On the other hand, even approximate analytical solution like variational one is much more desired as it can represent the useful orthogonal set, which can be further employed for the solutions of the problems, not obligatory dealing with fractional quantum mechanics."Michelle’s history and past experience as a true technology leader speaks for itself," said John Riccitiello, President and Chief Executive Officer, Unity.
We note here, that if for the states with \(l=0\), the function R( k) has a maximum at \(k=0\), the corresponding function \(\chi \) is zero by obvious reason.Īs we mentioned above, the Eq. 11) permits to avoid divergencies at \(k=0\).
#K 3d nd unity trial#
This form will be used below to construct trial functions for variational method as well as for numerical calculations, where the form (Eq. Here \(\Gamma (x)\) is \(\Gamma \)-function 14, \(\nabla \) is (also n- dimensional) gradient operator and \(U(\mathbf(k)\) Which at \(\mu =2\) yields the ordinary one 6, 7. In dimensionless units (diffusion coefficient and particle mass are set to unity) it reads
#K 3d nd unity pdf#
The pdf of a typical Lévy flight is usually determined by the fractional Fokker–Planck (FP) Eq. In the infinite space, it is profitable to define the corresponding pdf in terms of its characteristic function, i.e. The most prominent example here is so-called Lévy flights 8, 9, 10, 11, 12, 13, which are Markovian random processes whose probability density functions (pdf) are Lévy stable laws, characterized by the Lévy index \(0<\mu \le 2\). In other words, these distributions decay at infinities (sometimes much) slower, that Gaussian one. 6, 7) that strong disorder is well described by Non-Gaussian probability distributions, having so-called long tails. This is the case for so-called multiferroics, where ferroelectric and magnetic orders coexist 5. The above disorder influences phonon and electron spectra of a substance, leading to the distribution of the internal magnetic, electric and elastic fields. Disordered semiconductors and dielectrics like organometallic halide perovskites 1, 2 are widely used in photovoltaic cells, light-emitting devices, and nanolasers 3, 4. Specifically, by varying the type and concentration of different imperfections (like point or extended defects), we can customize the characteristics of such material to meet specific requirements, needed, for instance, in electronics. These features can have a strong impact on the physical properties of many solids, ranging from multiferroics to oxide heterostructures, which, in turn, are usable in modern microelectronic devices.Īlthough disorder (especially strong like substance amorphization) in a solid is usually considered to be a trouble for its possible device applications, its constructive properties have become increasingly clear in recent years as they give an additional possibility to fine tune their physical properties. Combining analytical (variational) and numerical methods, we have shown that in the fractional (disordered) 3D oscillator problem, the famous orbital momentum degeneracy is lifted so that its energy starts to depend on orbital quantum number l. that without fractional derivatives) 3D quantum harmonic oscillator. In this case, \(\mu \rightarrow 0\) corresponds to the strongest disorder, while \(\mu \rightarrow 2\) to the weakest so that the case \(\mu =2\) corresponds to “ordinary” (i.e. To solve the obtained 3D spectral problem, we pass to the momentum space, where the problem simplifies greatly as fractional Laplacian becomes simply \(k^\mu \), k is a modulus of the momentum vector and \(\mu \) is Lévy index, characterizing the degree of disorder. To be specific, this is accomplished by the introduction of a so-called fractional Laplacian (Riesz fractional derivative) to the Scrödinger equation with three-dimensional (3D) quadratic potential. This disorder may be described phenomenologically by a fractional generalization of ordinary quantum-mechanical oscillator problem. We study the role of disorder in the vibration spectra of molecules and atoms in solids.