5 - Postulates of Quantum Mechanics

Summary

A postulate is an idea that is suggested as, or assumed to be, the basis for a theory, argument or calculation. In quantum mechanics, the postulates concern the atomic and molecular properties, which are quite far removed from everyday experience. Consequently, in this regard, these may be difficult to understand. The important point is that the postulates are justified by their ability to predict. They should also have the ability to correlate experimental facts by their general applicability.

Before discussing the postulates of quantum mechanics, it will be useful to understand the meaning of two important terms: (a) dynamical variable and (b) observable.

First of all, we should know about the dynamical variable. Any property of a system of interest is known as the dynamical variable. Examples are the position r, the energy E, the x-component of linear momentum px, and so on.

Generally, any quantity of interest in classical mechanics is a dynamical variable. A very useful dynamical variable, which will be used later, comprise three components of the momentum vectors, which a particle in a system has when it remains at a fixed point.

Now, we shall know about observable. An observable is defined as any dynamical variable that can be measured. It should be kept in mind that in classical mechanics, all dynamical variables are observables, but it is not so in quantum mechanics.

In quantum mechanics, certain fundamental restrictions are imposed on simultaneously measurable variable quantities. To measure the component of the momentum vector, it is essential to make a simultaneous measurement of the position and momentum of the particle. We are aware of the fact that there exists an uncertainty relation for such kind of simultaneous measurement on microscopic particle, and the dynamical variable ‘the momentum at a point’ is not an observable. With this background in mind, we are going to introduce the basic postulates of quantum mechanics.

Postulate 1

The state of the quantum mechanical objects is described by a wave function.

or

A quantum mechanical system of n particles is described as fully as possible by a function, ψ (x, y, z, t) called the wave function, which determines all the measurable quantities of the system, where x, y, z, t are spatial co-ordinates.