Assignment 7

Due: 10:00 a.m., Wednesday, Nov. 4

Noble gas atoms are attracted to each other at long distances by weak induced dipole-induced dipole forces and repelled at short range by repulsion of their electron clouds. An approximate form for the potential energy of two noble gas atoms is the Lennard-Jones (LJ) potential:

where r is the distance between the two atoms. In this assignment, we will study the Lennard-Jones potential for two argon atoms trapped in one dimension. For argon, and . Use SI units throughout your calculations.

1. Plot the LJ potential for argon. Because the LJ potential becomes infinite at r=0, the following plot command may be useful: plot(V(r),r=0..?,??..???). You should replace ?, ?? and ??? by values which allow you to clearly see the shape of the function. The second range tells Maple which part of the y axis to show.

   Maple hints: Define and (using :=) in your Maple session before plotting. Note that defines a typical energy scale for this problem so that reasonable values for the maximum and minimum y axis values to plot are small multiples of this quantity. Similarly, defines a typical length scale so that the maximum r should probably be a small multiple of .

2. Write down the Hamiltonian for the relative-coordinate problem for two particles in one dimension subject to the LJ potential. is the most stable isotope of argon. The mass of one atom of this isotope is 39.962384amu. Calculate the reduced mass of two argon atoms and enter this value in your Maple session. Also enter any other constants which appear in your Hamiltonian.
3. Use the variational method to find the best approximate ground-state wavefunction of the form

   

   If you're curious about the form of this wavefunction, note that the LJ potential ``blows up'' at r=0 so the wavefunction must die off sufficiently quickly there.

   Hints: Recall that r is a distance so its range is . Maple will need to know that b is positive before you try to compute any integrals. Maple will give you a whole bunch of solutions to the variational problem, only two of which are even remotely reasonable. Use subs(b=?,Evar) (replacing ? by one of Maple's solutions) to substitute in values to your variational energy.

4. Normalize your wavefunction.
5. Plot the probability density.

Bonus 1:

Find the best variational wavefunction for the ground state of the form

Is this wavefunction better or worse than the one obtained in the main part of the assignment?

Bonus 2:

Use your imagination to find another form for the variational wavefunction. Do the variational calculation with your variational wavefunction and clearly state whether your form is better or worse than others used in this assignment.