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The paper outlines the development of a new, spectral method of simulating the Schr ödinger equation in the momentum domain. The kinetic energy operator is approximated in the momentum domain by exploiting the derivative property of the Fourier transform. These results are compared to a position-domain simulation generated by a fourth-order, centered-space, finite-difference formula. The time derivative is approximated by a four-step predictor-corrector in both domains. Free-particle and square-well simulations of the time-dependent Schr ödinger equation are run in both domains to demonstrate agreement between the new, spectral methods and established, finite-difference methods. The spectral methods are shown to be accurate and precise.

This paper outlines the development of simulations of the time-dependent Schrödinger equation produced in both position and momentum domains. In the position domain this is the x-y plane. In the momentum domain this is the k_{x}-k_{y} plane, as it is the Fourier transform of the position domain [

All simulations are advanced in time using a four-step predictor-corrector method. The predictor-corrector can be applied independently in both position and momentum domains to step the simulation forward in time. The predictor-corrector is generated using Lagrange polynomials, outlined by [

The position-domain approximation of the kinetic energy operator is derived using Lagrange polynomials and consistent with results from [

Given an initial state

The first simulation is a free particle with no imposed boundary conditions, when the Hamiltonian consists only of the kinetic energy operator. This simulation demonstrates the difference in boundary conditions of each domain. In the position domain, this is equivalent to an infinite square-well potential, or particle-in-a-box. When the wave function reaches one boundary, it is reflected back. In the momentum domain, this is equivalent to periodic boundary conditions. When the wave function disappears into one boundary it will reappear in the opposite boundary, travelling in the same direction. This is to establish a relative performance benchmark when only the kinetic energy operator is applied.

Second, a finite square-well potential of 100 eV is imposed in both domains. This simulation demonstrates the computational burden associated with imposing the same initial and boundary conditions in both domains. Application of the potential function in the position domain is carried out by entry-wise multiplication of the wave function and potential function lattices. In the momentum domain, this operation is equivalent to convolution. Rather than carry out this time-consuming operation, the wave function in the momentum domain is transformed back to the position domain at every time-step in the simulation in order to apply the potential function. The kinetic energy operator is applied when the wave function has been transformed forward into the momentum domain.

Each of the simulations begins with an initial state of a two-dimensional wave packet with a Gaussian envelope. The simulations are stepped forward in time and the complex-valued wave function components and densities, as well as some expectation values, are captured incrementally. The wave function components and densities are converted to image format and animated [

The following subsections outline the numerical methods used to generate solutions to the Schrödinger equation.

The same four-step predictor-corrector method is used to step the position and momentum domain simulations forward in time. The predictor and corrector start with the basic form of the differential equation

The predictor uses the integral form of the differential equation.

The function f is approximated by Lagrange polynomials [

The corrector uses the original form of the differential equation.

For the corrector, y is approximated by Lagrange polynomials [

The predicted value

In the position domain, the Schrödinger equation is written as follows using operator notation as shorthand. The Hamiltonian operator

At every time step, the predictor calculation is carried out.

That result is plugged in to the corrector calculation to advance the simulation forward one time-step.

The following sections outline the development of the Hamiltonian operator

The first simulations in the position and momentum domains assume free particle conditions. The particle is given the mass of an electron. Only the kinetic energy operator applies in the Hamiltonian.

Before the simulations are started, the position- and momentum-domain lattices must be constructed. This requires fixing the position-domain step sizes

number of rows

is defined by

of the y-direction accounts for the fact that computer storage increments the row index as the row moves down.

The momentum domain lattices are constructed according to the relationship

high frequencies have been shifted into the negative frequencies. Use of the FFTW library requires applying a phase-shift to the position domain before transforming into the momentum domain if negative frequencies are used instead of high frequencies [

The approximation to the kinetic energy operator in the position domain was generated using Lagrange polynomials. This was accomplished by approximating the second derivative in one dimension, as the same formula can be applied to all dimensions. This is a centered-space formula accurate to fourth order, requiring five sample points. In terms of the generalized coordinate

The real and imaginary parts of the wave function

Denote spatial sample points with subscripts:

The approximation to the kinetic energy operator in the momentum domain was generated using the transform of the derivative operator. Because the momentum domain is the Fourier transform of the position domain, the derivative operator is transformed as well. In terms of the generalized position-domain coordinate

function

The initial position-domain state

yielding a pair of coupled equations. The simulation can be stepped forward in time once the approximation to the Hamiltonian has been substituted into the predictor corrector formula.

