This is a short review of the main results on the oscillators considered basic models in quantum mechanics.Since the potential is polynomial, it is possible to extend the stationary states to entire functions on the complex plane where the semiclassical theory works better.The control on the energy levels is based on the isolation of the nodes, the zeros stable at the unperturbed limit.A new and simple model for the process of racemization of chiral molecules is added.
This is a short review of the main results on the oscillators considered basic models in quantum mechanics.Since the potential is polynomial, it is possible to extend the stationary states to entire functions on the complex plane where the semiclassical theory works better.The control on the energy levels is based on the isolation of the nodes, the zeros stable at the unperturbed limit.A new and simple model for the process of racemization of chiral molecules is added.
In classical mechanics the harmonic oscillator (HO) provides the generic example for the use of angle and action variables and I > 0 which played a prominent role in the “old” Bohr-Sommerfeld quantum theory. However, already classically there is a problem which has essential implications for the quantum mechanics of the (φ,I)-model for the HO: the transformation is only locally symplectic and singular for (q,p) = (0,0). Globally the phase space {(q,p)} has the topological structure of the plane ℝ2, whereas the phase space {(φ,I)} corresponds globally to the punctured plane ℝ2 -(0,0) or to a simple cone with the tip deleted. From the properties of the symplectic transformations on that phase space one can derive the functions h0 = I, h1 = Icos φ and h2 = -Isin φ as the basic coordinates on {(φ,I)}, where their Poisson brackets obey the Lie algebra of the symplectic group of the plane. This implies a qualitative difference as to the quantum theory of the phase space {(φ,I)} compared to the usual one for {(q,p)}: In the quantum mechanics for the (φ,I)-model of the HO the three hj correspond to the self-adjoint generators Kj, j = 0,1,2, of certain irreducible unitary representations of the symplectic group or one of its infinitely many covering groups, the representations being parametrized by a (Bargmann) index k > 0. This index k determines the ground state energy of the (φ,I)-Hamiltonian . For an m-fold covering the lowest possible value for k is k = 1/m, which can be made arbitrarily small by choosing m accordingly! This is not in contradiction to the usual approach in terms of the operators Q and P which are now expressed as functions of the Kj, but keep their usual properties. The richer structure of the Kj quantum model of the HO is “erased” when passing to the simpler (Q,P)-model! This more refined approach to the quantum theory of the HO implies many experimental tests: Mulliken-type experiments for isotopic diatomic molecules, experiments with harmonic traps for atoms, ions and BE-condensates, with charged HOs in external electric fields and the (Landau) levels of charged particles in external magnetic fields, with the propagation of light in vacuum, passing through strong external electric or magnetic fields. Finally it may lead to a new theoretical estimate for the quantum vacuum energy of fields and its relation to the cosmological constant.
Abstract In classical mechanics the harmonic oscillator (HO) provides the generic example for the use of angle and action variables and I > 0 which played a prominent role in the “old” Bohr‐Sommerfeld quantum theory. However, already classically there is a problem which has essential implications for the quantum mechanics of the (φ,I)‐model for the HO: the transformation is only locally symplectic and singular for (q,p) = (0,0). Globally the phase space {(q,p)} has the topological structure of the plane ℝ2, whereas the phase space {(φ,I)} corresponds globally to the punctured plane ℝ2 ‐(0,0) or to a simple cone with the tip deleted. From the properties of the symplectic transformations on that phase space one can derive the functions h0 = I, h1 = Icos φ and h2 = ‐Isin φ as the basic coordinates on {(φ,I)}, where their Poisson brackets obey the Lie algebra of the symplectic group of the plane. This implies a qualitative difference as to the quantum theory of the phase space {(φ,I)} compared to the usual one for {(q,p)}: In the quantum mechanics for the (φ,I)‐model of the HO the three hj correspond to the self‐adjoint generators Kj, j = 0,1,2, of certain irreducible unitary representations of the symplectic group or one of its infinitely many covering groups, the representations being parametrized by a (Bargmann) index k > 0. This index k determines the ground state energy of the (φ,I)‐Hamiltonian . For an m‐fold covering the lowest possible value for k is k = 1/m, which can be made arbitrarily small by choosing m accordingly! This is not in contradiction to the usual approach in terms of the operators Q and P which are now expressed as functions of the Kj, but keep their usual properties. The richer structure of the Kj quantum model of the HO is “erased” when passing to the simpler (Q,P)‐model! This more refined approach to the quantum theory of the HO implies many experimental tests: Mulliken‐type experiments for isotopic diatomic molecules, experiments with harmonic traps for atoms, ions and BE‐condensates, with charged HOs in external electric fields and the (Landau) levels of charged particles in external magnetic fields, with the propagation of light in vacuum, passing through strong external electric or magnetic fields. Finally it may lead to a new theoretical estimate for the quantum vacuum energy of fields and its relation to the cosmological constant.
