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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

    |⟨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 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)⟩

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.



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