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In quantum mechanics, all of these are resolved by describing the instantaneous state of a system with a wave function that encodes the probability distributions of all observables.
In quantum mechanics, all of these are resolved by describing the instantaneous state of a system with a wave function that encodes the probability distributions of all observables.
Changed: 17,40c17
The mathematically correct formulation was achieved by John von Neumann in 1932. 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.
When a system is in a state |ψ>, a measurement of an observable A produces an eigenvaluea 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]?.
In the formal mathematical theory, the state of a system is described by an element of a Hilbert space and observables are modeled as self-adjoint operators on this Hilbert space. Given a state and such an operator, the probabilities for the various outcomes of the corresponding observation can be calculated. The time evolution of a system is described by the Schrödinger equation, in which the Hamiltonian?, the operator corresponding to the energy observable, plays a prominent role.
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