In the governing thought, I find an equivalence between the classical information in a quantum system and the integral of that system’s energy and time, specifically , in natural units. I solve this relationship in four ways: the first approach starts with the Schrodinger Equation and applies the Minkowski transformation; the second uses the Canonical commutation relation; the third through Gabor’s analysis of the time-frequency plane and Heisenberg’s uncertainty principle; and lastly by quantizing Brownian motion within the Bernoulli process and applying the Gaussian channel capacity. In support I give two examples of quantum systems that follow the governing thought: namely the Gaussian wave packet and the electron spin. I conclude with comments on the discretization of space and the information content of a degree of freedom.
In the governing thought, I find an equivalence between the classical information in a quantum system and the integral of that system’s energy and time, specifically , in natural units. I solve this relationship in four ways: the first approach starts with the Schrodinger Equation and applies the Minkowski transformation; the second uses the Canonical commutation relation; the third through Gabor’s analysis of the time-frequency plane and Heisenberg’s uncertainty principle; and lastly by quantizing Brownian motion within the Bernoulli process and applying the Gaussian channel capacity. In support I give two examples of quantum systems that follow the governing thought: namely the Gaussian wave packet and the electron spin. I conclude with comments on the discretization of space and the information content of a degree of freedom.
The Heisenberg uncertainty principle (Δx·Δp ≥ ℏ/2) has been interpreted for nearly a century as arising from the disturbance caused by measurement. We argue that this epistemic framing is unnecessary. We demonstrate that Δx·Δp ≥ ℏ/2 follows as a logical necessity from three independently observed facts: (1) position is quantized at the Planck length lₚ ≈ 1.616 × 10⁻³⁵ m; (2) time is quantized at the Planck time tₚ ≈ 5.391 × 10⁻⁴⁴ s; (3) mass-energy is quantized. Using the discrete Fourier transform (DFT), we show that a discrete Gaussian wave packet on a Planck lattice saturates the uncertainty bound at ℏ/2 exactly, with the factor of 1/2 emerging from the minimum-uncertainty property of the Gaussian distribution, which is preserved under discretization. Furthermore, we show that the canonical commutation relation [̂x, ̂p] = iℏ is the continuum limit of a discrete difference commutator, identifying the Schrödinger equation as an effective theory of an underlying Planck-scale discrete structure. Finally, we address the relationship of our result to Lorentz invariance, arguing that the Planck-scale bounds apply as observer-independent lower limits and are consistent with available evidence for Lorentz Invariance Violation (LIV) at the Planck scale.
We derive the equations of quantum mechanics and quantum thermodynamics from the assumption that a quantum system can be described by an underlying classical system of particles. Each component ϕj of the wave vector is understood as a stochastic complex variable whose real and imaginary parts are proportional to the coordinate and momentum associated to a degree of freedom of the underlying classical system. From the classical stochastic equations of motion, we derive a general equation for the covariance matrix of the wave vector which turns out to be of the Lindblad type. When the noise changes only the phase of ϕj, the Schrödinger and the quantum Liouville equation are obtained. The component ψj of the wave vector obeying the Schrödinger equation is related to stochastic wave vector by ∣ψj∣2=⟨∣ϕj∣2⟩.
We study information measures in quantum mechanics, with the particular emphasis on providing a quantification of the notion of predictability and classicality. Our primary tool is the Shannon-Wehrl entropy I. We give a precise criterion for phase space classicality of states and argue that in view of this (a) I provides a good measure for the degree of deviation from classicality in closed systems and (b) $I\ensuremath{-}S (S$ the von Neumann entropy) plays the same role in open quantum system. We examine particular examples in nonrelativistic quantum mechanics. Finally we generalize the discussion into the field theory case, and (this being one of our main motivations) we comment on the field classicalization in early universe cosmology.
