they exhibit natural or “inherent” or “characteristic” frequencies. The mathematical equations can be solved for those specific frequencies. The German word “eigen” is used to describe such values. For the physical models, the solutions to eigenequations are eigenvalues and eigenvectors (typically written in matrix format), which describe the vibration modes of the system. Understanding eigenfunctions and eigenstates tells us more about the systems being analyzed. We can make predictions. 
When we get to microscopic systems, much of that classical mathematical framework applies to quantum mechanics. In new ways. Objects are no longer tiny balls in an exactly characterized (certainly valued) state of space and time — in definite eigenstates. The math is even more daunting, but we’re still looking at an eigen landscape because it’s all about vibrations (in fields).
These days we know that the electrons don’t really “orbit” at all, because they don’t really have a “position” or “velocity.” Quantum mechanics says that the electrons persist in clouds of probability known as “wave functions,” which tell us where we might find the particle if we were to look for it. — Carroll, Sean (2012-11-13). The Particle at the End of the Universe: How the Hunt for the Higgs Boson Leads Us to the Edge of a New World (Kindle Locations 646-648). Penguin Publishing Group. Kindle Edition.
[https://en.wikipedia.org/wiki/Introduction_to_eigenstates] When an object can definitely be “pinned down” in some respect, it is said to possess an eigenstate. … when the wavefunction collapses because the position of an electron has been determined, the electron’s state becomes an “eigenstate of position”, meaning that its position has a known value, an eigenvalue of the eigenstate of position.
1. Eigenvalue aka characteristic value or characteristic root associated with the eigenvector
The prefix eigen- is adopted from the German word eigen for “proper”, “inherent”; “own”, “individual”, “special”; “specific”, “peculiar”, or “characteristic”. … In essence, an eigenvector v of a linear transformation T is a non-zero vector that, when T is applied to it, does not change direction. Applying T to the eigenvector only scales the eigenvector by the scalar value λ, called an eigenvalue. This condition can be written as the equation
T(v) = λv
referred to as the eigenvalue equation or eigenequation. In general, λ may be any scalar. For example, λ may be negative, in which case the eigenvector reverses direction as part of the scaling, or it may be zero or complex. … 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.
An example of an eigenvalue equation where the transformation T is represented in terms of a differential operator is the time-independent Schrödinger equation in quantum mechanics:
HψE = EψE
where H, the Hamiltonian, is a second-order differential operator and ψE, the wavefunction, is one of its eigenfunctions corresponding to the eigenvalue E, interpreted as its energy.
2. This Khan Academy video is a helpful overview of eigenvectors and eigenvalues (eigenvalues are associated with a corresponding eigenvector), providing you are somewhat familiar with vector spaces (fields) and matrix mathematics (as a way to encode linear maps between vector spaces). In particular the Hilbert vector space. Fourier analysis (which involves the superposition of sine waves) usually uses the Hilbert space.
The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known.
The simple mass–spring damper model is the foundation of vibration analysis, but what about more complex systems? The mass–spring–damper model described above is called a single degree of freedom (SDOF) model since the mass is assumed to only move up and down. In more complex systems, the system must be discretized into more masses that move in more than one direction, adding degrees of freedom. … This is referred to an eigenvalue problem in mathematics …
5. “The simple harmonic oscillator … is an excellent model for a wide range of systems in nature.”
6. Using eigenvalues and eigenvectors to study vibrations
… a brief introduction to the use of eigenvalues and eigenvectors to study vibrating systems for systems with no inputs. MatLab code is also included on the “Vibrating Systems” page. Analyzing a system in terms of its eigenvalues and eigenvectors greatly simplifies system analysis, and gives important insight into system behavior. For example, once the eigenvalues and eigenvectors of the system above have been determined, its motion can be completely determined simply by knowing the initial conditions and solving one set of algebraic equations.
Eigenvalues are a special set of scalars associated with a linear system of equations (i.e., a matrix equation) that are sometimes also known as characteristic roots, characteristic values (Hoffman and Kunze 1971), proper values, or latent roots (Marcus and Minc 1988, p. 144).
