Guide to HW 6 & solutions

Please turn in HW 6 to Michael Saccone's mailbox in the physics department mailroom by 6 PM on Monday,  May 13

This guide discusses each of the problems in HW6, including a rough estimate of how long they could take to complete. This HW involves 3 parts:
A) problems 1 and 2 are directed creating a basis for understanding the meaning of the
wavefunction, and the related questions: what is \(\Psi^*(x,t)\Psi(x,t)\)? Why is it called a probability density?  You seemed very interesting in that question at the beginning of the quarter and I hope that you still are. It is an important question. I did not feel like I could answer it in a satisfactory and accurate way until we had more of a shared experience working with wave-functions. Anyway, I plan to answer that question and these problems (1&2) will be an important part of that answer.
My time estimate for problems 1 & 2 is as follows:  one hour or thereabouts to ruminate and think. 30 minutes of actual calculation. The actual calculations are overlap integrals*.  These integrals can be quickly done on wolfram alpha. If you find yourself doing spending a lot of time on the integral, that is not good. There is no real value in that.
     The point of the approximation, which is valid when the delta function ground state is very localized, is so that you don't have to even use wolfram alpha to do the integral, and, much more importantly, so that you can see how \(c_1\) is proportional to \(\psi^{ho}_1(x_o)\) and \(|c_1|^2\) is proportional to \(\psi^{*ho}\psi^{ho}\).
Added May 11 (Saturday):  I am thinking that when you apply the approximation you will get
\(c_1 = \sqrt{4a} \: \psi^{ho}_1(0)\) where \(a=\hbar/(m\alpha)\) is the shorter length scale, the one associated with the DF ground state. Does that make sense?
* The probability of the electron being trapped in the ground state of the delta function is \(c^*_1 c_1\). The coefficient \(c_1\) is obtained via an overlap integral involving: 1) the wave-function of the electron at t=0 just when the potential changes, and, 2) the GS wave-function of the delta function potential which comes into existence at t=0. That is: \(c_1 = \int_{-\infty}^{+\infty} \psi^*_1(x) \Psi(x,0) dx\) where \(\psi_1(x)\) is the ground state wave-function of the delta function.
B) Problem 3 may help you understand and think about what an eigenfunction is. If there is any problem you might want to skip due to a time crunch, this would be the one. On the other hand, it might help you feel more grounded in understanding what an energy eigenstates is and it should take only about 1/2 hour, mostly just thinking about it.
May 9. I fixed a typo in the energy \(E_4\) for the extra-credit part of problem 3.

C) Problem 4, about 30 minutes.
Problem 5 & 6. About an hour or so. Mostly for thinking about what to do.
Is it an eigenstate? In problems 5-10, there are wave-functions that are presented in a slightly cryptic way and you are asked to discern whether it is an energy eigenstate or not? This helps you test for yourself whether you understand the structure and nature of quantum mechanics.
In problem 7, it will really help you to factor out the time dependence of the ground state. It is not wrong if you don't do that, but if you do do that, it will help you in two ways. First, it make the problem easier to do. Secondly, it will allow your result to emerge more clearly and you will likely  gain a better understanding of the time dependence of quantum mechanics. If all goes well, you may say, "ah, now I see how it works".
In problems 9 and 10, factor out the time dependence of the ground state! That is, factor out \(e^{-iE_1 t/\hbar}\), and you will be left with time dependence of the form \(e^{-i(E_2-E_1) t/\hbar}\). I think that factoring may help you to more easily see the process by which the time-dependence of quantum mechanics takes place.
Added: also \(E_2= 2 \hbar \omega\),  \(E_3= 3 \hbar \omega\)

Solutions link:
https://drive.google.com/file/d/1HhP5uXyQ3vHf6r77wddhy5AReYQbGfsH/view?usp=sharing


** I will add more to this as more things come up. Today a student and I looked at some interesting contour plots and 2D plots using the command Plot on wolfram alpha. I'll try to add more on that later. Here is a preview which shows a method you can use to explore all sorts of 2D plots and contours.