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Register a new userSchrodinger Equation with a Non-Central Potential: Some Statistical Quantities
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In this paper, we search the dependence of some statistical quantities such as the free energy, the mean energy, the entropy, and the specific heat for the Schrodinger equation on the temperature, particularly the case of a non-central potential. The basic point is to find the partition function which is obtained by a method based on the Euler-Maclaurin formula. At first, we present the analytical results by supporting with some plots for the thermal functions for one- and three-dimensional cases to find out the effect of the angular momentum. We also search then the effect of the angle-dependent part of the non-central potential. We discuss the results briefly for a phase transition for the system. We also present our results for three-dimesional harmonic oscillator.
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The aim of this paper is to study, in dimensions 2 and 3, the pure-power non-linear Schrodinger equation with an external uniform magnetic field included. In particular, we derive a general criteria on the initial data and the power of the non-linearity so that the corresponding solution blows up in finite time, and we show that the time for blow up to occur decreases as the strength of the magnetic field increases. In addition, we also discuss some observations about Strichartz estimates in 2 dimensions for the Mehler kernel, as well as similar blow-up results for the non-linear Pauli equation.
The Schr{o}dinger equation is solved exactly for some well known potentials. Solutions are obtained reducing the Schr{o}dinger equation into a second order differential equation by using an appropriate coordinate transformation. The Nikiforov-Uvarov method is used in the calculations to get energy eigenvalues and the corresponding wave functions.
We solve rigorously the time dependent Schrodinger equation describing electron emission from a metal surface by a laser field perpendicular to the surface. We consider the system to be one-dimensional, with the half-line $x<0$ corresponding to the bulk of the metal and $x>0$ to the vacuum. The laser field is modeled as a classical electric field oscillating with frequency $omega$, acting only at $x>0$. We consider an initial condition which is a stationary state of the system without a field, and, at time $t=0$, the field is switched on. We prove the existence of a solution $psi(x,t)$ of the Schrodinger equation for $t>0$, and compute the surface current. The current exhibits a complex oscillatory behavior, which is not captured by the simple three step scenario. As $ttoinfty$, $psi(x,t)$ converges with a rate $t^{-frac32}$ to a time periodic function with period $frac{2pi}{omega}$ which coincides with that found by Faisal, Kaminski and Saczuk (Phys Rev A 72, 023412, 2015). However, for realistic values of the parameters, we have found that it can take quite a long time (over 50 laser periods) for the system to converge to its asymptote. Of particular physical importance is the current averaged over a laser period $frac{2pi}omega$, which exhibits a dramatic increase when $hbaromega$ becomes larger than the work function of the metal, which is consistent with the original photoelectric effect.
New exact analytical bound-state solutions of the radial Dirac equation in 3+1 dimensions for two sets of couplings and radial potential functions are obtained via mapping onto the nonrelativistic bound-state solutions of the one-dimensional generalized Morse potential. The eigenfunctions are expressed in terms of generalized Laguerre polynomials, and the eigenenergies are expressed in terms of solutions of equations that can be transformed into polynomial equations. Several analytical results found in the literature, including the Dirac oscillator, are obtained as particular cases of this unified approach.
For the two-dimensional Schrodinger equation, the general form of the point transformations such that the result can be interpreted as a Schrodinger equation with effective (i.e. position dependent) mass is studied. A wide class of such models with different forms of mass function is obtained in this way. Starting from the solvable two-dimensional model, the variety of solvable partner models with effective mass can be built. Several illustrating examples not amenable to the conventional separation of variables are given.
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