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The Error in Solving Schrdinger Equation of Wave Function of Particle Spin with 4-order Symposium

Yuzhen Liu

Published Jan 1, 2007• 0 citations

Abstract

The numerical solution of wave function is obtained by solving schrdinger equation of partical spin with symposium.We investigated the error in the solution of 4-order symposium,which was compared with that of 2-order symposium.The precision of the solution of wave function with 4-order symposium is 7 quantitative ranks higher than that with 2-order symposium,namely its absolute error7 quantitative ranks lower than that of 2-order symposium.However,their error changes are similar,all of which oscillate periodically between positive and negative numbers,and are similar to those of sine or cosine function,while the amplitudes increasing continuously.The changeable error graph of 4-order symposium slightly delays that of 2-order symposium.It is discovered that the absolute error of 2-order symposium of wave function is exactly proportional to the rate of time change of the wave function.In other words,the ratior of the absolute error of a wave function to the rate of its time change is linear,while it is not true for 4-order symposiom.However,considering the delay of 4-order absolute error,we can also obstain similar linear results.

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The numerical solution of wave function is obtained by solving schrdinger equation of partical spin with symposium.We investigated the error in the solution of 4-order symposium,which was compared with that of 2-order symposium.The precision of the solution of wave function with 4-order symposium is 7 quantitative ranks higher than that with 2-order symposium,namely its absolute error7 quantitative ranks lower than that of 2-order symposium.However,their error changes are similar,all of which oscillate periodically between positive and negative numbers,and are similar to those of sine or cosine function,while the amplitudes increasing continuously.The changeable error graph of 4-order symposium slightly delays that of 2-order symposium.It is discovered that the absolute error of 2-order symposium of wave function is exactly proportional to the rate of time change of the wave function.In other words,the ratior of the absolute error of a wave function to the rate of its time change is linear,while it is not true for 4-order symposiom.However,considering the delay of 4-order absolute error,we can also obstain similar linear results.

The varying regularity of relative errors of wave function is uncover ed while the Schrodinger equation of harmonic oscillator is solved with symposiu m. It has been found through calculation that relative errors of wave function i n different X points are all the same a t the same time. The relative errors of real number part and imaginary number pa rt of wave function change periodically between positive and negative numbers wi th the passage of time. The period is 628 steps or π seconds. The relation betw een the relative error of real number part and the one of imaginary number part of wave function is similar to the characteristic of Uncertainty Principle. Whe n one relative error approaches infinitely small, the another one approaches inf initely great. The product of the two relative errors is a fixed small number. T he absolute value of the small number increases with the passage of time slowly.

For the general time-dependent linear potential, the exact and complete solution of the Schr?dinger equation was obtained by assuming a form of wave function. Meanwhile, it was pointed out that the matrices φ(t)〉和ψ(t)〉 defined by the two wave functions 〈φ(t)xψ(t)〉和〈φ(t)pψ(t)〉 satisfy classical equations of motion. According to the ensemble theory of quantum mechanics, such classical equations describe the motion of fluid. A further research shows that similar conclusions apply to any linear system.

We solve the solution of the Time-Dependent Schrodinger Equation (TDSE) using the higher-order Trotter-Suzuki Decomposition. We use The Gaussian function as a stationary state in a linear potential system. The TDSE solution using Baker - Campbell - Hausdorff was used to validate the results and to measure the accuracy of the Trotter - Suzuki decomposition. So that the difference between the Baker - Campbell - Hausdorff result and the Suzuki - Trotter decomposition is considered an error. The error of the TDSE solution by the Trotter - Suzuki second-order decomposition was lower than the first order. Meanwhile, the error of the TDSE solution by second-order hybrid will be lower than the second-order Trotter - Suzuki decomposition when the value of dx=0.1 and 0.05 with dt < 0.0001. The error comparison of these three methods is only valid when time t<1.

An important feature of the wave equation is that its solutions q(r, t) are uniquely specified once the initial values q(r, 0) and (del)q(r, 0)/@t are specified. As was mentioned before, if we view the wave equation as describing a continuum limit of a network of coupled oscillators, then this result is very reasonable since one must specify the initial position and velocity of an oscillator to uniquely determine its motion. It is possible to write down other “equations of motion” that exhibit wave phenomena but which only require the initial values of the dynamical variable — not its time derivative — to specify a solution. This is physically appropriate in a number of situations, the most significant of which is in quantum mechanics where the wave equation is called the Schrodinger equation. This equation describes the time development of the observable attributes of a particle via the wave function (or probability amplitude) . In quantum mechanics, the complete specification of the initial conditions of the particle’s motion is embodied in the initial value of . The price paid for this change in the allowed initial data while asking for a linear wave equation is the introduction of complex numbers into the equation for the wave. Indeed, the values taken by are complex numbers. In what follows we shall explore some of the elementary features of the wave phenomena associated with the Schrodinger equation.