Simulation Schroedinger Maple

8/13/2019 Simulation Schroedinger Maple

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Computer Simulations ofSchroedinger Equation using Maple

Revised 4/14/09

The purpose of this lab project is to:

1) solve Schroedinger equation numerically.2) learn how to solve differential equations using Maple.

Introduction

1. In this lab we will solve Schroedinger equation

)()()()(2 2

22

x E x xU xdxd

m =+

h.

It will help to change units so that we can apply the results to real world examples. We canrewrite Schroedinger equation units where length is measured in nm and energy is measured ineV by the following

0)()()()(22

=+ x x xu xdxd

c , wheree

U u = ,

e E = , 2

22

2mae

c h

= , and nma = .

First it makes Schrdingers equation simpler and more like a mathematical equation. But, italso makes the units appropriates for nano size objects.

To start the solution we need the initial conditions or boundary conditions. There are two wayto do this. The first used the boundary conditions at the origin. For a symmetric potential thesolutions are always symmetric or anti-symmetric about the origin. We can start the solution on

the axis of symmetry or for us x = 0. On the axis of symmetry the symmetric solutions all havea slope of zero or d (0)/dx = 0. We can choose any value for (0) because the wavefunctionsare not normalized. So choose (0) = 1. For the anti-symmetric solution all the solutions arezero or (0) = 0 on the axis of symmetry. We can choose any value for the slope again becausethe wavefunctions are not normalized. Choose d (0)/dx = 1.

The second way uses the boundary conditions at infinity. At infinite the wavefunction,

0)( = and the derivative of the wavefunction, 0)(

== xdx

xd has to approach zero. Of

course, if we start the integration of Schrdingers equation with these boundary conditions we

will get a null wavefunction. Thus, we want to start with a small value for the wavefunction andits derivative (something like 0.00005). Well how far is infinity? All we need to do is be fiveor six times the distance to the turning point. That will work. The real test is that if you try adistance and move a little further away the wavefunction should not change.

Now we make a guess as to the energy and use the initial conditions on the wavefunction to finda mathematical solution. Thus, there is a solution for each energy but for a physical solution thewavefunction must be finite. We see that all solutions diverge unless the energy is correct. It iseasy to find the correct energy because as you cross the correct energy the solution diverges in


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the opposite direction (ie if it was diverging negatively then it will change to diverge positively).

2. In this lab we will use Maple to solve Schrdingers equation. You can use Maple to solvedifferential equations without knowing how it works but if you would like to know read chapter8 section 1 on "Ordinary Differential Equations" in Numerical Mathematics and Computing by Ward Cheney and David Kincade. You can find the pdf on the course web page.

Infinite Square Well

3. I have provided a simple Maple program to get you started. The program called Finite_Square_Well amd will solve Schrdingers equation for the finite square well withdifferent widths and potential energy walls. We can also use the program for the infinite squarewell. I have entered a large potential energy so that the solutions closely approximate an infinitesquare well. Now lets use it to find the ground state energy. Calculate the ground state energyfor a well 20 nm wide. Do this in your notebook . Now run the program and see that it works. Ihave set the energy so that it is close to the eigenvalue. Also I have set the potential energy wallsto 200 eV so that it approximates an infinite square well. This is not infinity but close enough togive us the same answer. Using numeric solution it is too slow if the potential energy gets too

large. Next vary the energy a little above and below to see the behavior of the wavefunction.What do you observe? Scroll down through the program and make sure you understand all thecommands.

4. First lets find the first six energy levels and eigenfunctions for the infinite well. We dont wantto just guess so calculate the energy for the first six eigenvalues using the results found in classfor the infinite square well. Do they agree? Are the wavefunctions the same?

5. Use your Maple program

6. Now we will change the well potential energy height to explore the finite square well. Keep the

same well width of 20 nm and change the potential energy to U 0 = 3eV. We can make finitewells in semiconductor wires at Harvard this size. If you are interested ask me.

7. Now find the first six energy levels. You should find the energy levels just below the values forthe infinite square well. Start the search at the value for the infinite square well and lower theenergy until you find the eigenvalue for the energy.

8. Compare the infinite to the finite square well and comment on and differences in your notebook.Compare the wavefunctions you found to the expressions from class. Check to see that thewavefunctions are normalized by comparing the normalization constant found in class to the oneyou found using Maple. Do this in your notebook .

9. Change this section . Next we should change the program so that it will find the energiesfor us. You should have been using the bisection algorithm. This is the algorithm we will use.Open and resave the Schroedinger_General program and rename it Finite_Square_Well_with_Search . Enter the potential energy function for the finite squarewell where the program has U:= potential energy function . Find the eigenvalues (energy) andeigenfunctions (wavefunction) for the first six states. Draw an energy level diagram for thefinite and infinite square well next to each other. Comment on any differences in yournotebook.


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10. As the depth of the well is increased you would expect the energy levels to approach the valuefor the infinite square well. Look at the quantized energy for the infinite square well and makesure you understand why the energy changes. We will come back to this question later. Findthe energy for the first (n = 1) eigenvalue for different well depths (up to 500 eV). Plot theenergy for the first eigenvalue as a function of well depth using about 10 different values. Tryto see what value the first eigenvalue extrapolates too for an infinite well depth. Compare thisto the value we found for the infinite square well in class.

11. In this part of the program we can find the expectation value for different operators. Forexample we can find the momentum expectation value

or the potential energy expectationvalue. This is the average value you would get if you measured the momentum or potentialenergy of the electron a number of times.

First we will find the expectation value for the electrons position in the ground state. Thisis

dx x x x x nnn )()(*

=

Can you think of a simple classical argument to find the value for the position expectation?Write your answer in your lab book.

