Factorization in normal order form provides positive energy solutions, while the alternative anti-normal order form provides negative energy solutions of the time-independent Schroedinger equation for a linear harmonic oscillator. The positive and negative energy state spaces are related by quantum conjugation effected through sign-reversal of Planck's quantization constant . � Photons occupying negative energy states have the same displacement xand Hamiltonian H, but opposite momentum p ˆ - compared to photons occupying the corresponding positive energy partner states. Emission of positive energy photons from a positive energy quantum state is equivalent to absorption of negative energy photons into a negative energy quantum state, leading to state lowering, while absorption of positive energy photons into a positive energy quantum state is equivalent to emission of negative energy photons from a negative energy quantum state, leading to state raising. The complete positive- negative energy spectrum of the quantized oscillator may then be
Factorization in normal order form provides positive energy solutions, while the alternative anti-normal order form provides negative energy solutions of the time-independent Schroedinger equation for a linear harmonic oscillator. The positive and negative energy state spaces are related by quantum conjugation effected through sign-reversal of Planck's quantization constant . � Photons occupying negative energy states have the same displacement xand Hamiltonian H, but opposite momentum p ˆ - compared to photons occupying the corresponding positive energy partner states. Emission of positive energy photons from a positive energy quantum state is equivalent to absorption of negative energy photons into a negative energy quantum state, leading to state lowering, while absorption of positive energy photons into a positive energy quantum state is equivalent to emission of negative energy photons from a negative energy quantum state, leading to state raising. The complete positive- negative energy spectrum of the quantized oscillator may then be
The Hamiltonian of harmonic oscillator is symmetric under the replacement of canonically conjugate variables and a canonical transformation to ladder operators maintains this symmetry.The Hamiltonian is expressed through a symmetrized product of the ladder operators and, as a result, at quantization there arise a zeropoint energy.Therefore, for quantized fields, canonically conjugate variables of which enter into the Hamiltonian unsymmetrically, the zero-point energy could not arise.A new symmetry of harmonic oscillator is found: a wave equation and its solutions do not vary at a joint changing of signs of frequency, energy and mass of a particle.It is shown that the problem of the negative norm for negative-frequency states appears at taking positive mass at negative energy and, contrary, the problem disappears at taking the same sign of mass and energy as it is required by relativistic kinematics.In the nonrelativistic theory, considered as a limiting case of relativistic theory, the states of a particle with negative frequency, energy and mass are described consistently as evolving only backward in time and representing the states of its antiparticle with positive frequency, energy and mass evolving forward in time.For such charge-conjugation symmetric system of oscillators an extended space of states with generalized operators is constructed.
The Weber-Hermite differential equation, obtained as the dimensionless form of the stationary Schroedinger equation for a linear harmonic oscillator in quantum mechanics, has been expressed in a generalized form through introduction of a constant conjugation parameter according to the transformation , where the conjugation parameter is set to unity () at the end of the evaluations. Factorization in normal order form yields composite eigenfunctions, Hermite polynomials and corresponding positive eigenvalues, while factorization in the anti-normal order form yields the partner composite anti-eigenfunctions, anti-Hermite polynomials and negative eigenvalues. The two sets of solutions are related by an reversal conjugation rule . Setting provides the standard Hermite polynomials and their partner anti-Hermite polynomials. The anti-Hermite polynomials satisfy a new differential equation, which is interpreted as the conjugate of the standard Hermite differential equation.
We formulate a second-order organization of quantum dynamics in which physical energy is identified with the magnitude of the temporal oscillation frequency, whereas the algebraic sign of the frequency specifies temporal phase orientation rather than the sign of energy. For any time-independent self-adjoint first-order evolution generator G^, the associated second-order parent wave equation naturally defines a nonnegative physical-energy operator through the spectral modulus E^pe=∣G^∣=G^2. Through the polar decomposition G^=S^E^pe, this construction decouples the nonnegative physical-energy magnitude from the signed spectrum of the first-order generator. Applied to the free Dirac equation, the Dirac Hamiltonian H^D retains its conventional unbounded spectrum (−∞,−mc2]∪[mc2,∞), whereas the physical-energy spectrum is bounded below by the rest energy, σ(E^pe)=[mc2,∞). For generators with a spectral gap around zero, the second-order dynamics admits a positive-definite conserved wave norm and an equivalent two-component unitary evolution. We further prove a positive-frequency spectral completeness theorem in L2(R) and show that the temporal Hilbert transform Ht provides a natural real-linear operator realization of the modulus mapping via ℏHt∂t↔ℏ∣Ω∣. Direct numerical diagonalization and high-precision time-domain integration of the one-dimensional Dirac equation confirm both the spectral modulus mapping and the exact conservation of quadratic wave energy across opposite temporal orientations. While this kinematic framework places quantum energy on the same nonnegative footing as classical field and signal energies, promoting the two temporal orientations to full particle and antiparticle sectors requires a second-quantized field theory incorporating charge conjugation, spin-statistics, locality, and interactions.
