I left off last time with a brief introduction to uncertainty, followed by two classes worth of background, both mathematical and Mathematica.
Class 15 picked up the physics again, starting with an explanation of the connection between the Fourier theorem and uncertainty, namely that any attempt to construct a wavefunction that has both particle and wave propertied will necessarily involve some uncertainty in both position and momentum. This is basically Chapter 2 of the book-in-progress, with a bit more math.
After that, I start laying out quantum mechanics in a more formal way, stating four fundamental elements of the theory:
- Everything is described by a wavefunction
- Only certain wavefunctions, the eigenstates are mathematically allowed
- The wavefunction describes the probability of finding each of the eigenstates
- Measurement is an active process, and will return one and only one eigenstate.
This is a good chunk of Chapter 3 of the book-in-progress, with a bit more math.
Finally, I introduce the Schrödinger Equation, after drawing analogies between the way it's used and the solutions to Newton's Second Law (which determines how a particle's position and velocity evolve in time), and the classical wave equation (which you solve to find constraints and relationships between the wave parameters). This is not remotely like anything in the book-in-progress, to the great relief of all my prospective readers.
Class 16 dealt with the mathematical apparatus for solving the Schrödinger Equation and interpreting the results. I introduce the Born probability rule (the squared norm of the wavefunction is the probability density) and the idea of normalization, and grind through an example. Then I talk a bit about the properties that must be shared by all valid wavefunctions, basically that they need to be normalizable and differentiable.
Having established that wavefunctions give probability distributions, I introduce the idea of expectation values by an analogy to the center of mass. You can sort of think about the process of calculating an expectation value as taking a weighted average of some property, using the probability distribution as the weighting function. This isn't exactly correct, but it's a nice plausibility argument for the odd operation that is taking an expectation value.
I go through a couple of examples, which leads into another mathematical interlude, this one on parity. In particular, I show that a function of odd parity (that is, an f(x) that changes sign when you change x to -x) integrated over all space will integrate to zero. Since this is exactly the sort of integration that crops up in expectation value problems, this is a critically important technique.
(I put at least one exam problem on the second midterm and the final that is an ungodly slog for anybody who doesn't use parity to eliminate some integration. I'm not sure this actually has the desired result of fixing the importance of the technique in their minds, as opposed to the undesired result of convincing them I'm a total bastard.)
Class 17 introduces the time-independent Schrödinger Equation, and whips through the boring solutions: constant potential, step potential with E>V, step potential with E<V, barrier potential with E<V, and barrier potential with E>V. Actually, I didn't get to the last one, but it's not that important, and I just let them read about it in the notes.
This includes a bit of boundary matching, calculation of transmission and reflection coefficients, and a brief discussion of tunneling. It goes by really fast, though.
Class 18 goes through the solutions of the Infinite Square Well potential, and shows how it is that the Schrödinger Equation very naturally leads to quantization: only certain standing wave modes will be valid solutions of the equation, giving you a discrete set of possible energies. I go through this one in a good deal of detail, taking most of the class.
At the end of class, I set up the simple harmonic oscillator potential, and introduce the solutions. We don't go through the process of solving it, but just assert that Hermite polynomials are the solutions, and leave it at that. I talk very, very briefly about how anything in the universe can be made to look sort of like a harmonic oscillator, which makes this a critically important solution.
After that, there's supposed to be a really short discussion of the correspondence principle, the idea that as a quantum system gains more energy, it starts to look more classical. I ran out of time, though, so it wound up being left in the notes.
Class 19 was lost to post-Boskone bronchitis. I couldn't swallow when I woke up Monday morning, let alone speak well enough to do a lecture, so I cancelled class.
Class 20 was spent on using Mathematica to solve for the eigenstates of a finite square well potential. I give them a notebook with most of the code in place, and ask them to select appropriate boundary conditions. Most of the class figured it out fairly quickly, and then it's just a matter of trial-and-error, putting in various test energies, and trying to find ones that give reasonably good wavefunctions.
Thursday was the second mid-term exam, and Friday's class (which was supposed to be Wednesday) will be a quick survey of weird quantum stuff-- the measurement problem, a bit about cat states and interpretations and other stuff that turns up in Chapter 3 of the book-in-progress. I like this as a last-class-before-the-midterm thing, as it ties a lot of stuff together in a way that doesn't lend itself to homework problems or test questions. It ought to do equally well as a first-class-after-the-test, too. If nothing else, it will put an end to this groggy, phlegmy week.
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"I left off last time with a brief introduction to uncertainty"
Are you sure?
