---
tag: J1YDW3BE56
kind: concept
name: Mathematical formulation of quantum mechanics
msc: [81P05, 46N50, 81Q10]
sources:
  - url: https://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics
same_as:
  wikidata: Q606855
  wikipedia: Mathematical formulation of quantum mechanics
---

## Summary

The mathematical formulation of quantum mechanics describes a physical system by a complex [[Hilbert space|HM2HHE3J51]].
States are unit vectors or density operators, and measurable quantities are [[self-adjoint operators|0NEGN5CRSC]].
Measurement probabilities come from the spectral theorem through [[projection-valued measures|ATGT25E5FF]], and time evolution is a one-parameter unitary group generated by the Hamiltonian.
Dirac and von Neumann set out this framework around 1930, and it is also stated as the Dirac–von Neumann axioms.
It made quantum theory a part of functional analysis and gave rise to operator algebras.

## Explanations

### L1 human

In classical mechanics, the state of a system is a point, and a quantity such as energy is a function of that point.
In quantum mechanics, the state is a vector in a space of possibly infinite dimension, and a quantity is an operator that acts on these vectors.
The possible results of measuring the quantity are the points of the spectrum of the operator, and the chances of each result are computed from the state vector.
For example, a spin-$\tfrac12$ particle lives in $\C^2$, and its spin along one axis is the matrix $\operatorname{diag}(1, -1)$, with possible results $+1$ and $-1$.
Because operators need not commute, some pairs of quantities cannot both have sharp values, which is the source of the uncertainty principle.