2011 Homework

HW 1: Crystal Structures

Regarding the Nanohub exercise .... We still have problem with Miller plane for simple cubic system. Instead do the following:

1. Choose 'Material' in the 'Material and Crystal' system option. 
2. For material,  choose NaCl. If we do not distinguish between Na and Cl, then this lattice can be viewed as a Cubic lattice. 
3.  Show Miller Plane 'yes',  Select atomic plane (111) from the drop down menu, choose size '4'.  
Once the simulation is done,  choose 'Miller Plane' from the 'Results' plots.  You may want to magnify the figure. Also if you click on the blue arrow on the right hand side next to the plots, it will bring up a set of options. You can reduce the size of the sticks that connect the atoms and the increase the size of the spheres to make the red atoms on the surface clearly visible. 

HW 2: Quantum Mechanics

Problem 1. 

You want to distinguish between Intensity (shown in the HW figure)  and Intensity per wavelength or Power per unit volume (The formula discussed in nanohub lectures).  If you want to use the nanohub formula, you must convert it to right units, otherwise the temperature will be off by a factor of 2 or so. 

 Problem 3.

I mentioned four rules for solving the bound level problems: (1) divide up the space in regions defined by constant potential and write the solution of the Schrodinger equation in respective regions, (2) Use the boundary conditions at plus and minus infinity to eliminate two unknown coefficients, (3) use the boundary conditions at each interface to obtain two equations, (4) Set the determinant to zero to find the final characteristic equations.

Now if you follow the rules for HW3 ...

a) You will realize that there are three regions for this problem -- every region with (E < V). Can you therefore write down the solution for these regions?
b) Now you have 3 regions and 6 unknowns. Can you use the boundary conditions at plus/minus infinity to get rid of the two constants.
c) At each interface, the wave function is continuous -- but the derivative is not continuous -- but is given by part (a). You will get four equations.
d) Set the determinant to zero to get the final equation.