The Operator Postulate

With every physical observable there is associated a mathematical operator which is used in conjunction with the wavefunction. Suppose the wavefunction associated with a definite quantized value (eigenvalue) of the observable is denoted by Ψ_n (an eigenfunction) and the operator is denoted by Q. The action of the operator is given by

Hermitian Property Postulate

The quantum mechanical operator Q associated with a measurable propertu q must be Hermitian. Mathematically this property is defined by

where Ψ_a and Ψ_b are arbitrary normalizable functions and the integration is over all of space. Physically, the Hermitian property is necessary in order for the measured values (eigenvalues) to be constrained to real numbers.

Basis Set Postulate

The set of functions Ψ_j which are eigenfunctions of the eigenvalue equation

form a complete set of linearly independent functions. They can be said to form a basis set in terms of which any wavefunction representing the system can be expressed:

This implies that any wavefunction Ψ representing a physical system can be expressed as a linear combination of the eigenfunctions of any physical observable of the system.

Expectation Value Postulate

For a physical system described by a wavefunction Ψ , the expectation value of any physical observable q can be expressed in terms of the corresponding operator Q as follows:

It is presumed here that the wavefunction is normalized and that the integration is over all of space. This postulate follows along the lines of the operator postulate and the basis set postulate. The function can be represented as a linear combination of eigenfunctions of Q, and the results of the operation gives the physical values times a probability coefficient. Since the wavefunction is normalized, the integral gives a weighted average of the possible observable values.

Time Evolution Postulate

If Ψ is the wavefunction for a physical system at an initial time and the system is free of external interactions, then the evolution in time of the wavefunction is given by

where H is the Hamiltonian operator formed from the classical Hamiltonian by substituting for the classical observables their corresponding quantum mechanical operators. For a mechanical system, the classical Hamiltonian would be just the kinetic energy plus the potential energy, i.e., the expression for energy. The role of the Hamiltonian in both space and time is contained in the Schrodinger equation.