2 - Fundamentals

Summary

Summary 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 q₁, q₂, … q_n is specified by a state (or wave) function ψ(q₁, q₂, … 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 q₁, q₂, q₃ 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:

1. (i) The operator corresponding to the Cartesian position coordinate x is x × similarly for the coordinates y and z.

2. (ii) The operator corresponding to p_x, the x component of linear momentum, is (ħ/i)∂/∂x – similarly for the y and z components.

3. (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.