An earlier study of quantum-mechanical time has suggested that, from the quantum-theoretic point of view, time is a random variable. The present work explores how such a description of time can be used to bring greater detail into the analysis of the one-dimensional harmonic oscillator. Specifically, the classical equations for harmonic motion are used as a point transformation between phase space and time-energy space, permitting probability distributions in the two spaces to be different representations of the same thing. The familiar energy-quantization hypothesis is supplemented with the hypothesis that quantized energy values are means (averages) of energy distributions characteristic of each energy state. With the probability distribution of time determined from the previously cited study, the desired quantized energy distributions follow from the requirements: (1) that they should be normalized, non-negative, and with means equal to the quantized energy values; (2) that the equivalent representation of distribution in position-momentum space should have marginal densities that are proportional (except near the origin) to the densities in position and momentum, ${|\ensuremath{\phi}(q)|}^{2}$ and ${|\ensuremath{\psi}(p)|}^{2}$, which are postulated in quantum theory. Requirement (2) is a weakening of the Born postulate. In the time-energy space, the probability distributions exclude energies smaller than the mode (energy value with maximum probability). In position-momentum space this exclusion requirement becomes the dynamical restriction: $\frac{{q}^{2}m{\ensuremath{\omega}}^{2}}{2}+\frac{{p}^{2}}{2m}\ensuremath{\ge}\frac{{\ensuremath{\Lambda}}_{n}\ensuremath{\hbar}\ensuremath{\omega}}{2}$, where, for the first three energy states, ${\ensuremath{\Lambda}}_{0}=0$, ${\ensuremath{\Lambda}}_{1}=\frac{3}{2}$, ${\ensuremath{\Lambda}}_{2}=2+\ensuremath{\surd}\frac{3}{2}$. In general terms it is thus shown that the quantized harmonic oscillator can be described as a statistical ensemble of classical oscillators. Some comparisons with work on phase-space formulations of quantum mechanics (especially with the work of Wigner) are made and some aspects of the theory that should submit to experimental verification are pointed out.
The first application of quantum ideas to a system with a nonuniform potential energy is to the harmonic oscillator: a pendulum, or a mass on a spring. The result is a quantization of energy with equally spaced energy levels. The wave function is localized in the potential well but can do quantum tunneling beyond the turning points into classically forbidden regions. A wave function can be a negative quantity, but the related position probability is always positive. The wave function shows the particle is not in any single place; in a sense it is in many places simultaneously. The fact that the lowest energy quantum state does not vanish is an example of zero-point energy. States of very high energy show a strong analogy to classical oscillator behavior, an example of the Bohr correspondence principle.
The simple harmonic oscillator has a well-known normalizable, positive energy, bound state spectrum. We show that degenerate with each such positive energy eigenvalue there is a non-normalizable positive energy eigenstate whose eigenfunction is orthogonal to that of the standard energy eigenfunction. This class of states is not built on the vacuum that a annihilates, but is instead built on the vacuum that a†a annihilates. These non-normalizable but nonetheless stationary energy eigenstates are accompanied by yet another set of non-normalizable states, states whose wave functions however are not stationary but instead grow linearly in time. With these states not being energy eigenstates, the eigenbasis of the Hamiltonian is incomplete; with the full Hilbert space containing states that are not energy eigenstates. Thus each energy eigenvalue of the harmonic oscillator is an exceptional point at which the Hamiltonian becomes of non-diagonalizable, and thus manifestly non-Hermitian, Jordan-block form. Such non-Hermitian structures occur for Hamiltonians that have an antilinear PT symmetry. As is characteristic of such systems, one can construct a probability conserving inner product that despite the linear in time growth is nonetheless time independent, and not only that, it leads to states with zero norm. In addition, as is again characteristic of PT symmetry, these non-normalizable states can be made normalizable by a continuation into a so-called Stokes wedge domain in the complex plane. In this domain one has a completely consistent quantum theory, one that lives alongside the standard normalizable energy eigenspectrum sector. This thus not quite so simple harmonic oscillator provides an explicit realization of our general contention that antilinearity is more basic to quantum theory than Hermiticity.