Pdf numerical simulation of nonlinear schrodinger equation in. This was done by rst using the algorithm to solve in the case of the potential in an in nte square well and comparing the results with the analytical solutions. Using the numerov algorithm, the numerical solution of the time independent onedimensional schr odinger equation was found. Pdf solutions of the schrodinger equation by spanning the wave function. There are two basic forms of the equation, a timedependent form that gives the timedependent wavefunction showing how properties of the system change with position and time, and a timeindependent form that gives the timeindependent wavefunction, showing how properties of the system depend upon position, but. Pdf solving the twodimensional schrodinger equation using. But the logical roles of the two schr odinger equations are completely di erent. In two dimension, nonlinear schrodinger equation can be considered. Time dependent schrodinger equation the time dependent schrodinger equation for one spatial dimension is of the form for a free particle where ux 0 the wavefunction solution can be put in the form of a plane wave for other problems, the potential ux serves to set boundary conditions on the spatial part of the wavefunction and it is helpful to separate the equation into the time. The timeindependent schroedinger equation a very important special case of the schroedinger equation is the situation when the potential energy term does not depend on time. In fact, this particular case will cover most of the problems that well encounter in ee 439. I know this has to do with dimensional analysis, but i was sick when we had that class, and ive been searching for. Make clear which, if any, of the levels is degenerate.
Schrodinger equation with noninteger dimensions citeseerx. It is satisfying to nd the reduced mass in this equation. L x 2 is the wavefunction, vx is the potential energy function and mis the mass of the particle. The schrodinger equation the previous the chapters were all about kinematics how classical and relativistic particles, as well as waves, move in free space. For a particle of mass munder a potential vx, the onedimensional, timeindependent schr odinger equation is given by. Eigenvalues of the twodimensional schrodinger equation with nonseparable potentials h. Numerical simulation of the nonlinear schrodinger equation with. We shall study the case of a twodimensional quantum system. I have spent many years working with this equation, and have learned that, with a little guidance, it is not so daunting after all. The more fundamental is the law, however, the more important it is to conduct precise, quantitative tests of this law. The nls equation is the main governing equation in the area of optical solitons. By changing three parameters via sliders provided, slit width, obstacle width, and initial position of the wave packet, different behaviors can be explored. Newtons laws, the schrodinger equation does not give the trajectory of a particle, but rather the wave function of the quantum system, which carries information about the wave nature of the particle, which allows us to only discuss the probability of nding the particle in di erent regions of space.
In one dimension there is no special advantage in the former procedure, but in more dimensions this is the method of choice. Since two terms contain 1r era, and two terms contain only era, for the equation to work at all values of r, the terms with the same variables have to equal each other. Also, a particle has a non zero probability to be reflected at any boundary regardless. The schroedinger equation can not be derived from classical mechanics.
We note that earlier numerical studies on closely related problems can be found in 3, 5, 11, 22, 23, relying on di. Schr odinger equation equation 3 or 7, because both involve the same hamiltonian operator. Notes on quantum mechanics physics weber state university. A general solution to the schrodinger equation for a particle approaching from the.
To be precise, we found that the constant of proportionality was equal to h2. Therefore, this equation describes the quantum mechanical motion of. Similar analogies led schrodinger to postulate that the distribution of electron density and other properties of electrons in and out of molecules were determined from solutions to an equation similar to. Chapter 4 schroedinger equation mit opencourseware. Then the eigenvalue equations are l z f bottom lf bottom. Eid department of mathematics, middle east technical university, 06531 ankara, turkey received june 6, 1995. The timedependent schr odinger equation is a partial di erential equation, 1st order in time, 2nd order in the spatial variables and linear in the solution r. Because the splitstep error between the periodic and nonperiodic parts is relatively.
The tise is the equation that we must solve to nd the energy eigenfunctions and their corresponding eigenvalues, while. The case of e ux kinetic energy is positive, bound state and scattering 2. Some notes on understanding the schrodinger wave equation. Solution of schrodinger equation for a double well potential, article pdf available in acta physica polonica series a 70.
Show full abstract case of three identical rings that will gradually merge into a single one, representing the equivalent of a. The two dimensional schrodinger equation model simulates the time evolution of a twodimensional wave packet as it moves towards a slit with an obstacle in it, both with variable widths. If we first start by considering a particle in one dimension. The 3dimensional schrodinger equation before we analyze this result, let us do the analogue computation for the eigenfunction f bottom, corresponding to the lowest possible eigenvalue of l z for a xed value of, which we assume to be l. On the coulombtype potential of the onedimensional.
The proton mass is much larger than the electron mass, so that. For several reasons, our discussion of electrons led us to a discussion of waves. Write down the schrodinger equation for the system. We consider again the time dependent schrodinger equation prop. In this chapter, we start with the timedependent schrodinger equation with non. Some examples of its applications are demonstrated. Also, solving problems in one dimension is usually. Write down the ground state wave function and find the lowest four energy levels in terms of the quantities h, k, m etc. You can gain valuable insight into quantum mechanics by studying the solutions to the onedimensional time independent schrodinger equation. The applet has been designed primarily as a pedagogical tool. Solving one dimensional schrodinger equation with finite. The rst of the two separated equations describes the relative motion of the proton and electron. Use any of the results you need from the onedimensional harmonic oscillator without proof. But when considering discretized system, this state will reappear in the finitedimensional hilbert space consisting of piecewise.
The schrodingers schrodingers equation is the basic equation of quantum mechanics. The solution of the schr odinger equation via the secular equation is an alternative to numerical integration of ch. Cancelling out the common terms on both sides of the equations, a and e are easily solved for the hydrogen atom. Pdf solution of schrodinger equation for a double well. Solution for the ground state now, we should solve the differential equation 1.