Quantum Mechanics and Statistical Description of Results of Measurement

Abstract

Quantum mechanics and its meaning have been discussed in a large number of publications from many different points of view (see e.g. books (Auletta, 2001; Wheeler & Zurek, 1981)). It shows that quantum mechanics is, despite its successful applications, difficult to understand. In this chapter, we discuss quantum mechanics from the point of view of mathematical statistics and show that the most important parts of the mathematical formalism of quantum mechanics can be derived from the statistical description of results of measurement. Various aspects of this approach can be found for example in (Frieden, 1998; 2004; Frieden & Soffer, 1995; Kapsa & Skala, 2009; 2011; Kapsa et al., 2010; Reginatto, 1998; 1999; Skala & Kapsa, 2005a;b; 2007a;b; 2011; Skala, Cižek & Kapsa, 2011). One of the main differences between classical and quantum mechanics is consistent statistical description of results of measurement in quantum mechanics. In contrast to classical mechanics according to which physical measurement can be made in principle arbitrarily exact, quantummechanics takes into consideration physical reality confirmed by experiments and describes physical measurement statistically. The most important points of the statistical description of measurement of the space coordinate x are summarized in Section 2. An important quantity appearing in this approach is the probability density ρ(x, t) of obtaining the value x in measurement made at time t. For the sake of simplicity, only one spatial coordinate x is taken here. Due to the normalization condition for the probability density corresponding to the fact that the measured system must be somewhere in space the probability density ρ must obey the continuity equation analogous to that known from classical continuummechanics. Therefore, except for ρ, we have to take into account also the corresponding probability density current j(x, t) appearing in the continuity equation. We note that the density current j is also necessary for describing the motion in space. To describe the statistical state of the system, both quantities ρ and j are necessary. It is shown in Section 3 that instead of two real quantities ρ and j, we can use also two real functions s1(x, t) and s2(x, t) given by equations ρ = exp(−2s2/h) and j = ρ v = ρp/m = ρ(∂s1/∂x)/m, where s1 corresponds to the Hamilton action S in the expression p = ∂S/∂x known from the Hamilton–Jacobi theory of classical mechanics. More compact way of describing the statistical state of the system is to use the complex wave function ψ = exp[(is1 − s2)/h] as it is done in quantum mechanics. We note that the expression for the probability density current j = ρ(∂s1/∂x)/m is equivalent to the expression for the probability density current known from quantum mechanics. 10