Mathematical formulation of quantum mechanics

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[[fr:Postulats de la m�canique quantique]]

The mathematical formulation of quantum mechanics in general use is based on an identification of the bra-ket notation of Dirac, with the abstract notion of Hilbert space used in functional analysis. This formulation is often attributed to von Neumann.

Contents

Setting

There is given a separable Hilbert space H. The states are the projective rays of H. An operator in this setting is an unbounded operator, i.e. a linear map from a dense subspace of H to H. We cannot assume that the operator is defined on the whole of H: the Hellinger-Toeplitz theorem) that an operator is continuous, then it is a bounded linear map from H to H. But the operators of quantum theory are certainly in general not bounded. This can be seen from spectral theory: a bounded operator will have a finite spectral radius

By tradition, observables are thus identified with operators, although this is rather questionable, especially in the presence of symmetries leading to superselection sectors. This is why some people prefer the density state formulation.

Basic postulates (Schr�dinger picture)

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
    <math> | \lang a| \psi \rang |^2 </math>
    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 [[Schr�dinger equation]]:
    <math> i \hbar \ {d \over dt} | \psi \rang = H | \psi \rang </math>
    This is called the [[Schr�dinger picture]].

Other formulations

These include the Heisenberg picture, Born principle, relative state interpretation.

In the Heisenberg framework, the 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, there is given a C* algebra, the associative algebra of operators. Positive elements of its dual vector space 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.

List of mathematical tools

See also: list of mathematical topics in quantum theory.


References

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