Note that for problem 5 the most important part is the last step where we use the eigenvectors to match initial conditions and thereby infer time dependence (see last paragraph of #5).
These review-related problems (0-5) are due in class Thursday, April 2. You can have more time for problem 6 if you like. Comments and questions are welcome and encouraged.
PS. (added 3-29) If you put your picture in your profile, that will help me get to know people. (You can always change it later, after a few weeks?, if you want.)
#0. What is your sense of the basic postulates or starting point of quantum physics? (no need to do a big research project. Just what you know now is fine.) What is quantum physics? What does it tell us? What is it important to?
#1. (Wave eqn problem) a) Write the wave equation for a string under tension. (Wave eqn refers to the thing with the spatial and temporal derivatives and the wave-speed.) b) To explore a little, express the wave-speed in terms of T (tension) and m/L (mass density) and c) rewrite the wave equation with just m or m/L on the "right-hand side" (with the time derivatives) (and T moved over to the other side).
d) Compare this to the 1-dimensional, time-dependent Schrodinger wave equation (with no potential), if you are familiar with that.
#2. (String problem) Consider a string of mass m and stretched between two posts a distance L apart.
a) What are the solutions of the wave equation for the transverse motion of this string? (I guess what I am really asking here is what are the normal modes.)
b) For an initial string displacement y(x) (at t=0), write an expression for the time dependent displacement, y(x,t).
c) Create and simple, but non-trivial, example of time-dependent displacement.
Discuss and elaborate.
3. (Linear algebra) Write one or more well-constructed paragraphs expressing your understanding of linear algebra. Emphasis on concepts and words such as "basis", linear independence, spanning, orthogonality, inner-product and normalization is encouraged.
4. (Hermitian) Discuss Hermitian matrices. What are they? What are they characteristics of their eigenvalues and eigenvectors? What is an eigenvector? What is an eigenvalue?
Please discuss and explain, and provide functional definitions and give an example or two.
5. Consider a linear system (1 dimensional) of 2 equal masses, m, and three springs.
||-----------m--------m-----------||
For our purposes, let us assume that the two outer springs are identical (spring constant k) but that the inner spring, the one connecting the two masses together, can be stronger, weaker or the same as k.
a) (What are our purposes? Why is it helpful for the masses and the outer springs to be the same? (The answer can be one word, though more nuance is welcome.)
Assume that there is an equilibrium configuration for the system such that all 3 springs are at their natural length and let us call the deviations from that: x1(t), for the position of the left hand mass, and x2(t) for the right-hand mass. Suppose that at t=0 the left-hand mass at x1(o)=1 meter (or x_0 if you prefer) and the other mass is at either zero or also 1 meter. (These are two different problems with 2 different solutions.)
b) Solve for x1(t) and x2(t).
(hint: How do you approach/begin this problem? Please comment below.)
6. What is your understanding of the origin of the size of the hydrogen atom. (Just H. No discussion of multi-electron atoms please.) (Size, not mass.)
Question 0: I started writing this out, then noticed you want problems 1-5. I'm going to finish writing it out so I can look at it again at the end of the quarter and see how much my response would have changed...but is this something you want turned in?
ReplyDelete-Edolfo
Yes. You are doing the right thing.
ReplyDeleteI should have said 0 to 5; I added in #0 last, and forgot to change that. I'll edit the post to change that.
for the string problem can we assume the initial condition of the velocity is zero at time zero (d/dt)y(x,0)=0.
ReplyDeleteRicardo: Good point. Yes. Without that the problem is ambiguous. (Also, this is a good thing to take note of since it is different for this wave eqn and the Sch. wave eqn that we use in the quantum theory.)
ReplyDeletePS. Does everyone know how to do 5? It is quick and easy if you know how. A 2x2 matrix which reflects the symmetry of the system should emerge and... well, you know what to do. (Because of the symmetry the eigen vectors are "nice".)
#5: http://mathbin.net/7467
ReplyDeleteNot sure how much detail I should post here.
You can post any level of detail.
ReplyDeleteTo be clear, for 5b you are supposed to have a clear, precise (and simple) solution, not just start the problem.
ReplyDeleteFor 2b, are we supposed to make up some initial y(x) for a plucked string or just say it's amplitude*sin(pi*x/L)? Then do we do Fourier analysis to find the coefficients and write the expression?
