These are the key mathematical ideas to understand!!
The state of a quantum system is a vector of unit length in a Hilbert space. TODO why Hilbert Space.
"Making a measurement" means applying a self-adjoint operator to the state, and after a measurement is done
- the state 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. Schrodinger equation solution for a free 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 probabiilty over the projection on a continous 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 57. ""Visualization of Quantum Physics (Quantum Mechanics)" by udiprod (2017)"
Self adjoint operators are chosen because they have the following key properties:
- their eigenvalues form an orthonormal basis
- they are diagonalizable
This naturally generalizes to Schrodinger equation solution for the hydrogen atom.
This formulation was apparently called more precisely Dirac-von Neumann axioms, but it because so dominant we just call it "the" formulation.