Guide to Homework 4, updated.

This was updated Friday (Feb 1) with some extra graphs at the end related to problem 10, which is an important problem and not really so difficult. (Much less difficult than 9, for example.) 

These are notes related to preparing for, understanding and doing HW 4:

For the quantum systems we have studied so far: the infinite square well, the harmonic oscillator, and the -1/r potential ("H-atom"), it is possible to find exact solutions to the differential equation that the wave equation (Schrodinger) equation provides. Specifically we have look in some detail at the first 3 states of the infinite square well and 1DHO and 2DHO, and at the ground state and 1st excited states of the -1/r potential using the {x, y, z, r} choice for the 4 excited states for the latter system (H-atom).

In a lot of quantum systems one would like to study and understand, one can't solve the differential equation in a normal way. One of the methods that is very widely used is to guess a solution. The art of this method lies in making educated guesses for wave-functions. This approach of making educated guesses is call the "variational method", because in some of the guessed wave-functions there is a parameter that can be varied (like "a" for the wave-functions in problems 4, 5 and 6.) Generally what one does with a guessed or variational wave function is to calculate the energy of that wave-function. One can do this by calculating the expectation value of the kinetic and potential energies and adding them together. That is what you are asked to do in problems 4-6.

Problems 3 is not of this type (and neither is 7). In problem 3 you are asked to calculate the expectation value of the kinetic energy, T, and the potential energy, U, using the exact solution for the ground state of the infinite square well, which you know already.

Problem 7 is also not a variational method problem. In 7 you are asked to find the ground state of a finite square well using a fairly standard boundary matching approach (where you require continuity and smoothness at the boundary). The forms of the solution in the two regions in this problem are not guesses, but rather come from solutions to the time independent Schrodinger equation. It is from the Schrodinger equation, which is different inside and outside the well because inside U(x)= -V and outside U(x)= 0, that one gets a relationship between $\gamma$ and k. So the method in problem 7 is quite different from that in 4-6. The most important part of 7 are the questions in c) and d) (is this value of k larger or smaller…?, why?), though you have to do the whole thing to get to that.
     Regarding the method in 7, I think you have some experience in boundary condition matching from 101a. The key thing for that method is that E is the same inside and outside the well. that is what makes the endeavor of matching conclusive; otherwise there would be one too many parameters. The graph in part c) represents a visual way of finding a solution. Where the two curves intersect is a value of k that satisfies both the differential equations and the boundary conditions. I think one can tell from that if the value of k is larger or smaller than that for the infinite square well (which is pi/L, right?)

Problems 1 and 2 are introducing a form of wave-function that will be used for guessed molecular states, probably next week. The wave-function of problem 2 is also related to what one typically uses for electron states in semiconductors (and other crystalline solids). (Imagine repeating this over an over with different "rb" values and getting a wave-function with lots and lots of peaks (each one centered over a different atom). (Note however that this is a one-electron wave-function. We will need lots of states like this (an entire band of them) and then consider a description of electrons in semiconductors analogous to the approach to the periodic table: namely filling one-electron states one after another.)

Here I set b=2 and a =1, since otherwise WA might treat b as a 2nd variable...

Check these graphs out also. They illustrate the nature of the integrand for problem 10, in which you were asked to graph the integrand associated with the T calculation for an electron in the ground state wave-function if the 1DHO.