I don't understand what your fundamental element 2 means... It's certainly not true that only eigenstates of the Hamiltonian, or of position, for example, are acceptable as particle wavefunctions. (Indeed, your elements 3 and 4 imply that the initial, pre-measurement state might not be an eigenstate of the operator you are measuring.) Perhaps you mean to say that any state must be expressible as a *sum* of eigenstates of, for example, the Hamiltonian. But if I remember my QM (and the associated math) correctly, this will be true for any sufficiently well-behaved (square-integrable?) function. Is this what you were trying to get at?
But if I remember my QM (and the associated math) correctly, this will be true for any sufficiently well-behaved (square-integrable?) function.
Your point regarding Chad's #2 is correct: only when you make a measurement will the wavefunction be collapsed into some eigenstate of the operator corresponding to your measurement. But I would qualify the above quoted statement: the domain of the function must also match the domain of the eigenfunctions you are considering. All of the eigenfunctions for the particle-in-a-box problem vanish outside the box, so any function which does not vanish outside the box cannot be represented as a sum of particle-in-a-box eigenfunctions. Likewise, you can express any function on the range [-1,1] as a sum of Legendre polynomials, but there is no guarantee of said Legendre series representing the function outside that range.
Ditto comment #2.
And I'm against the qualifiers of comment #3: the particle-in-a-box potential is merely an approximation for a potential depth >> the particle energy and a wall "stiffness" >> the k vectors involved in the wavefunction. For any real potential, any initial wavefunction should be possible (if experimentally inconvenient to create).
This sounds like an excellent curriculum!
I enjoyed the 2-day Western States Mathematical Physics Meeting, the start of this week, hosted at Caltech by the amazing Barry Simon. It was more than half about Schrödinger Equation, as seen from a dozen different approaches. Kooman's theorem and point perturbation; Coupled random matrices and biorthogonal polynomials; Random matrices with Poisson eigenvalue statistics; Random matrices with external source; A family of Schrödinger operators; Vortices and spontaneous symmetry breaking in rotating Bose gases; One-dimensional Anderson model with nonhomogeneous disorder; Asymptotic expansion of the integrated density of states of a
two-dimensional periodic Schrödinger operator...
It's still a few steps from your course to the students being able to read cutting-edge papers in the field, but you've taken them the huge first step: getting them to think like a quantum physicist.
In 1929, Dirac sailed from America to Japan with Werner Heisenberg. During the trip, Heisenberg spent the evenings dancing while Dirac looked on, puzzled. Eventually Dirac asked his friend why he danced.
Heisenberg replied, "Well, when there are nice girls it is a pleasure to dance."
After thinking for 5 minutes, Dirac said: "But how do you know beforehand that the girls are nice?"
The list of elements as it appears here is a rapidly generated copy of what I say at more length in class and in the book. Element #2 is intended to refer to the fact that the system will only ever be observed to be in one of the eigenstates of whatever basis you're measuring in. Totally arbitrary wavefunctions can, of course, be constructed as a sum of eigenstates, but when a measurement is made, only one eigenstate will be found.
The list ends up feeling a little arbitrary, particularly in its ordering-- why are eigenstates before measurement?-- but it's an attempt to distill out the essential elements and present them in an order that makes some sense, providing a frame for the rest of the material. I've waffled about the order of 2 and 3 over the years (as seen in different versions of my lecture notes). The current order puts the idea of eigenstates earlier because that's the bit that puts the "quantum" in "quantum mechanics," though I probably ought to flip 2 and 3 because I talk about the Born rule before talking about eigenstates.
I, too, am concerned about the residual uncertainty in your assertion about eigenstates. ;-)
The standard approach to QM is to posit a thing called a wave function. For any given space, there are typically a number of different bases for decomposing the wave function. For example, a particle in some state | Ï > in 1d can be decomposed in position basis < x| Ï > or in momentum basis < p| Ï > . I think it is misleading to called the basis elements (e.g. |p > ) "eigenstates" before you introduce the concept of "operator". It's probably less confusing to call these eigenstates "basis states". When you say "only eigenstates are allowed", I think that is confusing -- what does "allowed" mean? Ever? You mean particles can only exist in eigenstates? (That would be incorrect, I think). Eigenstates for what operator?
The point is that a measurement corresponds to applying an operator to a ket (say, applying the position operator to | Ï > ), and the operation spits out an eigenvalue of that particular operator and projects the wave function into that eigenstate (and the probability of getting the result x depends on the amplitude | < x | Ï > |² ). If the operator does not commute with the Hamiltonian, then the eigenstate is not stationary and the different components (in the energy basis!) evolve at different rates....
BBB
Re: "1. Everything is described by a wavefunction. 2. Only certain wavefunctions, the eigenstates, are mathematically allowed. 3. The wavefunction describes the probability of finding each of the eigenstates. 4. Measurement is an active process, and will return one and only one eigenstate."
Sounds rather decoherent to me...