WHAT TO LOOK FOR

Although the classical theory developed in Chapter 2 can be used to describe a great many phenomena in the real world, there is little doubt that the proper description of the world is quantum. To a large measure we perceive the world classically and we must find ways to uncover the underlying quantum theory. The process of “quantization” normally consists of turning a classical theory into a corresponding quantum theory. In turn, the “classical limit” is the process by which a quantum theory is brought to its associated classical theory. The parameter ħ = 1.0545 × 10−27 (≃ 10−27) erg-seconds (cgs units), referred to as Planck's constant, sets the scale of quantum phenomena. In many applications it will prove useful to use “natural” units in which ħ = 1 just so that formulas are less cumbersome. For most typical formulations of quantum mechanics the classical limit means the limit that ħ → 0, and in this text we shall have many occasions to use this definition as well. In addition, we shall also learn to speak about the classical and quantum theories simultaneously coexisting – as they do in the real world – without the need to take the limit ħ → 0.

There are three generally accepted “royal” routes to quantization, due principally to Heisenberg, Schrödinger, and Feynman, respectively, and we shall exploit them all. In each case we begin with a single degree of freedom (N = 1).