Heisenberg’s Uncertainty Principle

Abstract

We establish in this section an integral

$$\begin{array}{*{20}{c}} \hfill {\left[ {\int_{\mathbb{R}} {{{{(x - b)}}^{2}}|\psi (x)} {{|}^{2}}dx} \right]\left[ {\int_{\mathbb{R}} {|\psi \prime (x){{|}^{2}}dx - 2ia} \int_{\mathbb{R}} {\psi \prime (x)\bar{\psi }(x)dx} } \right.} \\ \hfill {\left. { + {{a}^{2}}\int_{\mathbb{R}} {|\psi (x){{|}^{2}}dx} } \right] \geqslant \frac{1}{4}{{{\left[ {\int_{\mathbb{R}} {|\psi (x){{|}^{2}}dx} } \right]}}^{2}},} \\ \end{array}$$

(6.1.1)

to as Heisenberg’s inequality, for a sufficiently nice function Ψ : ℝ → ℂ where a, b ∈ ℝ. Even though Ψ is allowed to be complex-valued the second factor on the left-hand side of (6.1.1) will be non-negative. This inequality, which is a purely mathematical result of course, will be shown to form the basis for a precise formulation of Heisenberg’s uncertainty principle. The discussion of this principle in Chapter 1 was brief and non-rigorous. For now, we proceed strictly along mathematical lines.