The goal of this assignment is to help focus your review and understanding of 1D quantum bound states and their time dependence. Please feel free and encouraged to post comments and questions here.
Problems 1-9 are really short. You can think of 1-4 as 4 parts of one problem, and 5-9 as 5 parts of a 2nd problem. 14 is the most important problem. Problems 1-11 are pretty much warm-ups for 14. Feel free to skip 12 (and 13) in order to get to 14 if you are running low on time.
PS. (1-10-12) We are going to set up a box in the physics department mailroom where you can turn in your homework up to 3 PM Friday. Don't be late as the reader will come and pick it up just after the 3 PM deadline.
If you post questions here, I will try to answer them.
ReplyDeleteFor number 14 can we use the ground state and first excited state wave functions you gave in class today for the simple harmonic oscillator?
ReplyDeleteThanks
yes. were they normalized?
ReplyDeleteYes, they were.
DeletePerfect then!
DeleteDoes anyone want a tif or a png (snapshot in a still picture) of the HW?
ReplyDeleteyes please
Deletewhen looking at number 7 and 8 i keep finding that the indefinite integral involves a Gaussian error function that seems to keep repeating. when i plugged into wolfram from negative infinity to infinity it gave me a very definite answer but i cannot figure out how to take the integral using methods we already know. can you point me in the right direction?
ReplyDeleteIt has been a while (since grad school) that I tried doing those sorts of integrals by hand, so I am not really sure. Yes, those error functions appear whenever one does integrals of Gaussian-type functions over finite intervals. However, I think that the error function has the integration limits in its argument, and that the error function would tend to go to zero in the limit that the integration limits go to infinity?
Deletei dont know if i quite understand what you mean. can you explain a little more?
Deletehttp://en.wikipedia.org/wiki/Gaussian_integral
DeleteCheck this out of you like. I think it explains the Gaussian integrals. But really for our purposes in this class, this is a side issue. All we need is the result (e.g., sqrt(pi)). Our focus is on the phenomenology of the normalized quantum states.
A method of solving 7 is taught in multivariable calculus. It is not an error function since it is a definite integral over the region -inf to inf. You can do it by squaring the integral and solving it in polar coordinates (you rotate the x-axis so that it creates a plane (this is what squaring does) and integrate under the resulting 3-dimensional gaussian which is possible to do with a transformation to polar coordinates).
Delete8 should be solvable by changing the integrand into -1/2(x/a) * -2(x/a)*exp([x/a]^2) and doing integration by parts.
However, since this is physics, I am guessing that we are not expected to be mathematically thorough in our solutions.
Trus. You do not need to show any of the work involved in solving the integrals.
DeleteIt wasn't clear to me what wave functions we were supposed to use in problems 14, 15 and 16 on the homework. There doesn't seem to be a consensus as to a standard form in the different resources I've looked at.
ReplyDeleteExcellent question. I will post response addressing this later today.
DeleteWell I think a number of other people were also unclear about this point (from looking at the HW turned in Friday), so it is good that you asked. I only wish someone had asked before the HW is due, but since these are important problems that we will refer back to throughout the quarter, it is still very helpful that you asked this.
DeleteThe short answer is that the wave functions you were supposed to use were presented in the post "What this class is all about and review of 1D quantum physics" (the 2nd video). Also, they were presented and discussed in exactly the form you want on the first day of class.
I can see why you might have been confused looking at a variety of resources, because even though the wave functions are actually always the same, they can look different. Many references do not use a length scale to simplify and some use omega instead of k, etc. So they can look different although actually they are not.