= Schrödinger picture
{c}
To better understand the discussion below, the best thing to do is to read it in parallel with the simplest possible example: <Schrödinger picture example: quantum harmonic oscillator>.
The state of a quantum system is a vector of unit length in a <Hilbert space>. TODO why Hilbert Space.
"Making a measurement" for an <observable> means applying a <self-adjoint operator> to the state, and after a measurement is done:
* the state <wave function collapse>[collapses] to an <eigenvector> of the self adjoint operator
* the result of the measurement is the <eigenvalue> of the self adjoint operator
* the probability of a given result happening when the spectrum is <discrete> is proportional to the modulus of the projection on that eigenvector.
For continuous spectra such as that of the <position operator> in most systems, e.g. <Schrödinger equation for a free one dimensional particle>, the projection on each individual eigenvalue is zero, i.e. the probability of one absolutely exact position is zero. To get a non-zero result, measurement has to be done on a continuous range of eigenvectors (e.g. for position: "is the particle present between x=0 and x=1?"), and you have to integrate the probability over the projection on a continuous range of eigenvalues.
In such continuous cases, the probability collapses to an uniform distribution on the range after measurement.
The continuous position operator case is well illustrated at: <video Visualization of Quantum Physics (Quantum Mechanics) by udiprod (2017)>
Those last two rules are also known as the <Born rule>.
Self adjoint operators are chosen because they have the following key properties:
* their eigenvalues form an orthonormal basis
* they are diagonalizable
See also: https://en.wikipedia.org/wiki/Measurement_in_quantum_mechanics
Perhaps the easiest case to understand this for is that of <spin (physics)>, which has only a finite number of eigenvalues. Although it is a shame that fully understanding that requires a <special relativity>[relativistic] quantum theory such as the <Dirac equation>.
The next steps are to look at simple 1D bound states such as <particle in a box> and <quantum harmonic oscillator>.
This naturally generalizes to <Schrödinger equation solution for the hydrogen atom>.
The solution to the <Schrödinger equation for a free one dimensional particle> is a bit harder since the possible energies do not make up a <countable set>.
This formulation was apparently called more precisely <Dirac-von Neumann axioms>, but it because so dominant we just call it "the" formulation.
<Quantum Field Theory lecture notes by David Tong (2007)> mentions that:
\Q[if you were to write the wavefunction in quantum field theory, it would be a functional, that is a function of every possible configuration of the field $\phi$.]