12. Change this section if 9 changes . Reopen your program Finite_Square_Well . At the endof the program after you find the energy and wavefunction and add a command to calculate theposition expectation. Use the program to find the position expectation for the ground state.Does it agree with your argument?

Find the expectation value for the position for two or three other eigenvalues. Is this what you

expected?

13. Lets find the momentum expectation for the finite square well. First can you use classicalarguments to find the average momentum? Write your argument in your lab book.

Add a command to calculate the momentum expectation after the position expectation. Youwill recall that the momentum operator is

[ ]dxd

mi

pe2

2h= dx x

dxd

m x p n

enn

)(2

)(2

* =

h

Again we will set 1=h to keep the momentum value from being extremely small. Thus to findthe actual momentum you need to multiple you value by h .

Find the momentum expectation for the ground state. Do you get the classical results? Whatabout higher energy states? Does the classical argument work? Check it out for a couple ofother states.

14. Lets find the expectation for the energy and see if it is the same as the eigenvalue energy. Whatdo you think? Use the momentum expectation and position expectation to find the kineticenergy expectation, , potential energy expectation,

, and total energy expectation, .


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Does the expectation for the total energy agree with the energy value you found earlier?

15. Using the wave functions for the infinite square well calculate the kinetic energy expectation.How does it compare to the kinetic energy expectation for the finite square well? Is thisdifference about the same as the total energy difference between the finite and infinite squarewell?

16. Look at the probability of finding the electron outside the well for all eigenstates. Which statehas the largest probability of being found outside the well? Does this agree with the energyvalues you found in section 5? You should be able to explain the difference in energy for thefinite and infinite square well using the probability plots. Recall

= +

The kinetic energy is related to the curvature of the wave function. Does it change? Can youestimate the change in curvature? From the probability calculate the change in potential energy.This difference should be the difference between the energy for the infinite and finite squarewell.

The region outside the well is called the classically forbidden region. Why is this regionreferred to in this way?

17. you find the potential energy expectation for the region outside the well?

18. Finally, lets calculate x and p and see if x p is equal to h/2 . We will use the root-mean-squared for x and p. The root-mean-squared is defined for a continuous function as

( ) =

dx x x x nn

2*

19. where (x) is a normalized probability density and is the average value. From this showthat

22 x x x =

20. Use this expression for the root-mean-squared to find the uncertainty in x for the ground state.Use a similar expression to find the uncertainty in the momentum in the ground state. What isx p? This is the explanation for the zero point energy or why the ground state energy is notand cannot be zero.


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Quantum Ball for a Linear Restoring Potential Energy

21. Lets think about a new example. Consider a constant restoring force toward the origin. Canyou use the definition for the potential energy in terms of the work? For a constant restoringforce you would find a linear potential energy function as shown.

x

V(x)

k|x|

22.

Quantum Ball in a Gravitational Field

23. Lets think about a new example. Consider a constant force toward the origin. The force wouldbe F = kx/|x|. The term x/|x| causes the force to point toward the origin. The potential energy isV(x) = k|x|. This is shown below. From your understanding of Schroedinger equation draw thefirst four wavefunctions.

Try to guess how the energy levels will compare with the harmonic oscillator. Higher, lower,why?.

7. Use Maple to solve Schrdingers equation for a constant force or u(x) = k|x| for x > 0. Findthe first four wavefunctions and energy levels. Compare them with the harmonic oscillator.Explain why the energies are higher or lower than the harmonic oscillator. For thewavefunctions think about what ahalf quantum harmonic might look like.


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Finite Square Well with an Electric Field

12. Lets think about a new example. Consider an electric field applied to the electron in the box.The force would be F = -e E and the potential energy is U(x) = eEx. This is shown below. Fromyour understanding of Schroedinger equation draw the first four wave functions. Try to guesswhat the energy will be for the first four levels compared to the finite square well..

Use your Maple program to solve Schrdingers equation for a constant force F = -e E or V(x) =eEx. Find the first four wave functions and energy levels. Compare them with the finite squarewell.

x

V(x)

Two Finite Square Well: A Diatomic Molecule

9. The potential energy for a double well is shown in the figure on the next page. Try to guess thefirst six wavefunctions for the double well. Draw them in your notebook. Now try to guess theenergy for the first six levels. Think about the kinetic and potential energy changes.

10. Use your program with graphics to plot the wave function and find the energy levels for asquare well with a bump in it. Again we will look at this problem in class. We will useperturbation theory to solve this in class. How do the energy levels and wave functionscompare to the same well with out the bump?

Quantum Harmonic Oscillator

3. First we will use Maple on a problem we can solve analytically (we know the solutions and


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energies from class), the harmonic oscillator. The potential energy for the harmonic oscillatoris U(x) = (1/2)kx 2 = (1/2)m 2 x2. In our units u(x) = (1/2)m 2 x2 /C. Dont forget

You can use the Maple commands diff and dsolve or the command that does it all DEplot.Dont forget to add the plotting functions (with (plot): and with (DEtools) :

Find the energy levels (just the first six) for a harmonic oscillator. How do the energy levelscompare with the analytical values, E n = _ (n+1/2) Show both on an energy leveldiagram.

4. How do the wave functions compare?

5. Lets compare the analytic solution to the approximation (computer solution) found withMaple. Plot the wavefunction for n = 3 and overlay the analytical solution (y) = Aexp(-y2 /2)H 3(y), where y= x??. How do they compare.

Radial Hydrogen Equation

radial dimension for Hydrogen

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Simulation Schroedinger Maple