Book contents
- Frontmatter
- Contents
- 0 Introduction
- 1 What is Fisher information?
- 2 Fisher information in a vector world
- 3 Extreme physical information
- 4 Derivation of relativistic quantum mechanics
- 5 Classical electrodynamics
- 6 The Einstein field equation of general relativity
- 7 Classical statistical physics
- 8 Power spectral 1 / ƒ noise
- 9 Physical constants and the 1/x probability law
- 10 Constrained-likelihood quantum measurement theory
- 11 Research topics
- 12 EPI and entangled realities: the EPR–Bohm experiment
- 13 Econophysics, with Raymond J. Hawkins
- 14 Growth and transport processes
- 15 Cancer growth, with Robert A. Gatenby
- 16 Summing up
- Appendix A Solutions common to entropy and Fisher I-extremization
- Appendix B Cramer–Rao inequalities for vector data
- Appendix C Cramer–Rao inequality for an imaginary parameter
- Appendix D EPI derivations of Schrödinger wave equation, Newtonian mechanics, and classical virial theorem
- Appendix E Factorization of the Klein–Gordon information
- Appendix F Evaluation of certain integrals
- Appendix G Schrödinger wave equation as a non-relativistic limit
- Appendix H Non-uniqueness of potential A for finite boundaries
- Appendix I Four-dimensional normalization
- Appendix J Transfer matrix method
- Appendix K Numerov method
- References
- Index
Appendix D - EPI derivations of Schrödinger wave equation, Newtonian mechanics, and classical virial theorem
Published online by Cambridge University Press: 03 February 2010
- Frontmatter
- Contents
- 0 Introduction
- 1 What is Fisher information?
- 2 Fisher information in a vector world
- 3 Extreme physical information
- 4 Derivation of relativistic quantum mechanics
- 5 Classical electrodynamics
- 6 The Einstein field equation of general relativity
- 7 Classical statistical physics
- 8 Power spectral 1 / ƒ noise
- 9 Physical constants and the 1/x probability law
- 10 Constrained-likelihood quantum measurement theory
- 11 Research topics
- 12 EPI and entangled realities: the EPR–Bohm experiment
- 13 Econophysics, with Raymond J. Hawkins
- 14 Growth and transport processes
- 15 Cancer growth, with Robert A. Gatenby
- 16 Summing up
- Appendix A Solutions common to entropy and Fisher I-extremization
- Appendix B Cramer–Rao inequalities for vector data
- Appendix C Cramer–Rao inequality for an imaginary parameter
- Appendix D EPI derivations of Schrödinger wave equation, Newtonian mechanics, and classical virial theorem
- Appendix E Factorization of the Klein–Gordon information
- Appendix F Evaluation of certain integrals
- Appendix G Schrödinger wave equation as a non-relativistic limit
- Appendix H Non-uniqueness of potential A for finite boundaries
- Appendix I Four-dimensional normalization
- Appendix J Transfer matrix method
- Appendix K Numerov method
- References
- Index
Summary
EPI is by construction a relativistically covariant theory (Sec. 3.5). Thus, non-covariant uses of EPI, specifically with Fisher coordinates whose dimension is less than four, give approximate answers. However, these are often easier to obtain than by a fully covariant EPI approach. We use EPI in this manner to derive the following effects: the one-dimensional, stationary Schrödinger wave equation, Newton's second law of classical mechanics, and the classical virial theorem. All are recognizably approximations in one sense or another.
Schrödinger wave equation
Here, a one-dimensional analysis is given. The derivation runs parallel to the fully covariant EPI derivation in Chap. 4 of the Klein–Gordon equation. We point out corresponding results as they occur.
The position θ of a particle of mass m is measured as a value y = θ + x (see Eq. (2.1)), where x is a random excursion whose probability amplitude law q(x) is sought. Since the time t is ignored, we are in effect seeking a stationary solution to the problem. Notice that the one-dimensional nature of x violates the premise of covariant coordinates as made in Sec. 4.1.2.
Since the approach is no longer covariant, it is being used improperly. However, a benefit of EPI is that it gives approximate answers when used in a projection of fourspace. The approximate answer will be the non-relativistic, Schrödinger wave equation.
Assume that the particle is moving in a conservative field of scalar potential V(x). Then the total energy W is conserved. This is assumed in the derivation below.
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- Science from Fisher InformationA Unification, pp. 445 - 451Publisher: Cambridge University PressPrint publication year: 2004