The postulates of
quantum mechanics, written in the
bra-ket notation, are as follows:
- The state of a quantum mechanical system is represented by a unit vector, called a state vector, in a complex separable Hilbert space.
- An observable is represented by a Hermitian linear operator in that space.
- When a system is in a state |ψ>, a measurement of an observable A produces an eigenvalue a with probability
- |<a|ψ>|2
where |a> is the eigenvector with eigenvalue a. After the measurement is conducted, the state is |a>.
- There is a distinguished observable H, known as the Halmiltonian?, corresponding to the energy of the system. The time evolution of the state vector |ψ(t)> is given by Schrodinger's equation:
- i (h/2π) d/dt |ψ(t)> = H |ψ(t)>
In this framework, Heisenberg's uncertainty principle becomes a theorem about noncommuting operators. Furthermore, both continuous and discrete observables may be accomodated; 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]?.