Operator-Observable Correspondence

David J. Ross

Gordon Leslie Squires

of theory What you need to know Definitions and properties Operator, linear operator, functions of operators, commuting and noncommuting operators, eigenfunction, eigenvalue, degeneracy, normalised function, orthogonal functions, Hermitian operator. Postulates of quantum mechanics (1) The state of a system with n position variables q 1 , q 2 , … q n is specified by a state (or wave) function ψ( q 1 , q 2 , … q n ). All possible information about the system can be derived from this state function. In general, n is three times the number of particles in the system. So for a single particle n = 3, and q 1 , q 2 , q 3 may be the Cartesian coordinates x, y, z , or the spherical polar coordinates r , θ, φ, or some other set of coordinates. (2) To every observable there corresponds a Hermitian operator given by the following rules: (i) The operator corresponding to the Cartesian position coordinate x is x × similarly for the coordinates y and z . (ii) The operator corresponding to p x , the x component of linear momentum, is (ħ/i)∂/∂ x – similarly for the y and z components. (iii) To obtain the operator corresponding to any other observable, first write down the classical expression for the observable in terms of x, y, z, p x , p y , p z , and then replace each of these quantities by its corresponding operator according to rules (i) and (ii). (3) The only possible result which can be obtained when a measurement is made of an observable whose operator is A is an eigenvalue of A .

Lev Deych

K. Kong Wan

Many important quantum observables, e.g., angular momentum, have their origin in corresponding classical observables. Postulate 26.1(OV) does not tell us how to find the operators to describe these observables. We can tackle this problem by formulating a process of quantisation to establish the operators for the corresponding quantum observables. Before we can formulate a quantisation process we must choose an appropriate description of classical observables. This is done in the following section.

C. Abugattas Chacoff

The proposed formulation establishes a formal correspondence between the Heisenberg representation and the Bayesian formulation of inverse problems. The likelihood function is first expressed in Hilbert space through a quadratic norm of the difference between the noisy observations and the response of a forward model depending on the unknown parameters, weighted by the inverse of the observational noise covariance. Based on the common mathematical structure of this quadratic form and the observable operator in the Heisenberg representation, the inverse observational noise covariance operator is identified with the initial observable. To establish this correspondence, a stochastic field with Matérn covariance is considered and represented through the Karhunen-Loève expansion. The eigenvalues and eigenfunctions of this expansion are then used to define the state $|Ψ\rangle$ within the proposed formalism. The main results are summarized by four theorems: the unitarity of the evolution operator, the self-adjointness of the Hamiltonian, the conservation of the trace of the observable, and the formal correspondence between the Bayesian likelihood and the Heisenberg observable structure. Limiting cases of the Matérn covariance are analyzed. Finally, numerical examples illustrate the analytical results.