The model required that one of the sets of oscillators have positive energy, whilst the other, negative energy. There was never a requirement that the origin/location of the energies be explained; this work seeks to address this.
The analysis proceeds from the Time-independent Schrödinger equation for a simple quantum harmonic oscillator and shows that the positive energy resides in real space, whilst the negative energy resides in imaginary space.
It is considered that the work has relevance to super-luminal travel employing deformation of space as typified by the Alcubierre drive. It is calculated that the current vacuum energies of the universe (with their appropriate algebraic sign) are of the order of 10^72 J.
Table of Contents
1. Introduction
2. The analysis
3. Discussion
4. References
Objectives & Core Topics
The primary objective of this work is to provide a theoretical explanation for the simultaneous existence and equality of positive and negative energy within a model of the universe based on frequency-quantised, simple quantum harmonic oscillators.
- Mathematical derivation of negative energy residing in imaginary space.
- Exploration of the vacuum as a source of energy for super-luminal travel.
- Modeling the universe as a channel of mutually orthogonal planes representing energy states.
- Calculation of vacuum energy magnitudes relevant to theoretical propulsion concepts like the Alcubierre drive.
Excerpt from the Book
The Analysis.
For simplicity we consider the Time-independent Schrödinger equation in one spatial dimension, i.e. [-h^2 / 2μ d^2 / dx^2 + U(x)]ψ(x) = E ψ(x) -------------- (1).
The symbols have their usual connotation. Following convention we write equation (1) as: Ĥ ψ(x) = E ψ(x) ----------- (2), where, Ĥ is the Hamiltonian operator. We now consider a simple, classical harmonic oscillator where the potential function, U(x) is given by: U(x) = 1/2 k x^2, where k is the spring constant. It may be noted that equation (1) contains the reduced mass μ and hence also applies to a simply-connected system of two oscillating masses, m1 and m2 reduced to the oscillation of a single entity of reduced mass, μ = m1m2/(m1 + m2) moving about a fixed point in one of the other masses.
Hence, we may write the Hamiltonian operator as: Ĥ = [-h^2 / 2μ d^2 / dx^2 + 1/2 k x^2]. We now transform all of the elements of the Hamiltonian to imaginary space by multiplying each by the operator, i , so that ( x = ix, k = ik, h = ih and μ = iμ ) ---------------- (a). The Hamiltonian is now written out, in extenso, so that the reader may verify the end result. i.e. -(ih)^2 / 2iμ d^2 / d(ix)d(ix) + 1/2 ik (ix)^2.
Summary of Chapters
Introduction: This chapter reviews previous research regarding a universe modeled as a system of quantum harmonic oscillators and identifies the need to explain the coexistence of positive and negative energy.
The analysis: Using the Schrödinger equation, the author demonstrates that negative energy resides in imaginary space, which is orthogonal to the real space containing positive energy.
Discussion: The findings are applied to calculate the magnitude of vacuum energy and its potential relevance for high-energy propulsion systems such as the Alcubierre drive.
References: This section lists the scientific publications and external data sources used to support the theoretical model.
Keywords
Orthogonal Universe, Negative Energy, Quantum Harmonic Oscillator, Vacuum Energy, Imaginary Space, Schrödinger Equation, Alcubierre Drive, Cosmology, Hubble's Law, Energy Density, Hamiltonian Operator, Space-time Deformation.
Frequently Asked Questions
What is the central focus of this publication?
The work investigates how positive and negative energy can coexist within a model of the universe built upon frequency-quantised quantum harmonic oscillators.
What are the key research areas addressed?
The study bridges quantum mechanics, cosmology, and theoretical physics, specifically focusing on the properties of the vacuum and its energy states.
What is the primary objective of this research?
The main goal is to prove that negative energy resides in an imaginary domain that is orthogonal to our real space, establishing a symmetric model of energy.
Which scientific methodology is employed?
The author uses a mathematical transformation of the Time-independent Schrödinger equation to map energy states into imaginary space.
What is covered in the main body of the text?
The text provides the mathematical derivation of the orthogonal space model and discusses the practical implications for massive energy requirements in space travel.
Which terms best characterize this work?
The work is defined by concepts such as the "Orthogonal Universe," energy-quantisation, and the application of vacuum energy to super-luminal propulsion.
How does this model explain the "coexistence" of energies?
By transforming the Hamiltonian operator into the imaginary domain, the author shows that positive and negative energies occupy mutually orthogonal planes, allowing them to exist simultaneously.
Why is the "Alcubierre drive" mentioned in the discussion?
The Alcubierre drive serves as a practical context for the study, as it theoretically requires vast quantities of both positive and negative energy-density to function.
Does the author suggest that vacuum energy is "free"?
No, the author clarifies that extracting energy from the vacuum is not a "free lunch," as energy must be expended to gather and concentrate it.
- Quote paper
- William Fidler (Author), 2020, The Orthogonal Universe, Munich, GRIN Verlag, https://www.grin.com/document/904068
