10.1: Nature of Quantum Mechanics

Quantum mechanics is weird. There seems to be no way to make it appear otherwise. Many of its predictions are not what one would expect from everyday experience with objects of the size normally encountered.

Quantum mechanics is mysterious, even to very good physicists. The underlying philosophy and interpretation of its equations and techniques are controversial.

Quantum mechanics is difficult to use. Relatively advanced mathematical skills are needed. The basic equation, although linear, is a partial differential equation that cannot be solved analytically except in a very few simple situations. Usually numerical solutions are necessary.

Quantum mechanics, like other physical theories, requires skill and judgment both in modelling and in mathematics. It is not generally taught in any depth before the graduate or advanced undergraduate level.

Quantum mechanics comes in different forms. It has many alternate formulations. Generally these are equivalent in the sense that they predict the same results of experiments, but are not equally easy to learn or to use for particular purposes.

In light of these disadvantages, why is quantum mechanics important? Because it works. It is the ONLY fundamental physical theory that works over such a wide range of situations. Its predictions have been verified experimentally time after time. It applies to everyday size objects, and to astronomical objects (although it is usually not necessary for them). It applies to atomic-size objects, to electromagnetic waves, and to sub-atomic objects. There is a version that is compatible with the theory of special relativity. About the only physical phenomenon not handled well at this time is gravity; quantum mechanics has not yet been extended to be compatible with the theory of general relativity.

In these notes we cannot cover quantum mechanics in much depth. For the purpose of examining information processing in physical systems, we only need to understand a few of the general features such systems must have. In particular, we need a model of physical systems in which there are many possible states, each with its own probability of being the one the system is actually in (i.e., the state “occupied”). These states all have physical properties associated with them, and energy is one of these. Quantum mechanics justifies this model.

We will use this model in two situations. The first (below) is one with many states, where the objective is to understand how the information associated with the occupancy of these states affects the flow of energy. The second (in a later chapter of these notes) is one with a very small number of states, where information is represented using the occupancy of these states, and the objective is to understand both the limits and opportunities in information processing afforded by quantum mechanics.

The next two sections, Section 10.2 “Introduction to Quantum Mechanics” and Section 10.3 “Stationary States,” may be skipped by readers who are prepared to accept the state model without justification. They may proceed directly to Section 10.4 “Multi-State Model.” Other readers may glean from the next two sections some indication of how quantum considerations lead to that model, and in the process may find some aspects of quantum mechanics less mysterious.