Classical Limit of Quantum Physics

Abstract

For open systems, the transition from classical physics to quantum physics is carried out by means of a quantization process taking into account the effects caused by the appearance of the Planck constant. In turn the classical limit is a physical phenomenon characterized by a dequantization process in which the Planck constant also plays a fundamental role. Mathematically, we can represent such a physical situation through the limiting procedure

$$\hbar \to 0,$$

(5.1)

although physically ћ is a constant fixed by nature. We make the following requirements to ensure that the mathematical limit (5.1) makes some physical sense:

• The limit ћ → 0 should be a mathematically well-defined procedure. For example, taking ћ → 0 directly for the differential momentum operator

  $$\hat p = \frac{\hbar }{i}\frac{\partial }{{\partial x}}$$

  leads to the nonsensical identity $\hat p = 0$, since the number zero is not an operator. As another example, the classical limit of the fine-structure constant

  $$\alpha = \frac{{{e^2}}}{{\hbar c}}$$

  where —e is the charge of the electron and c the velocity of light, gives rise to a divergence as ћ → 0. To avoid these difficulties, we must elaborate a method to make sure that the notion of classical limit makes sense.

• How should we describe the underlying physics during the domain transition quantum → classical?