Geometric Proof of Exact Quantization Rules in One Dimensional Quantum Mechanics

In terms of the construction of vector field with momentum and logarithmic derivative of wavefunction as its components, a geometric proof of an exact quantization rule in one dimensional quantum mechanics systems is given. The quantization rule arises from the SO(2) gauge transformation. In addition, the quantization rule is generalized to the case when the potential function is piecewise continuous between the two turning points. This work was supported by doctoral foundation of HPU.     

Quantum adiabatic approximation to interacting boson-fermion systems

We present an adiabatic approximation method for the path integral of the Fermi field in the presence of a Bose field. The adiabatic phenomenon recently found by Berry and Simon is used for evaluating the Grassman path integral. Then we obtain the path integral of the effective action analogous to a magnetic field, and the quantization rule is derived by applying the semiclassical quantization method.     

The quantization of Fermi systems and the Dirac bracket

This work is devoted (a) to discussing some problems related to the quantization rule for constrained classical models of Fermi type, and (b) to working with some detail a specific model which is a classical analogue of the quantum Fermi systems. The quantization of this model is shown to depend on the addition of a total time derivative to the corresponding Lagrangian.     

Hydrogen-like atom with nonnegative quantum distribution function

Among numerous approaches to probabilistic interpretation of conventional quantum mechanics (CQM), the closest to N. Bohr’s idea of the correspondence principle is the Blokhintzev-Terletsky approach of the quantum distribution function (QDF) on the coordinate-momentum (q, p) phase space. The detailed investigation of this approach has led to the correspondence rule of V.V. Kuryshkin parametrically dependent on a set of auxiliary functions. According to investigations of numerous authors, the existence and the explicit form of QDF depends on the correspondence rule between classical functions A(q, p) and quantum operator A. At the same time, the QDF corresponding to all known quantization rules turns out to be alternating in sign or overly complex valued. Finally nonexistence of nonnegative QDF in CQM was proved. On the other hand, from this follows the possibility to construct quantum mechanics where a nonnegative QDF exists. We consider a certain set of auxiliary functions to construct explicit expressions for operators O(H) for the hydrogen atom. Naturally, these operators differ from the related operator Ĥ in CQM, so that spherical coordinates are no longer separable for a hydrogen-like atom in quantum mechanics with nonnegative QDF. The text was submitted by the authors in English.     

On the superfluidity of classical liquid in nanotubes,II. Case of odd number of neutrons

We show that the superfluidity effect in nanotubes arises in a classical liquid (regarded as the limit as h → 0 of the quantum liquid) and involves not only the Bogolyubov “running waves,” but also a “standing wave.” This is obtained from the variational equations in the context of ultrasecondary quantization. We consider cases of Bose and Fermi liquids.     

Dirac Superfield Quantization in Odd Superspace

We generalize the superfield method of Dirac quantization to the odd sector of superspace for N = 2 extended models. We discuss the mechanism of generating constraints in the odd sector of supersymmetric classical mechanics and then Dirac quantization in the odd superspace in superfield version. An example of supersymmetric system, defined by odd superfields, and the application of the method in the odd superspace is given.     

Quantum dynamics of theSU(2)-Skyrme model in Nelson stochastic mechanics

In the framework of Nelson stochastic mechanics the SkyrmeSU(2) model is quantized. A new term is added to a classical skyrmion mass. It coincides with the term obtained by Fujiiet al. by modifying the canonical quantization. This example illustrates that stochastic mechanics as an alternative method of quantization is convenient for theories with collective coordinates and for nonlinear theories, as some problems related to operator ordering and modification of canonical formalism are naturally solved.     

Weyl-Wigner-Moyal formalism for Fermi classical systems

The Weyl-Wigner-Moyal formalism of fermionic classical systems with a finite number of degrees of freedom is considered. The Weyl correspondence is studied by computing the relevant Stratonovich-Weyl quantizer. The Moyal -product, Wigner functions and normal ordering are obtained for generic fermionic systems. Finally, this formalism is used to perform the deformation quantization of the Fermi oscillator and the supersymmetric quantum mechanics.     

Departures from the Fermi Golden Rule

This paper shows that exact calculation for the transition probability in quantum mechanics gives rise to breaking the Fermi golden rule, and energy conservation, and gives two examples.     

