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Progress in mathematical physics

Heisenberg’s Uncertainty Principle

Floyd L. Williams

Published Jan 1, 2003• 7 citations
Publisher

Abstract

We establish in this section an integral 6.1.1 $$\begin{array}{*{20}{c}} \hfill {\left[ {\int_{\mathbb{R}} {{{{(x - b)}}^{2}}|\psi (x)} {{|}^{2}}dx} \right]\left[ {\int_{\mathbb{R}} {|\psi \prime (x){{|}^{2}}dx - 2ia} \int_{\mathbb{R}} {\psi \prime (x)\bar{\psi }(x)dx} } \right.} \\ \hfill {\left. { + {{a}^{2}}\int_{\mathbb{R}} {|\psi (x){{|}^{2}}dx} } \right] \geqslant \frac{1}{4}{{{\left[ {\int_{\mathbb{R}} {|\psi (x){{|}^{2}}dx} } \right]}}^{2}},} \\ \end{array}$$ 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.

DOI
10.1007/978-1-4612-0009-3_7

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We establish in this section an integral 6.1.1 \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} 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.
A mathematical uncertainty principle is an inequality or uniqueness theorem concerning the joint localization of a function or system and its spectrum. A Fourier uncertainty principle is thus a statement that a function and its Fourier transform cannot both decay too rapidly. The most familiar form is the Wiener-Heisenberg inequality, 5.1 $$ \left\| {(x - x_0 )f(x)} \right\|_2 \left\| {(\xi - \xi _0 )\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{f} (\xi )} \right\|_2 \geqslant \frac{{\mathop {\left\| f \right\|}\nolimits_2^2 }} {{4\pi }},$$ which becomes an equality only for multiples of Gaussians of the form exp(−πα(x − x0)2 + 2πiξ0x). In this chapter, D will denote the normalized differentiation or momentum operator D = (1/2πi)d/dx. The inequality (5.1) is often written 5.2 σX(f)σD(f)≡∥Xf−⟨Xf,f⟩f∥2∥Df−⟨Df,f⟩f∥2⩾1, \sigma _X (f)\sigma _D (f) \equiv \left\| {Xf - \left\langle {Xf,f} \right\rangle f} \right\|_2 \left\| {Df - \left\langle {Df,f} \right\rangle f} \right\|_2 \geqslant 1,σX​(f)σD​(f)≡∥Xf−⟨Xf,f⟩f∥2​∥Df−⟨Df,f⟩f∥2​⩾1, where X is the multiplication operator f(x) ↦ xf(x). Barnes [22] attributes the first proof of (5.1) to a lecture delivered by Wiener in Hilbert’s mathematical physics seminar in Göttingen in 1925. In fact, Wiener discusses this lecture, though not the explicit proof of (5.1), at length in his autobiography (see [365], p. 105). What distinguished Wiener’s approach, however, was his macroscopic interpretation of (5.1), a point of view that led to further important achievements, including the Landau, Pollack and Slepian “Bell Labs” uncertainty principles in Chapter 3
Since the publication of Heisenberg’s paper1 on the “anschaulichen Inhalt” of quantum mechanics, discussions of the fundamental limitations on the accuracy of physical measurements have been much in the foreground. According to Heisenberg, the quantum mechanics implies that it is impossible to measure simultaneous values of a coordinate and its conjugate momentum with unlimited precision. Instead, if Δp be the estimated error or uncertainty in a momentum and Δq that in the associated coordinate one must have the inequality, ΔpΔq>h2π \Delta p\Delta q>\frac{h}{{2\pi}} ΔpΔq>2πh​ (1) This inequality has come to be known quite generally as Heisenberg’s uncertainty relation.
ABSTRACT. The well-known second moment Heisenberg-Weyl inequality (or uncertainty relation) states: Assume that f: R → C is a complex valued function of a random real variable x such that f ∈ L2 (R), where R = (−∞, ∞). Then the product of the second moment of the random real x for |f | 2 and the second moment of the random real ξ for ∣ f ∣ 2 is at least E 2 R,|f | 4π, where f is the Fourier transform of f, f (ξ) = ∫ R e−2iπξxf (x) dx and f (x) = ∫ R e2iπξx ˆ f (ξ) dξ, and E 2 R,|f | = ∫ R |f (x)|2 dx. This uncertainty relation is wellknown in classical quantum mechanics. In 2004, the author generalized the afore-mentioned result to the higher order moments for L2 (R) functions f. In this paper, a refined form of the generalized Heisenberg-Weyl type inequality is established.
Abstract The aim of this paper is to unify the partial results, which up to now, have been dispersed in various publications in order to show the importance of the functional form of parallelogram identity in mathematics and physics. We study vector spaces admitting a real non-negative functional which satisfies an identity analogous to the parallelogram identity in normed vector spaces. We show that this generalized parallelogram identity also implies an equality analogous to the Cauchy-Schwarz inequality. We study the consequences of this identity in real and complex vector spaces, in generalized Riesz spaces and in abelian groups. We give a physical interpretation to these results. For vector spaces of observables and states, we show that the parallelogram identity implies an inequality analogous to Heisenberg’s uncertainty principle (HUP), and we show that we can obtain the standard structure of quantum mechanics from the parallelogram identity, without assuming from the beginning the HUP. The role of complex numbers in quantum mechanics is discussed.
The uncertainty principle was first formulated by W. Heisenberg in the year 1927, which showed that certain physical values such as the position and the momentum of a particle, and also the time and the energy, cannot be specified with an arbitrary precision, something that was far from experiences in physics before the principle was discovered. In the same year, Kennard showed the same principle using a rigorous mathematical formalism, which has motivated further research of the principle in many different mathematical contexts. Using the fundamental results of Fourier analysis, and respecting the mathematical formalism of quantum mechanics, we present many different formulations of the uncertainty principle. We observe the principle for a single function – its qualitative, quantitative, operator and entropic formulations, for Gabor and wavelet systems of functions, and finally, generally for finite abelian groups, and more specifically for cyclic groups of prime order.