bidsoli.blogg.se

A density operator is a positive-semidefinite operator on the Hilbert space whose trace is equal to 1. The eigenvectors of a von Neumann observable form an orthonormal basis for the Hilbert space, and each possible outcome of that measurement corresponds to one of the vectors comprising the basis. : 79 These issues can be satisfactorily resolved using spectral theory : 101 the present article will avoid them whenever possible. Indeed, introductory physics texts on quantum mechanics often gloss over mathematical technicalities that arise for continuous-valued observables and infinite-dimensional Hilbert spaces, such as the distinction between bounded and unbounded operators questions of convergence (whether the limit of a sequence of Hilbert-space elements also belongs to the Hilbert space), exotic possibilities for sets of eigenvalues, like Cantor sets and so forth. Many treatments of the theory focus on the finite-dimensional case, as the mathematics involved is somewhat less demanding. Alternatively, the Hilbert space may be finite-dimensional, as occurs for spin degrees of freedom. The predictions that quantum physics makes are in general probabilistic.The mathematical tools for making predictions about what measurement outcomes may occur were developed during the 20th century and make use of linear algebra and functional analysis. The dimension of the Hilbert space may be infinite, as it is for the space of square-integrable functions on a line, which is used to define the quantum physics of a continuous degree of freedom. In quantum physics, a measurement is the testing or manipulation of a physical system to yield a numerical result. : 17 These observables play the role of measurable quantities familiar from classical physics: position, momentum, energy, angular momentum and so on.

The approach codified by John von Neumann represents a measurement upon a physical system by a self-adjoint operator on that Hilbert space termed an "observable". In quantum mechanics, each physical system is associated with a Hilbert space, each element of which represents a possible state of the physical system.