A square-well potential was chosen to test the effectiveness and relative performance of the simulations of both domains, when the particle interacts with an electrostatic potential. The particle is again given the mass of an electron. For the purposes of this demonstration, the chosen potential must be high enough to reflect most of the particle off the potential step, back into the region where the potential is zero.

The lattice describing the potential must have the number of columns

less than

The application of the potential operator is straightforward in the position domain. The lattices representing the real and imaginary parts of the wave-function

In the momentum domain, application of the potential operator transforms to the convolution operation,

For simplicity and readability the procedure below does not write the state functions

First, the potential operator is applied in the position domain and transformed to the momentum domain.

The kinetic energy operator is applied to

Following the example of Equation (8) in Section 2.1.3, the predictor formula is applied to produce the predicted value

At the corrector step, apply only the kinetic energy operator to the predicted value

Following the example of Equation (9) in Section 2.1.3, the corrector formula is applied to produce the corrected value

Both the corrected value

All simulations are run under the same constraints and initial conditions to illustrate similarities and differences in the particle’s position over time. The free particle simulations show differences in position due to the differences in boundary conditions between the two domains, despite having the same initial conditions. The square well simulations will show strong agreement in position due to the boundary conditions and initial conditions being the same. Regarding precision, all simulations were shown to be accurate to 8 decimal places. This value is stable for the duration of the simulations and was measured by finding the difference between the current state’s density function and 1.

The physical constants of electron mass

The row and column sizes are set to

seconds. Each simulation is run for 100,000 time steps. The lifetime of each simulation is 0.1 picoseconds. The components, densities and expectation values are measured and recorded every 500 time steps. This increment is referred to as a “frame” in the graphs below and each frame is equal to

All initial condition parameters are given in index units and the actual parameters are multiplied by the appropriate lattice step size. The initial conditions set the particle on the +x-axis at index +70.0, relative to the origin point and close to the vertical boundary at the end of the +x-axis. The particle is given positive momentum, which will propel the particle into the boundary. The particle’s spatial wavelength

The initial velocity of the particle in all simulations is

Based on the initial conditions and predicted velocity, the particle is expected to reach the boundary at time ^{th} frame of the simulation. The position-domain particle reflects off the boundary while the momentum-domain particle travels through the boundary and reappears in the opposite boundary, travelling in the same direction at the same velocity. ^{th} frame to show where the paths are expected to diverge.

Four free-particle animations are produced for the position domain. One overhead view and one cross- sectional view are produced for the finite-difference free-particle simulation and one overhead view and one cross-sectional view are produced for the spectral free-particle simulation.

Based on the initial conditions and predicted velocity, the particle is expected to reach the boundary at time ^{nd} frame of the simulation. The particles in both domains reflect off the boundary imposed by the electrostatic potential.

Four square-well animations are produced for the position domain. One overhead view and one cross-sectional view are produced for the finite-difference square-well simulation and one overhead view and one cross- sectional view are produced for the spectral square-well simulation.

One additional animation is produced for the momentum-domain, square-well simulation that shows the cross- sectional view of the momentum density function. The cross-section is taken along the ^{nd} frame. As the particle interacts with the electrostatic potential, the particle is reflected and reverses direction. In the momentum domain, this is indicated by the density function disappearing from the positive axis and reappearing on the negative axis.

^{nd} frame indicating when the particle interacts with the boundary and reverses direction.

Momentum-domain simulations of the time-dependent Schrödinger equation provide precise and accurate results; however, the application of these techniques is not limited to the Schrödinger equation. The methods

described in this paper are also suitable to simulate the heat equation

temperature in space and time and

These simulations were produced on single-core, AMD Athlon X2 processor. The spectral methods demonstrated faster performance for the free-particle simulation, while the finite difference methods demonstrated faster performance for the square-well simulation. None of the simulations employed parallel computing techniques due to the limitations of the hardware. Multiple cores would allow multiple Fourier transforms to be calculated at the same time. Because the spectral-method, square-well simulation requires multiple Fourier transforms at every time step, introducing parallel computing techniques would increase performance substantially.

Thanks to Craig Unrein for proofreading assistance.

MichaelJennings, (2015) Simulation of Time-Dependent Schrödinger Equation in the Position and Momentum Domains. American Journal of Computational Mathematics,05,291-303. doi: 10.4236/ajcm.2015.53027