Quantum Evolution and Anticipation Quantum Evolution and Anticipation by Hans-Rudolf Thomann March 2010 Author address Im Altried 1h, 8051 Zürich, [email protected] All rights reserved 1/22 Quantum Evolution and Anticipation CONTENTS 0 Introduction ................................................................................................................. 4 1 Quantum Evolution Systematic ................................................................................. 5 1.1 Basic Terminology and Definitions ................................................................................ 5 Definition 1: Evolution Scenario ...................................................................................... 5 Definition 2: Spectral measures and quantities ............................................................... 6 Proposition 1: Spectral Measures ................................................................................... 7 Theorem 1: Embedded Orthogonal Evolution ................................................................. 8 1.2 Characterization of Finite-dimensional Evolution ......................................................... 8 1.2.1 Amplitudes, Characteristic Polynomial, Eigenvalues and Spectra .................... 8 1.2.2 Domain of Positivity .......................................................................................... 9 1.2.3 Summary .........................................................................................................10 Theorem 2: Characterization of Finite-dimensional Evolution ........................................10 1.3 Characterization of Infinite-dimensional Evolution .................................................... 11 Theorem 3: Characterization of Infinite-dimensional Evolution ......................................11 2 Anticipation in Quantum Evolution ......................................................................... 12 2.1 Anticipation and Retrospection ................................................................................... 12 Definition 3: Anticipation Amplitudes and Probabilities ..................................................12 2.2 Anticipation Amplitudes and the Spectral Difference ................................................ 12 Definition 4: Spectral Difference ....................................................................................13 Lemma 1: Well-behaved measures ...............................................................................13 Proposition 2: Representation of Anticipation Amplitudes ..............................................13 2.3 Assessment Methodology ............................................................................................ 14 Definition 5: ...................................................................................................................14 2.4 Anticipation Strength .................................................................................................... 14 Theorem 4: Anticipation Strength ..................................................................................14 Lemma 2 .......................................................................................................................15 3 Conclusions .............................................................................................................. 16 4 Bibliography .............................................................................................................. 17 Appendix A: Spectral-analytic Lemmata ....................................................................... 18 A.1 Inversion of the Discrete Fourier Transform 퐝훎^퐧 퐧 ∈ ℤ ............................................. 18 Lemma A.1: Inversion of the Discrete Fourier Transform 퐝훎^퐧 퐧 ∈ ℤ ............................18 Examples ......................................................................................................................20 A.2 Representation of the Dirac -Functional .................................................................... 20 Lemma A.2: Representation of the Dirac -Functional ...................................................21 Corollary A.2 .................................................................................................................21 A.3 Time-averages of 풅흁^풕ퟐ .................................................................................................... 