Representations of Quantum Mechanics

Abstract

In ordinary quantum mechanics, the description of a particle is in terms of its wave function whose time development is determined by the Schrödinger equation. This is sometimes neither the best nor the most convenient way. In order to change the description we start from the basic concept that reality must be independent of its mathematical description. Reality corresponds to the measurable quantities, which in turn correspond to the expectation values. Consider a measurable quantity O; corresponding to this there is an operator Ô and the measured quantity, the expectation value <O>, is given by

$$ \langle O\left( t \right)\rangle = \int {{\psi ^{ + }}} \left( {{q_{i}},t} \right)\hat{O}\left( {{q_{i}}} \right)\psi \left( {{q_{i}},t} \right)d{q_{i}}] $$

([4.1])

Ψ(q _i ,t) is the wave function, q _i represent the set of commuting variables, and dq _i represents the integration over all such variables.