Abstract ................................................................................................. 164 3.1 Introduction .................................................................................. 165 3.2 The Wave-Function Continuity .................................................... 169 3.2.1 Quantum Tunneling and the Gamow Factor .................... 169 3.2.2 Nuclear Systems’ Lifetime by Alfa Disintegration .......... 176 3.3 Eigen-Energies and Eigen-Functions ........................................... 181 3.3.1 Atomic Hydrogenic States by Solving Laguerre Type Equations ................................................. 182 3.3.2 Molecular Vibrational States by Solving Hermite Type Equations .................................................. 200 3.3.3 Solid State Free Electronic States .................................... 209 3.4 Semiclassical Quantification of Energy ....................................... 216 3.4.1 Hydrogenic Atomic Systems ........................................... 219 3.4.2 Molecular Vibrational Systems ........................................ 222 3.4.3 Generalized |x|α Potential Systems ................................... 223 3.5 Variational Wave-Function and Energies ..................................... 228 3.5.1 Hydrogen’s Quantum Ground Level ................................ 228 3.5.2 Vibrational Ground Level ................................................ 231 3.5.3 Ground State Paradox of Free Electrons in Solids .......... 232 3.6 Causal Quantum Evolution .......................................................... 236 3.6.1 Schrödinger’s Picture ....................................................... 236 3.6.2 Unitary (“U”) Picture. Heisenberg Picture ...................... 246 3.6.3 Quantum Transitions: Interaction Picture ........................ 254 3.6.4 Green Function and Its Causal Properties ........................ 262 3.6.5 Free Particle’s Propagator ................................................ 271 3.7 Stationary Perturbations ............................................................... 274 3.7.1 General Perturbation Algorithm ....................................... 275 3.7.2 The Zeeman Effect ........................................................... 281 3.7.3 Nuclear Isotropic Corrections on Hydrogenic Atoms ...... 285 3.7.4 Harmonic Oscillator Perturbations .................................. 292 3.7.5 Quasi-Free Electronic Model of Solids ........................... 300 3.8 Quantum Measurements .............................................................. 304 3.8.1 Classical vs. Quantum Electric Resistance ...................... 306 3.8.2 Quantum Conservation Laws ............................................311 3.8.3 Quantum Elastic Scattering on Fixed Target ................... 318 3.8.4 More about Scattering: Born Approximation and Resonance Profile ..................................................... 332 3.8.5 From Yukawa Potential to Rutherford Scattering ............ 342 3.9 Conclusion ................................................................................... 353 Keywords .............................................................................................. 354 References ............................................................................................. 355 Author’s Main References ........................................................... 355 Specific References ...................................................................... 355 Further Readings .......................................................................... 355 ABSTRACT The phenomenological quantum mechanics is practically “re-storied” in a more formalized way, under the so-called extended quantum postulates, while opening and unfolding more complex analysis of the Nature’s phenomena, from nuclear, to atomic, to molecular, and to solid state and scattering observational quantum effects.
We study in the framework of the Schrodinger equation the effect of intermolecular interactions on the tunneling racemization of the active molecule. The active molecule is assumed as a two-level system and the left-right isomerism is viewed in terms of a double-bottomed harmonic potential well. The active molecule is assumed to be embedded in a gas, liquid or solid, submitted to a perturbing potential U created by the molecules of the sample. In our model we take into account the difference of energy E due to the weak interactions between the left (L) and right (R) configurations. We have shown that when E is equal to zero the system cannot be optically stable: the optical activity tends asymptotically to zero in the case of dilute gases or compressed gases and liquids or oscillates periodically around zero when the molecules are isolated or submitted to a static potential. Only when E is different of zero the system can be optically stable depending on the strength parameters of the potential U and on the magnitude of the spontaneous tunneling.