Eigenvalue physical meaning What is the physical significance of eigenvalues or eigenvectors?? Please try to explain in very simple language
simple  harmonic oscillator , potential  well
could  you  support  your answer by  examples
many  thanks.
 A: An eigenvector is simply a vector that is unaffected (to within a scalar value) by a transformation. Formally, an eigenvector is any vector $x$ such that for an operator $\Omega$, $\Omega x = \lambda x$ for some scalar constant $\lambda$. All operators of dimension $n$ have exactly $n$ eigenvectors/eigenvalues (though these are only all distinct if $\Omega$ is diagonalizable).
Eigenvectors (or really, eigen-things, as physics seems to love to slap the term "eigen" in front of any word it wants) show up everywhere. Generalizing the idea of an eigenvector to any thing that is affected only up to a scalar value by some operator, here are a few examples:


*

*In math, the set of exponential functions (e.g. $n^x$) are the eigenfunctions of the differentiation operator, and $e^x$ is the eigenfunction with eigenvalue 1. You can use eigenvector/value completeness in the differentiation operator to prove Euler's identity, that $e^{i \theta} = \cos\theta + i \sin \theta$, and to prove that $e^x$ corresponds to its Taylor series.

*In quantum mechanics, an "eigenstate" of an operator is a state that will yield a certain value when the operator is measured. The eigenvalues of each eigenstate correspond to the allowable values of the quantity being measured. For example, the energy eigenstates of an electron in a hydrogen atom (a simple harmonic oscillator), corresponding to energies of $E_n = -Ry/n^2$ will always give their corresponding energies if their energies are measured. However, a state composed of a linear combination of eigenstates, such as $\left| \psi \right>=\frac{1}{\sqrt{2}}\left|0\right>+\frac{1}{\sqrt{2}}\left|1\right>$ will give energies of either $E_0$ or $E_1$ with equal probabilities. In general, a state $\left| \psi \right>$ composed of a linear combination of eigenfunctions $\psi_n$ of any observable variable (energy, spin, momentum, etc.) given by
$$\left|\psi\right>=\sum_{n=0}^{N} a_n \left|\psi_n\right>$$ 
will always yield a measured value that is an eigenvalue of the observable, with the probability of each value being given by $a_n^2$.

*Also in quantum mechanics, two observables generate an uncertainty principle if they do not commute - meaning that their matrix representations are not simultaneously diagonalizable, so they don't share a set of eigenvectors/eigenvalues. The most common example for this is the Heisenberg uncertainty principle, given by $\sigma x \cdot \sigma p \ge \frac{\hbar}{2}$, since $x$ and $p$ do not commute and thus do not share a set of common eigenstates. However, there are an infinite number of these corresponding uncertainty principles, each corresponding to some set of incompatible physical quantities.

*Google's PageRank algorithm generates transition probabilities between webpages by looking at the number of links from each page to each other page, and makes a giant "matrix" from these values. The resulting eigenvectors and eigenvalues of this "operator" provide a very reliable metric for ranking pages by relevance and quality of content.

*The first generations of airplanes were prone to disintegrate mid-flight due to a phenomenon called "flutter", which is when turbulence from air passing over the plane drives the plane at its natural resonance frequencies (the "eigenfrequencies", or vibrational modes, of the object). Modeling the aircraft and computing the eigenfrequencies allows engineers to modify the design accordingly, which usually involves using slightly different materials with nearly relatively prime resonant frequencies in key parts of the aircraft.
There are many other applications, but these are just a few that come to mind. Many problems across disciplines are actually eigenvalue problems in disguise.
A: The meaning of eigenvalue and eigenvector (or eigenstate if you want) depends on what operator and what observables you are considering.
If the operator is now a Hamiltonian, the eigenvalue you get will be the energy of the system and the eigenvector tell you its "state".
So for the SHO system, the eigenvalue of the Hamiltonian is $(n+1/2)hf=E$ and $n$ depends on the system's quantum state. And the eigenvector in SHO is just the eigenstate, and in most case I think you can consider the linear combination of the vector as a wavefunction. That is, the eigenvector gives out about what its state, and usually will include some probability in its coefficient since we usually get more than one eigenvector.
And for the potential well, I take the infinite potential well for example:
If the operator is observable A now, the eigenvalue you get will be the possible outcome of A, and the eigenvector is the state corresponding to the outcome. While if you use Hamiltonian $H$ in the above case, you will get $E=(n\pi\hbar)^2/2ma^2$.
For all of the above case, I didn't use position or momentum for examples since you will find out these types of operator has no eigenfunction in Hilbert space if you didn't give it some kind of restrictions. And always note the wavedunction should live in Hilbert space.
And how to overcome the problems are another story, while I think the Hamiltonian is enough for you to understand the meaning of eigenvalue and eigenvector.
