Mathematical formulation of quantum mechanics

The postulates of quantum mechanics, written in the bra-ket notation, are as follows:

1. The state of a quantum-mechanical system is represented by a unit vector, called a state vector, in a complex separable Hilbert space.

2. An observable is represented by a Hermitian linear operator in that space.

3. When a system is in a state |ψ⟩, a measurement of an observable A produces an eigenvalue a with probability density

   |⟨a|ψ⟩|²

   which when integrated over all paths, yields a probability, where |a⟩ is the eigenvector with eigenvalue a. After the measurement is conducted, the state is |a⟩.

4. There is a distinguished observable H, known as the Hamiltonian, corresponding to the energy of the system. The time evolution of the state vector |ψ(t)⟩ is given by the Schrödinger equation:

   i (h/2π) d/dt |ψ(t)⟩ = H |ψ(t)⟩

   This is called the Schrödinger picture. See also Heisenberg picture.

In this framework, Heisenberg's uncertainty principle becomes a theorem about noncommuting operators. Furthermore, both continuous and discrete observables may be accommodated; in the former case, the Hilbert space is a space of square-integrable wavefunctions.

In the Everett many-worlds interpretation of quantum mechanics, postulate (3) is demoted to a phenomenological principle; see quantum decoherence.

C* formulation

In this formulation, we have a C* algebra, the associative algebra of operators. Positive elements of its dual algebra is are called states and they describe the quantum states. This is related to the density matrix. Given a state, we can construct a unitary representation of it using the Gelfand-Naimark-Segal construction. Two unitarily inequivalent representations are said to belong to different superselection sectors. Relative phases between superselection sectors are not observable.