Quantization by parts,self-adjoint extensions,and a novel derivation of the Josephson equation in superconductivity

There has been a lot of interest in generalizing orthodox quantum mechanics to include POV measures as observables, namely as unsharp obserrables. Such POV measures are related to symmetric operators. We have argued recently that only maximal symmetric operators should describe observables.¹ This generalization to maximal symmetric operators has many physical applications. One application is in the area of quantization. We shall discuss a scheme, to he called quantization by parts,which can systematically deal with what may be called quantum circuits. As a specific application we shall present a novel derivation of the famous Josephson equation for the supercurrent through a Josephson junction in a superconducting circuit. An interesting effect emerges from our quantization scheme when applied to a superconducting Y-shape circuit configuration. We also propose an experimental test for this effect which is expected to shed light on some conceptual problems on the quantum nature of the condensate.     

Effective potentials and the onset of quantization in ultrasmall MOSFETs

The connection between the minimum size of an electron wavepacket, and the introduction of an effective potential is discussed. The effective potential approach has a long history of use in trying to transition the gap between classical mechanics and quantum mechanics. An effective potential is one in which the quasi-classical regime is approximated through a density which arises from the effective potential W(x) through exp[ − βW(x)]. The generation of the effective potentialW (x) gives the effects of the onset of quantization in the system. In this paper, we study the use of the effective potential in a triangular well formed between the oxide and the depletion field of the semiconductor. We determine the quantization energy of the carriers in the potential well and their mean set-back from the interface. Finally, we show the connection between the effective potential and the Bohm-derived quantum potentials that have become of interest in simulations.     

Influence of magnetic quantization on the effective mass in semiconductor superlattices

An attempt is made to study the effect of a quantizing magnetic field on the effective electron mass in a semiconductor superlattice at low temperatures. It is found on the basis of the tight-binding approximation, taking GaAs-Ga_1–x Al _x As an example, that the effective mass at the Fermi level depends on the magnetic quantum number due to the cosine dependence of the wave-vector in the superlattice direction. The mass also exhibits oscillatory features in the presence of magnetic quantization because of its dependence on Fermi energy which oscillates with changing magnetic field.     

Quantization of the Sphere with Coherent States

Current views link quantization with dynamics. The reason is that quantum mechanics or quantum field theories address to dynamical systems, i.e., particles or fields. Our point of view here breaks the link between quantization and dynamics: any (classical) physical system can be quantized. Only dynamical systems lead to dynamical quantum theories, which appear to result from the quantization of symplectic structures.     

Effect of size quantization on the effective mass in ultrathin films ofn-Cd3As2

An attempt is made to investigate the effects of size quantization on the effective mass in ultrathin films ofn-Cd₃As₂. It is found that the effective mass at the Fermi level depends on the size quantum number due to the effect of crystal-field splitting, resulting in different effective masses at the Fermi level corresponding to different electric subbands. It is also observed that the different effective masses closely approach each other, for a given film thickness, with increasing electron concentration and, for a given electron concentration, with increasing film thickness.     

Quantum mechanics on Riemannian manifold in Schwinger's quantization approach III

Using the extended Schwinger quantization approach, quantum mechanics on a Riemannian manifold M with the given action of an intransitive group of isometries is developed. It was shown that quantum mechanics can be determined unequivocally only on submanifolds of M where G acts simply transitively (orbits of G action). The remaining part of the degrees of freedom can be described unequivocally after introducing some additional assumptions. Being logically unmotivated, these assumptions are similar to the canonical quantization postulates. Besides this ambiguity which is of a geometrical nature there is an undetermined gauge field of the order of (or higher), vanishing in the classical limit . Received: 19 February 2001 / Revised version: 10 May 2001 / Published online: 6 July 2001     

Fermi substructure of space-time

If space-time possesses a Fermi substructure, then the canonical quantization of the space-time and the Fermi coordinates of a relativistic point particle must be mutually consistent. We show that the Fermi substructure meets this requirement. We express the generators of the Lorentz group in terms of the Fermi coordinates and momenta and consider their coordinate representation.     

The Angular Momentum Dilemma and Born–Jordan Quantization

The rigorous equivalence of the Schrödinger and Heisenberg pictures requires that one uses Born–Jordan quantization in place of Weyl quantization. We confirm this by showing that the much discussed “ angular momentum dilemma” disappears if one uses Born–Jordan quantization. We argue that the latter is the only physically correct quantization procedure. We also briefly discuss a possible redefinition of phase space quantum mechanics, where the usual Wigner distribution has to be replaced with a new quasi-distribution associated with Born–Jordan quantization, and which has proven to be successful in time-frequency analysis.     

The New Type of Extended Uncertainty Principle and Some Applications in Deformed Quantum Mechanics

In this paper we present a new type of extended uncertainty principle (EUP) of the form [X, P] = i(1 − q|X|) and show that it has the non-zero minimal momentum. For this EUP we discuss the classical mechanics in the curved space, deformed calculus, deformed quantum mechanics and Bohr-Sommerfeld quantization.