Mathematical formulation of quantum mechanics

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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
    |<a|ψ>|2
    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 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.