##### Latest Research on Schrödinger Equation: Oct – 2019

**Fractional Schrödinger equation**

Some properties of the third Schrödinger equation ar studied. we tend to prove the Hermiticity of the third Hamilton operator and establish the parity conservation law for third quantum physics. As physical applications of the third Schrödinger equation we discover the energy spectra of a hydrogenlike atom (fractional “Bohr atom”) and of a third generator within the semiclassical approximation. AN equation for the third chance current density is developed and mentioned. we tend to conjointly discuss the relationships between the third and customary Schrödinger equations. **[1]**

**The recursive solution of the Schrödinger equation**

A new approach to the procedure answer of the Schrödinger equation relies on the partial transformation of the Hamiltonian to a tridiagonal matrix. the strategy is very suited to tight-binding Hamiltonians encountered in solid state physics and permits of the order of 104 degrees of freedom to be enclosed in an exceedingly calculation. freelance particle inexperienced functions are calculated naturally from the part tridiagonalized Hamiltonian. These cause easy computation of little energy variations, binding energies, transition matrix-elements and alternative helpful quantities. **[2]**

**A random‐walk simulation of the Schrödinger equation: H+3**

A simple random‐walk methodology for getting at the start solutions of the Schrödinger equation is examined in its application to the case of the molecular particle H+3 within the equiangular triangle configuration with facet length R=1.66 bohr. The method, that relies on the similarity of the Schrödinger equation and also the diffusion equation, involves the random movement of notional particles (psips) in lepton configuration area subject to a variable likelihood of multiplication or disappearance. The computation necessities for top accuracy in determinant energies of H+3 square measure bigger than those of existing LCAO–MO–SCF–CI strategies. For a lot of complicated molecular systems the tactic is also competitive. **[3]**

**A fast and adaptable method for high accuracy integration of the time-dependent Schrödinger equation**

We gift associate pliant, fast, and sturdy methodology for integration the time-dependent Schrödinger equation. we tend to apply the tactic to calculations of High Harmonic (HHG) and higher than Threshold Ionisation (ATI) spectra for one atomic lepton in an intense optical maser field. Our approach implements the stabilised bi-conjugate gradient methodology (BiCG-STAB) for determination a thin linear system to evolve the electronic wavefunction in time. the utilization of this established methodology makes the propagation theme less restrictive compared to alternative schemes which can have explicit needs for the shape of the equation, like use of a three-point finite-difference approximation for spatial derivatives. **[4]**

**Arbitrary l-state Solution of the Schrödinger Equation for q-deformed Attractive Radial Plus Coulomb-like Molecular Potential within the Framework of NU-Method**

The Schrödinger equation in one dimension for the q-deformed engaging radial and coulomb-like molecular potential (ARCMP) is solved more or less to get sure states chemist solutions exploitation the constant quantity Nikiforov-Uvarov (NU) methodology. The corresponding unnormalized chemist functions ar evaluated in terms of Jacobi polynomials. curiously, the ensuing chemist energy equations will be accustomed study the spectrometry of some hand-picked matter atoms and molecules. **[5]**

**Reference**

**[1]** Laskin, N., 2002. Fractional schrödinger equation. Physical Review E, 66(5), (Web Link)

**[2]** Haydock, R., 1980. The recursive solution of the Schrödinger equation. Computer Physics Communications, 20(1), (Web Link)

**[3]** Anderson, J.B., 1975. A random‐walk simulation of the Schrödinger equation: H+ 3. The Journal of Chemical Physics, 63(4), (Web Link)

**[4]** A fast and adaptable method for high accuracy integration of the time-dependent Schrödinger equation

Daniel Wells & Harry Quiney

Scientific Reports volume 9, Article number: 782 (2019) (Web Link)

**[5]** Iserom Ita, B., Hitler, L., Joseph, I., Alexander, I., Isa Amos, P. and Thomas Odey, M. (2017) “Arbitrary l-state Solution of the Schrödinger Equation for q-deformed Attractive Radial Plus Coulomb-like Molecular Potential within the Framework of NU-Method”, Physical Science International Journal, 16(4), (Web Link)