DOING PHYSICS WITH MATLAB

MATLAB RESOURCES

QUANTUM MECHANICS

                  

Ian Cooper

 

matlabvisualphysics@gmail.com

 

 

TIME DEPENDENT SCHRODINGER EQUATION

FINITE DIFFERENCE TIME DEVELOPMENT METHOD

FREE PARTICLE: MOTION OF A WAVEPACKET IN A UNIFORM ELECTRIC FIELD

 

 

DOWNLOAD DIRECTORY FOR MATLAB SCRIPTS

 

   GitHub

 

   Google Drive

 

simpson1d.m      Function for [1D] integration

QMG24E.m        Propagation of Gaussian wavepacket in a uniform electric field

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LINKS

link    [1D] time dependent Schrodinger Equation using the Finite Difference Time Development Method (FDTD).   

 

link   Expectation values 

 

link   Gaussian wavepacket propagation

 

 

Motion of an electron in a uniform electric field

 

We can simulate the motion of a wavepacket representing an electron in a uniform electric field. The force acting on the electron is derived from the potential energy function

          (1)    

 

For a uniform electric field, the force acting on the electron is constant, therefore, the potential energy is a linear function with position x of the form

          (2)    

 

where L is the width of the simulation region and is a constant.

 

We can solve the time dependent Schrodinger equation using the finite difference time development method (TDSE/FDTD) to compute the wavefunction of a Gaussian wavepacket as a function of time.  I will consider a single electron in the uniform electric field. The electron is represented by a wavepacket to localize it. An initial state described by the Gaussian function is

        (3)    

 

where A is a normalized constant and is calculated so that , xc is the centre of the Gaussian wavepacket, s determines its width and   is the nominal wavelength.

 

Once, the wavefunction is known than the expectation values of any observable quantities of the wavepacket can be evaluated:

      <x>    position

     <p>    momentum

     <K>   kinetic energy

     <U>  potential energy

    <E>   total energy

 

The momentum of the wavepacket (the electron) is calculated from the nominal wavelength  using the de Broglie relationship

     (4)    

 

From equations 1 and 2, we can calculate the uniform acceleration a of the wavepacket

     (5)    

 

The initial values (t = t1 = 0) are represented by the subscript 1 and final values at time

(t = t2).

The initial velocity are final velocities are

             

 

The initial and final displacements are

            

 

The initial and final momenta, and impulse are

           

 

The initial and final kinetic energies, and work done are

          

 

The initial and final potential energies are

          

 

The initial and final total energies are

         

 

 

We can apply Ehrenfest’s theorem to calculate the changes in expectation values, that is, we can apply the principles of classical physics only to expectation values and not instantaneous values or not for eigenvalues

          

 

Even though we don’t know exactly where the electron is or its exact velocity at any instant, we can predict with certainty the time evolution of the probability distribution function and expectation values. Table 1 shows that there is excellent agreement between the classical predictions and the numerical results from the solution of the Schrodinger Equation.

 

Table 1. Summary of the simulation parameters and computations.

The result summary in Table 1 shows the classical concept of conservation of energy holds in our quantum system. The total energy of the system is time independent and any change in the expectation value of the potential energy is accompanied by a corresponding change in the expectation value of the kinetic energy. For example, the electric force does work on the electron to increase its kinetic energy and this is shown by the increase in the expectation value of the kinetic energy and the decrease in the expectation value of the potential such that the expectation value of the total energy does not change (figure 1).

Fig. 1.  The time evolution of the expectation values for the kinetic energy, potential energy

 

Fig. 2. The time evolution of the expectation values for position and velocity, and the uncertainties in position and momentum.

 

Fig. 3.   Animation of the motion of the wavepacket.

 

The wavepacket undergoes wavepacket spreading and with increasing uncertainties in position and momenta. The wavepacket undergoes a reflection around x = 8 nm where the faster components of the wavepacket are travelling to the left while the slower components are still travelling to the right. The opposite travelling waves interfere with each other leading to rapid oscillations in both the real and imaginary parts of the wavefunction. However, the probability density function remains its Gaussian profile but with an increased width produced by the reflection. The reflection occurs around t = 1.2 fs and the interference of the forward and backward waves produces the interesting changes in the uncertainties in position and momentum (figure 2).

 

The electron can be though as a particle and the classical laws of physics can be used to predict the time evolution of expectation values. But the wave nature of the electron means that we do not know the precise values of position and velocity at an instant. We can’t predict the path of an electron, we can only predict the probability of finding the electron at each instant within a range of x values. Thus, we can conclude that the expectation value for position tracks the classical trajectory.