Operator methods in quantum mechanics

The quantum theory has dominated physics for over half a century, yet it retains some peculiar features. The basic concepts are observable and state. When a system is in a specified state, each observable is a well-defined random variable. However there is no sample space on which all of these random variables are simultaneously defined, and hence no notion of the outcome of the total experiment that underlies the observation one chooses to make. In short, there is no real world [2]. Nevertheless, there is a reasonably coherent mathematical formulation of quantum mechanics, based on the theory of operators in Hilbert space. Let % be a Hilbert space. The inner product of <J> and \p in % is denoted <<f>, i//>. An operator is a linear transformation A from a linear subspace tf){A) to %. If ty(A) is dense in %, then A has an adjoint operator A*. It is defined on the linear subspace tf)(A*) consisting of all <J> in % such that there exists a x in % (necessarily unique) with <<J>, Axpy = <x> ̂ ) f° * ^ * tf)(A). The operator itself is defined by A*<j> = xAn operator A is selfadjoint if A = A*. Selfadjoint operators have a spectral theory, and one consequence of this is a functional calculus that gives a natural meaning to f {A) for every Borel measurable function ƒ and every selfadjoint operator A. Thus selfadjoint operators resemble random variables in that one can form functions of them. There is a conventional dictionary relating the physics to the mathematics. It goes like this: