Encyclopedia > Mathematical formulation of quantum mechanics

  Article Content

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.



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
F-stop

... with f-stop; for an example of this relationship, visit the depth of field article. Picture sharpness[?] also varies with f-stop. The optimal f-stop vary with the lens ...