22 All rights reserved 2/22 Quantum Evolution and Anticipation Abstract A state is said to exhibit orthogonal evolution of order 퐿 and step size 푇 iff it evolves into or- 푇 thogonal states at times 0 < 푛 푇 ≤ 퐿. Measurement at time yields states from this ortho- 2 gonal set with certain probabilities. This effect, which we name anticipation, has been de- scribed in a previous paper in a special setting. Here we extend our analyzes to general type quantum evolutions and spectral measures and prove that quantum evolutions possessing an embedded orthogonal evolution are characte- rized by positive joint spectral measure. Furthermore, we categorize quantum evolution, assess anticipation strength and provide a framework of analytic tools and results, thus preparing for further investigation and experi- mental verification of anticipation in concrete physical situations such as the H-atom, which we have found to exhibit anticipation. All rights reserved 3/22 Quantum Evolution and Anticipation 0 Introduction This paper generalizes and extends the findings of (Thomann, 2008), where we studied the special case of orthogonal evolutions of maximal order with pure point and absolutely conti- nuous measures, to arbitrary types of spectral measures and evolutions. 푖퐻 − 푡 Under a Hamiltonian 퐻, a quantum state 푞 evolves into an orbit 푞푡 = 푈푡 푞, where 푈푡 = 푒 ℏ is defined on the closure of 퐷 퐻 . By spectral theory (Reed, 1980), the spectral measure 휇푞 휆 of 푞 and 퐻 is uniquely defined by 푞, 푓 퐻 푞 = ∫ 푓 퐻 푑휇푞 휆 for any analytic function 푓. For fixed step size 푇 > 0, 푞0 = 푞 and 푛 ∈ ℤ, an evolution of order 퐿 ≥ 0 is given by the prima- 푛 푛 ry sequence 푞푛 = 푈푇 푞0 and the dual sequence, 푟푛 = 푈푇 푟0, such that 푟푘 , 푞푙 = 훿푘푙 푘, 푙 ∈ ℤ , 푘 − 푙 ≤ 퐿 (see definition 1). The amplitudes 푟푡, 푞0 are determined by the spectral meas- ure 휌푑휇푞 (see definition 2). Positive evolutions are defined as those with positive 휌. Theorem 1 states that every positive evolution contains an embedded orthogonal evolution, driving the component 푠0 of 푞0 of size 휁 through 퐿 + 1 mutually orthogonal states. Theorems 2 and 3 prove the existence of (not necessarily positive) evolutions of any order for point, singular-continuous and absolutely continuous spectrum, respectively. Quantum-Mechanical anticipation (definition 3) is defined for the embedded orthogonal evo- lution in terms of the anticipation amplitudes 훼푛 = 푠푛 , 푈푇 2푠0 , and the anticipation probabili- 2 ties 푝푛 = 훼푛 , expressing the result of a measurement of 푠0 at time 푇 2, where 푛 ≤ 퐿. Quantum-mechanical retrospection is the time-reverse of anticipation. Measurements antici- pate future (or recall past) states 푠푛 with probability 푝푛 , which is the reason for the naming. Theorem 4 assesses the anticipation strength in various situations. This paper provides analyses and tools facilitating further research about anticipation. In chapter 1 we introduce the basic terminology and definitions, characterize finite-dimensional and infinite-dimensional evolutions and their spectral measures, and explore the topology of the manifold of positive evolutions. Chapter 2 explains and defines anticipation, provides a representation of anticipation amplitudes and analyzes anticipation strength. To cover not only point and absolutely continuous but also singular-continuous measures, we use some formalism from spectral theory an provide mathematical theorems in appendix A. Theorem A.1 establishes explicit formulae for the Fourier coefficients of continuous tions 퐹 supported on 퐾 whose second distributional derivatives yield a given measure 휈, in- verting the discrete Fourier transform 푑휈^ 푛 . Theorem A.2 proves a very general represen- tation of Dirac's 훿 functional. In a future paper we will present results from numerical simulations of the point spectrum of the H-atom as well as equidistant and random point spectra, demonstrating that anticipation and retrospection occur in common situations. The quantum anticipation simulator software running und Microsoft Windows is already downloadable from the author's homepage http://www.thomannconsulting.ch/public/aboutus/aboutus-en.htm. All rights reserved 4/22 Quantum Evolution and Anticipation 1 Quantum Evolution Systematic In this chapter we introduce the basic definitions and terminology, cast quantum evolutions into a set of categories and derive the characteristic spectral properties of each category. 1.1 Basic Terminology and Definitions Let 퐻 be an essentially self-adjoint Hamiltonian. The set of states we are considering is the Hilbert space ℋ which equals the closure of 퐷 퐻 . Throughout the paper all states will be assumed to have unit norm. New technical terms are introduced in italic type. For any state 푞 ∈ ℋ the finite spectral measure 휇푞 휆 is uniquely defined by the inner product 푞, 푓 퐻 푞 = ∫ 푓 퐻 푑휇푞 휆 for any analytic function 푓 (see e.g. (Reed, 1980)) This relation 2 induces a unique unitary mapping ℑ푞 : ℋ → ℒ 푑휇푞 such that the image of 푞 is the constant function 1. We will sometimes identify states with their images under this mapping. 푖퐻 − 푡 The evolution operator 푈푡 = 푒 ℏ is a unitary isomorphism of ℋ, under which