ReplyDeleteI know it's probably a lot more work, but the Lagrangian approach (like we did a hundred times in phys 105) to problem 5 is more familiar to me than the matrix approach even though it's probably pretty much the same thing. I'm just not that comfortable with matrix representations yet, although I'm working on it.
ReplyDeleteIn solving problem 5, the following may prove useful (taken from Boas p.134 [8.9]).
ReplyDelete"A system of n homogeneous equation in n unknowns has nontrivial solutions if and only if the determinant of the matrix of coefficients is equal to zero."
- Adi
So I found two different eigenvectors for the two masses, and I know how to describe the motion of the masses by inspecting the eigenvectors, but I'm not sure how to put the motion into formula form, or how to apply the initial conditions. I saw the form of solution Edolfo posted on Mathbin, But if I were to use that with my eigenfrequencies, my answer would turn out imaginary (from the exponent). Hmmm.
ReplyDeleteOh, and for the "how do you approach/begin this problem?" question:
ReplyDeleteI found the total kinetic energy of the system and the total potential energy of the system. I made two different matrices for the kinetic and potential energies, and used the relation:
det(A - [w^2]m) = 0
Where A is the potential energy matrix, w^2 are the eigenfrequencies, and m is the kinetic energy matrix.
Voila.
The WebCT page for PHYS105 is still up from fall quarter, and this problem is similar to Homework 9 Problem 5.
Beth,
ReplyDelete"do we do Fourier analysis to find the coefficients and write the expression?"
Yes exactly. No need to actually calculate the coefficients. You can leave them as integrals. So you can call the intitial displacement f(x); no need to actually specify it since you are not doing the integrals anyway. The important thing is the structure and the time dependence (and x dependence) of the individual terms of the fourier series. (Write it as an infinite series with undetermined coefficients.)
The important thing to recognize is that the normal modes span the space of "all" initial displacements, therefore at t=0 we can write any initial displacement as a superposition of normal modes (a fourier series expansion), and that that then determines the time dependence...
"or just say it's amplitude*sin(pi*x/L)"
Well no. then there would be just one term.
My approach to #5 would be to get the eigenvectors from equations of motion which could be based on either Lagrangian or Newtonian (F=ma) methods. Either way you will get (1,1) for the lower frequency mode*, in which the two masses move in concert("in phase"; and (1,-1) for the (higher frequency) mode in which the two masses move in opposition(180 "out of phase").
ReplyDelete*mode 1
(Amazingly) these two eigenvectors span a two-dimensional vector space that encompasses all possible motions of the system. For the initial conditions in which both masses are displaced (by equal amounts), that is a (1,1)-like displacement and is orthogonal to (1,-1). You get pure motion of mode 1, the low frequency mode. To represent the displacement, at t=0, of just the left-hand mass, you need an equal mix of (1,1) and (1,-1). That will have more complex time dependence.
I would suggest just using cos(wt) for the time dependence. The sin(wt) terms, which would also be there in general, are absent due to having initial speeds of zero. The e(iwt) notation is confusing because this motion is real. It also obscures something in the time dependence that is different for this problem and QM.
just something i found while thinking about #6
ReplyDeletehttp://www.phrenopolis.com/perspective/atom/
Oh, and by the way,I the matrix those come from is, I believe,
ReplyDelete| k+g, g |
| g, k+g |
within an overall minus sign.
(and then there is a mw^2 on the left...)
But really the most point on enphasize here is in how you use the eigenvectors to "represent" the initial conditions in each case.
Wilson, that is pretty cool and very interesting. There are sort of two ways to think about the electron: one is that it is this tiny object zipping (or orbiting) around a proton; the other is that it is an inherently extended object that adopts a size dependent on its environment. Whether matter is mostly empty, or not, depends on how we resolve that critical issue.
ReplyDeletein the sense that it is very different from how we might looking at the results from quantum theory
oops, never mind that last bit. editing mishap
ReplyDeleteOne last question, before I go to sleep. Is this assignment too long? If you want to agree on an appropriate way to modify it by some sort of consensus here, that is fine with me. Don't stay up too late. Tomorrow is a big day with lots of expectation (values), and I promise to try not to get all confused.
ReplyDeleteTo anyone watching:
ReplyDeleteOn 2c, are we supposed to solve the equation from 2b with our choice of expression for y(x), or draw some diagrams with wave superposition?
I think I heard both discussed in section, and the phrasing of the question is really broad.
I was just wondering what general approach other people took.
kcollier, I just set y(x) and found the coefficients.
ReplyDelete