The determination of the eigenvalues and eigenvectors of a system is extremely important in physics and engineering, where it is equivalent to matrix diagonalization and arises in such common applications as stability analysis, the physics of rotating bodies, and small oscillations of vibrating systems, to name only a few. Each eigenvalue is paired with a corresponding so-called eigenvector (or, in general, a corresponding right eigenvector and a corresponding left eigenvector; there is no analogous distinction between left and right for eigenvalues).
The decomposition of a square matrix A into eigenvalues and eigenvectors is known in this work as eigen decomposition, and the fact that this decomposition is always possible as long as the matrix consisting of the eigenvectors of A is square is known as the eigen decomposition theorem.
8. “If you get nothing out of this quick review of linear algebra you must get this section. Without this section you will not be able to do any of the differential equations work that is in this chapter.”
9. “The Schrödinger equation is the basic equation for obtaining the constant energy states of atoms, molecules, etc. … Boundary conditions [as in particle in a box] give additional equations, on top of Schrödinger’s equation, and this “narrows down” the number of acceptable wavefunctions. … One of the postulates … of quantum mechanics … is that to each physical property (energy, momentum, position, kinectic energy, number of particles, etc.) there is an associated operator. … [and] that the result of a measurement of that property must give one of the eigenvalues associated with that operator.”
http://www.yorku.ca/renef/eigen.pdf — re eigenvalue equation [Useful overview in 6 pages]
The wavefunction for a given physical system contains the measurable information about the system. To obtain specific values for physical parameters, for example energy, you operate on the wavefunction with the quantum mechanical operator associated with that parameter. The operator associated with energy is the Hamiltonian, and the operation on the wavefunction is the Schrodinger equation. Solutions exist for the time independent Schrodinger equation only for certain values of energy, and these values are called “eigenvalues*” of energy.
It is a general principle of Quantum Mechanics that there is an operator for every physical observable. A physical observable is anything that can be measured. If the wavefunction that describes a system is an eigenfunction of an operator, then the value of the associated observable is extracted from the eigenfunction by operating on the eigenfunction with the appropriate operator. The value of the observable for the system is the eigenvalue, and the system is said to be in an eigenstate.
 Dictionary definitions
(comb. form) proper; characteristic: eigenfunction. ORIGIN from the German adjective eigen ‘own.’
- each of a set of values of a parameter for which a differential equation has a nonzero solution (an eigenfunction) under given conditions.
- any number such that a given matrix minus that number times the identity matrix has a zero determinant.
eigenvector: a vector that when operated on by a given operator gives a scalar multiple of itself.
eigenfunction: each of a set of independent functions that are the solutions to a given differential equation.
 As an undergrad, I studied linear (matrix) algebra. But I “hit the wall” in math while taking a class about the calculus of complex variables in n-space. An applied mathematics class dealing with complex analysis or the theory of functions of a complex variable. Perhaps my struggle was due to the way the course was taught. The TA merely came into the classroom and started writing his notes on a roll-up section of blackboards, moving on to the next section when each filled up and erasing earlier ones (the blackboards wrapped around three sides of the room). Probably was an early morning class as well. I remember barely keeping up taking down those notes. No Q&A. When I met with the TA and asked why he just didn’t handout copies of his notes for discussion, he said something like “what would be the point of the class?” Sigh.
 At the macroscopic level, every object (rather than a dynamic system of objects) may be viewed as in its own “eigenstate.” A baseball, for example, is in a state of composite quantum decoherence. All its (emergent) properties appear definite. No quantum behavior is observable — there are no coherent aggregates (as in lasers and superconductors). And the de Broglie wavelength of a baseball is too small to ever measure (~10^–34 m).
Decoherence represents an extremely fast process for macroscopic objects, since these are interacting with many microscopic objects, with an enormous number of degrees of freedom, in their natural environment. The process explains why we tend not to observe quantum behavior in everyday macroscopic objects. It also explains why we do see classical fields emerge from the properties of the interaction between matter and radiation for large amounts of matter.