Download Introduction and finite square well

Quantum Mechanics 1
Schršdinger's Equation in One Dimension
Introduction to the software and the finite square well
In this part of the course we will use the CUPS (Consortium for Upper-level Physics) software to
solve Schršdinger's equation for potentials of various shapes. Access the software by clicking the
Start button, then choosing Teaching Software, Cups and finally Quantum Mechanics.
On running the program you will be presented with the main menu. This gives you the choice of 8
simulation programs, and an Exit option to terminate the program. We are primarily interested in
the first two of these options: One Dimensional Bound State Problems and Stationary Scattering
States in One Dimension.
It is important to note that this software is written to simulate electrons in potential wells. Later,
when we go on to scaling, you will see how the results for electrons can be generalised to other
particles in potential wells.

Prerequisite: The infinite square well
Before starting the computer based
exercises you should determine the energies
(in eV) of the first three eigenfunctions, 1,

Infinite square well
En 
3
 2 n 2 2
2
8ma
n=3 2nd excited state
2 and 3, for an electron in an infinite
n=2 1st excited state
2
potential well of width 0.2 nm, (i.e. 2a = 0.2
E3
nm) as shown in the diagram. Be aware that
E2
n=1
Ground state
1
the energy you calculate by this method is
E1
2a
the energy of the eigenstate above the
bottom of the potential well. Once you have
calculated E1, E2 and E3 you should go on to the computer based exercise.
Exercises: The finite square well
Select the One Dimensional Bound States Problems option from the main menu1. When this
option loads, you will see a green information box which tells you about the program, click on the
box to move on. By default you will see a square potential well with depth 300 eV and width 0.2
nm.
1
Select options either by using the mouse to point and click, or by using the arrow keys to highlight your preferred
option and then pressing the Enter key. If this option is not available, check that you are using Part 1 by clicking on the
Parts | Part 1 : Finding Eigenvalues drop-down menu. If you are unsure of drop-down menu's then read footnote 2.
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For illustration, select the Method | Try energy (with mouse) option from the drop-down menus2.
Then try clicking at a few well-spaced energies within the potential well. On doing so, you will see
that the computer draws curves of various shapes. The question now is which, if any, of these are
eigenfunctions for this potential well? The key is in the nature of the curve for large positive values
of x, i.e. on the right-hand-side of the screen. If the curve diverges from the x-axis for large values
of x, then the curve cannot be an eigenfunction. This is obvious when you consider what an
eigenfunction represents; the modulus squared of an eigenfunction, |2, is the probability of finding
an electron, with that eigenfunction's energy, at the various positions in and around the potential
well. If the curve diverges from the axis, then as x so the curve tends toward infinity. This
would mean that the electron has an infinite probability of being at infinity! The curves which
diverge on the right-hand-side are therefore not eigenfunctions. We are looking for curves which
approach the x-axis asymptotically. In the diagram of the finite square well an eigenfunction is
shown by the solid line, and two diverging functions are shown by the dotted lines.
Finite square well
Exercise A
0
V
2a
Now that you know what eigenfunctions look like, try to find all
the eigenfunctions for this potential well. A good estimate of the
positions of the first few eigenfunctions can be obtained by
comparing this finite well with the infinite well. Note that the
energies you calculated for the infinite well were measured from
the bottom of the well, and that the scale in the CUPS program
measures down from the top of the well. If we take the distance from the top of the well as a
positive number then this is known as the binding energy of the eigenstate.
Starting near the bottom of the potential well, try and find an eigenfunction with no nodes (i.e. one
that does not cross the x-axis). Then try and find eigenfunctions with 1, 2, and successive numbers
of nodes. You may not be able to get the exact location as the energy steps, dictated by the pixels,
may be too broad. Try to get an estimate of the energy of each of the eigenfunction that you find.
Exercise B
With the estimates gathered in the previous exercise you can now make the computer determine a
more accurate estimate of the energy of the eigenstates. Taking each of your estimates in turn use
the Method | Hunt for Zero option to home in on the eigenstate. Enter initial values of binding
energy that straddle your estimate by a few eV. The computer will then calculate the position of the
eigenstates and will show a plot of the eigenfunction.
2
At the top of the screen are boxes which act as drop-down menus. Clicking on these boxes reveals a list of options.
For example, to access the Method | Try energy (with mouse) option you should click on the Method box, then on the
Try energy (with mouse) option.
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Exercise C
The process you have used in finding the eigenstates up until now has involved a fair amount of
guessing. This can be avoided by allowing the computer to determine the position of the
eigenstates. Select Spectrum | Find Eigenvalues 3 to make the computer find the eigenvalues for
you. Now that the eigenvalues have all been calculated, use the Spectrum | See Wave Functions
menu to investigate the shapes of the various eigenfunctions. For each eigenstate, comment on:
the number of nodes
• the shape of the eigenfunction within the well and in the walls
• the symmetry of the wavefunction, note that a wavefunction which is symmetrical about the
origin, (x) = (-x), is said to have even-parity and an anti-symmetric wavefunction,
•
(x) = -(-x), is said to have odd-parity
From your observations, what conclusions can you draw about the eigenfunctions? State the number
of nodes, and the parity of each of the eigenfunctions.
Exercise D
An important property of eigenfunctionsis their orthonormality. This means that eigenfunctions
are orthogonal to each other and that they are normalised. Recall that an eigenfunction  is said to

be normalised if
  *  dx  1, where  *

is the complex conjugate of .

Two eigenfunctions, 1 and 2, are said to be orthogonal if .   1 *  2 dx  0

Using Part 2 of the program you can determine whether or not individual eigenfunctions are normalised and pairs of
Parts | Part 2 : Wavefunction
Properties. You can select which eigenfunctions appear in the top and middle graphs using the
Psi 1 and Psi 2 drop-down menus respectively4. The bottom graph shows the product of the two
eigenfunctions. Note the symmetry properties of this product when testing for orthonormality. Once
a product has been displayed you can check the integration using the Integrate menu. You should
satisfy yourself that each of the eigenfunctions is normalised and is orthogonal to each of the other
eigenfunctions.
eigenfunctions are orthogonal. Move to Part 2 of the program using the
3The
computer beeps every time it detects an eigenvalue. To turn this off click the Spectrum | Sound option. This
menu option toggles the sound on and off.
4
Notice that the middle graph displays 1*u2, the inclusion of the one allows for the placement of an operator - to
determine position, momentum etc. However, to check orthogonality and normalisation this operator should be one
(effectively not there).
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