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The general Schrodinger equation can be simplified by separation of variables to the time independent Schrodinger equation, without any loss of generality:

The general Schrodinger equation can be simplified by separation of variables to the time independent Schrodinger equation, without any loss of generality:

The time-independent Schrodinger equation is an eigenvalue equation

The general Schrodinger equation can be simplified by separation of variables to the time independent Schrodinger equation, without any loss of generality:

$$ \left[ \frac{-\hbar^2}{2m}\nabla^2 + V(\mathbf{r}) \right] \Psi(\mathbf{r}) = E \Psi(\mathbf{r}) $$

The left side of that equation is a linear operator (infinite dimensional matrix acting on vectors of a Hilbert space) acting on the vector $\Psi$ (a function, i.e. a vector of a Hilbert space). And since E is a constant (the energy), this is just an eigenvalue equation.

Have a look at: Real world application of Fourier series to get a feeling for separation of variables works for a simpler equation like the heat equation.

Heuristic argument of why Google PageRank comes down to a diagonalization problem

PageRank was mentioned at: at https://math.stackexchange.com/a/263154/53203 but I wanted to add one cute handy wave intuition.

PageRank is designed to have the following properties:

• the more links a page has incoming, the greater its score
• the greater its score, the more the page boosts the rank of other pages

The difficulty then becomes that pages can affect each other circularly, for example suppose:

• A links to B
• B links to C
• C links to A

Therefore, in such a case

• the score of B depends on the score A
• which in turn depends on the score of A
• which in turn depends on C
• which depends on B
• so the score of B depends on itself!

Therefore, one can feel that theoretically, an "iterative approach" cannot work: we need to somehow solve the entire system in one go.

And one may hope, that once we assign the correct importance to all nodes, and if the transition probabilities are linear, an equilibrium may be reached:

 Transition matrix * Importance vector = 1 * Importance vector 

which is an eigenvalue equation with eigenvalue 1.

Markov chain convergence

https://en.wikipedia.org/wiki/Markov_chain

This is closely related to the above Google PageRank use-case.

The equilibrium also happens on the vector with eigenvalue 1, and convergence speed is dominated by the ratio of the two largest eigenvalues.

See also: https://www.stat.auckland.ac.nz/~fewster/325/notes/ch9.pdf

Eigenvalues and eigenvectors are central to the definition of measurement in quantum mechanics

Measurements are what you do during experiments, so this is obviously of central importance to a Physics subject.

The state of a system is a vector in Hilbert space, an infinite dimensional space square integrable functions.

Then, the definition of "doing a measurement" is to apply a self-adjoint operator to the state, and after a measurement is done:

• the state collapses to an eigenvalue of the self adjoint operator (this is the formal description of the observer effect)

• the result of the measurement is the eigenvalue of the self adjoint operator

Self adjoint operators have the following two key properties that allows them to make sense as measurements as a consequence of infinite dimensional generalizations of the spectral theorem:

• their eigenvectors form an orthonormal basis of the Hilbert space, therefore if there is any component in one direction, the state has a probability of collapsing to any of those directions
• the eigenvalues are real: our instruments tend to give real numbers are results :-)

As a more concrete and super important example, we can take the explicit solution of the Schrodinger equation for the hydrogen atom. In that case, the eigenvalues of the energy operator are proportional to spherical harmonics:

Therefore, if we were to measure the energy of the electron, we are certain that:

• the measurement would have one of the energy eigenvectors

  The energy difference between two energy levels matches experimental observations of the hydrogen spectral series and is one of the great triumphs of the Schrodinger equation

• the wave function would collapse to one of those functions after the measurement

Bibliography: https://en.wikipedia.org/wiki/Measurement_in_quantum_mechanics

Eigenvalues and eigenvectors are central to the definition of measurement in quantum mechanics

Measurements are what you do during experiments, so this is obviously of central importance to a Physics subject.

The state of a system is a vector in Hilbert space, an infinite dimensional space square integrable functions.

Then, the definition of "doing a measurement" is to apply a self-adjoint operator to the state, and after a measurement is done:

• the state collapses to an eigenvalue of the self adjoint operator (this is the formal description of the observer effect)

• the result of the measurement is the eigenvalue of the self adjoint operator

Self adjoint operators have the following two key properties that allows them to make sense as measurements as a consequence of infinite dimensional generalizations of the spectral theorem:

• their eigenvectors form an orthonormal basis of the Hilbert space, therefore if there is any component in one direction, the state has a probability of collapsing to any of those directions
• the eigenvalues are real: our instruments tend to give real numbers are results :-)

As a more concrete and super important example, we can take the explicit solution of the Schrodinger equation for the hydrogen atom. In that case, the eigenvalues of the energy operator are proportional to spherical harmonics:

Therefore, if we were to measure the energy of the electron, we are certain that:

• the measurement would have one of the energy eigenvalues

  The energy difference between two energy levels matches experimental observations of the hydrogen spectral series and is one of the great triumphs of the Schrodinger equation

• the wave function would collapse to one of those functions after the measurement, which is one of the eigenvalues of the energy operator

Bibliography: https://en.wikipedia.org/wiki/Measurement_in_quantum_mechanics