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" for an observable means applying a self-adjoint operator to the state, and after a measurement is done:
Those last two rules are also known as the born rule.
- 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 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 67. ""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.
The solution to the Schrodinger 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:
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 .