Abstract In quantum mechanics, a classical particle is raised to a wave-function, thereby acquiring many more degrees of freedom. For instance, in the semi-classical regime, while the position and momentum expectation values follow the classical trajectory, the uncertainty of a wave-packet can evolve and beat independently. We use this insight to revisit the dynamics of a 1d particle in a time-dependent harmonic well. One can solve it by considering time reparameterizations and the Virasoro group action to map the system to the harmonic oscillator with constant frequency. We prove that identifying such a simplifying time variable is naturally solved by quantizing the system and looking at the evolution of the width of a Gaussian wave-packet. We further show that the Ermakov-Lewis invariant for the classical evolution in a time-dependent harmonic potential is actually the quantum uncertainty of a Gaussian wave-packet. This naturally extends the classical Ermakov-Lewis invariant to a constant of motion for quantum systems following Schrödinger equation. We conclude with a discussion of potential applications to quantum gravity and quantum cosmology.
Chapter 4 deals with Quantum Mechanical (QM) phenomena. The first section describes the first attempts at quantization of energy, momentum, angular momentum, etc. We start with Plank's solution to the black-body problem, introducing the concept of photons: discrete packets of electromagnetic energy. We show how Plank's description of black-body radiation is perfectly fit by the cosmic microwave radiation, providing the strongest evidence for the Big-Bang at the origin of the Universe. We then describe Bohr's solution to the paradoxical stability of atoms, introducing quantization of angular momentum (or action) and correctly predicting the energy levels of the hydrogen atom. We show how Einstein used Plank's idea to explain the photo-electric effect (an effect exploited in CCD cameras and solar cells) and how he used Bohr's understanding of energy levels in atoms to provide an intuitive explanation of Plank's law. Einstein insight opened the way to the field of spectroscopy and the development of the laser. In the second section, following the lead of Dirac, we show that QM is equivalent to Linear Algebra in the space of complex functions (which describe the state of the system). We show that Schrödinger's equation is a natural consequence of that description, once the relation between energy and frequency established by Plank (for photons) and de Broglie (for particles) is taken into account. In the context of the chiral ammonia molecule we discuss the queerness of QM which describes physical systems as superposition of states (such as left and right chirality state of a molecule, dead and alive cat). We show that this queerness is nonetheless supported by experiments and is behind much of our technology: masers and lasers, atomic clocks, semiconductors (diodes, transistors, etc.). In the following section we introduce the representation of operators in QM (the equivalent of matrices in linear algebra): momentum, position, energy, etc. which correspond to measurements on the system. We show that the non-commutability of some operators (common with matrix operations) leads to Heisenberg uncertainty principle: the position and momentum of a particle cannot be determined with absolute precision. We discuss the solution of Schrödinger's equation for free particles demonstrating their equivalence with EM phenomena: particles undergo refraction and diffraction, exemplifying the particle/wave duality inherent to QM systems. We discuss QM tunneling and its use in the Scanning Tunneling Microscope. In the following section we study Dirac's equation: the successful attempt by Dirac to unify QM with Einstein's relativity, with the resulting prediction of anti-particles and spin. We then solve for the energy levels of electrons in the hydrogen atom (from Schrödinger's equation) and show that the existence of spin profoundly affects the behavior of electrons in an atom: they cannot share the same energy level, a rule known as Pauli's exclusion principle. Pauli's principle is central to the understanding of the chemical properties of atoms and fully explains Mendeleev's periodic table of the elements. In the next section we discuss along with two exactly soluble systems (diatomic rotational and vibrational levels), various approximations used to study systems (e.g. molecules) for which there are no analytical solutions. In particular, we introduce Pauling's variational approach, Hückel molecular orbital theory and time independent perturbation theory. We apply those approaches to determine the energy levels in alkenes, in multi-electron atoms and in systems under constant electric or magnetic field (such as in Magnetic Resonance Imaging). In the last section we discuss time dependent perturbation theory. We use it to establish the rules allowing for transitions between energy levels and revisit Einstein description of absorption/emission phenomena.
We show that the main difference between classical and quantum systems can be understood in terms of information entropy. Classical systems can be considered the ones where the internal dynamics can be known with arbitrary precision while quantum systems can be considered the ones where the internal dynamics cannot be accessed at all. As information entropy can be used to characterize how much the state of the whole system identifies the state of its parts, classical systems can have arbitrarily small information entropy while quantum systems cannot. This provides insights that allow us to understand the analogies and differences between the two theories.