In this assignment, we continue to study bound state solutions of the time-indepdendent Schrödinger equation using a modification of Program eigen in Eigen2.java. Click here to run this applet. Note that Eigen2.java uses the comphys package.We also study the time dependent Schrödinger equation using TDSE.java, which is a translation of Program tdse. Click here to run this applet.
For this assignment, work and submit the following:
- Use Eigen2.java to
- Find the bound state spectrum of Argon molecules using the numerical values given on page 215 for the Lennard-Jones parameters. Note that the "quantum parameter" gamma in the program is defined to be hbar^2 / (2 m epsilon sigma^2).
- See if you can find a gamma that gives a reasonable fit to the vibrational spectrum of molecular hydrogen:
n En (eV) n En (eV) 0 -4.477 8 -1.151 1 -3.962 9 -0.867 2 -3.475 10 -0.615 3 -3.017 11 -0.400 4 -2.587 12 -0.225 5 -2.185 13 -0.094 6 -1.811 14 -0.017 7 -1.466 An additional piece of data that may be useful to you is the interatomic separation in the ground state which has been measured experimentally: d = 0.74166 Angstroms. This can be deduced from the rotational spectrum which gives the moment of inertia of the molecule I = 0.5 matom d2. Note: you need to relate the atomic mass to the reduced mass which appears in the Schrödinger equation.
If the fit does not seem to be very satisfactory, try the Morse Potential:
V(r) = V0 [ (1 - exp( - (r - r0) / a) )2 - 1 ],
or in code
V0 * ( Math.pow( 1 - Math.exp(-(r - r0)/a), 2 ) - 1 )
This potential has 3 parameters, V0, r0, and a. What is the physical significance of these parameters? See if you can combine 2 of them into a single quantum parameter so you can use the same program and just change the potential function. If you notice any difference in the quality of the fit, try to understand why. Hint: one of the potentials better describes molecules of inert-gas atoms which behave more like hard spheres at small values of r.
- Choose any one of Problems 18.5 - 18.9 on pages 640 - 641 which you find interesting. Use TDSE.java to solve the problem you have chosen.