Heisenberg algebra for noncommutative Landau problem

The Landau problem on non-commutative quantum mechanics is studied, where the Heisenberg algebra and the Landau energy levels as well as the non-commutative angular momentum are constructed in detail in non-commutative space and non-commutative phase space respectively.

Wigner Function for Klein-Gordon Landau Problem

First we calculate the Wigner phase-space distribution function for the Klein-Gordan Landau problem ona commmutative space.Then we study the modifications introduced by the coordinate-coordinate noncommuting andmomentum-momentum noncommuting, namely, by using a generalized Bopp’s shift method we construct the Wignerfunction for the Klein-Gordan Landau problem both on a noncommutative space (NCS) and a noncommutative phasespace (NCPS).

The Noncommutative Landau Problem in Podolsky’s Generalized Electrodynamics

The Landau problem in Podolsky＇s generalized electrodynamics is studied by the method of diagonalization in noncommutative phase space and we find that the different noncommutative effects for a certain system led by the nonuniqueness of generalized Bopp shift can be avoided. The exact energy eigenvalues are found, and the result shows that the energy spectra are generically non-degenerate, fhrthermore, we obtain the special energy spectra of noncommutative space and commutative space.

On Prime Numbers between kn and (k + 1) n

In this paper along with the previous studies on analyzing the binomial coefficients, we will complete the proof of a theorem. The theorem states that for two positive integers n and k, when n ≥ k - 1, there always exists at least a prime number p such that kn p ≤ (k +1)n. The Bertrand-Chebyshev’s theorem is a special case of this theorem when k = 1. In the field of prime number distribution, just as the prime number theorem provides the approximate number of prime numbers relative to natural numbers, while the new theory indicates that prime numbers exist in the specific intervals between natural numbers, that is, the new theorem provides the approximate positions of prime numbers among natural numbers.

Relativistic Oscillators in a Noncommutative Space and in a Magnetic Field

In this work, we study the relativistic oscillators in a noncommutative space and in a magnetic field. It is shown that the effect of the magnetic field may compete with that of the noncommutative space and that is able to vanish the effect of the noncommutative space.

Gauge Invariance, the Quantum Metric Tensor and the Quantum Fidelity

The quantum metric tensor was introduced for defining the distance in the parameter space of a system. However, it is also useful for other purposes, like predicting quantum phase transitions. Due to the physical information this tensor provides, its gauge independence sounds reasonable. Moreover, its original construction was made by looking for this gauge independence. The aim of this paper, however, is to prove that the quantum metric tensor does depend on the gauge. In addition, a real gauge invariant quantum metric tensor is introduced. A related concept is the quantum fidelity, which is also shown to depend on the gauge in this paper. The gauge dependences are explicitly shown by computing the quantum metric tensor and the quantum fidelity of the Landau problem in different gauges. Then, a real gauge independent metric tensor is proposed and computed for the same Landau problem. Since the gauge dependences have not been observed before, the results of this paper might lead to a new study of topics that are believed to be completely understood.

Super W_(1+∞) n-Algebra in the Supersymmetric Landau Problem

We analyze the super n-bracket built from associative operator products.Since the super n-bracket with n even satisfies the so-called generalized super Jacobi identity,we deal with the n odd case and give the generalized super Bremner identity.For the infinite conserved operators in the supersymmetric Landau problem,we derive the super W_(1+∞) n-algebra which satisfies the generalized super Jacobi and Bremner identities for the n even and odd cases,respectively.Moreover the super W_(1+∞) sub-2n-algebra is also given.

Klein-Gordon oscillators in noncommutative phase space

We study the Klein-Gordon oscillators in non-commutative （NC） phase space. We find that the Klein-Gordon oscillators in NC space and NC phase-space have a similar behaviour to the dynamics of a particle in commutative space moving in a uniform magnetic field. By solving the Klein-Gordon equation in NC phase space, we obtain the energy levels of the Klein-Gordon oscillators, where the additional terms related to the space-space and momentum-momentum non-commutativity are given explicitly.

Spin-1/2 relativistic particle in a magnetic field in NC phase space

This work provides an accurate study of the spin-l/2 relativistic particle in a magnetic field in NC phase space. By detailed calculation we find that the Dirac equation of the relativistic particle in a magnetic field in noncommutative space has similar behaviour to what happens in the Landau problem in commutative space even if an exact map does not exist. By solving the Dirac equation in NC phase space, we not only obtain the energy level of the spin-1/2 relativistic particle in a magnetic field in NC phase space but also explicitly offer some additional terms related to the momentum-momentum non-commutativity.