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Week of Dec 1
Readings: Shankar Ch. 3
After some preliminaries and logistics, I will discuss the reasons we need to go beyond classical mechanics to describe particles and waves. With the example of the double-slit experiment, I will show how a wavefunction describing a single particle allows us to describe something which has both wave-like and particle-like behavior. I will cover the Born interpretation of the wavefunction in terms of probabilities. I'll then briefly introduce a few issues of interpretation which we may come back to later in the course.
Readings: Shankar 1.1-1.4. Material we didn't cover in lecture that you are responsible for: Gram-Schmidt, Schwarz and triangle inequalities, subspaces.
We'll begin a comprehensive review of the math tools required for this course: vector spaces, linear independence, basis vectors, inner products, bras and kets, adjoint operations. We will follow Shankar's content and notation closely for this review. If any of this material is unfamiliar (or you are rusty), it will be worth your while to spend time on it until you are comfortable. If you know the material well, you may still want to read the chapter carefully to familiarize yourself with the specific notation and terminology we will be using.
Readings: Shankar 1.5-1.6
We'll continue with the math review, and discuss linear operators and their matrix representations, projection operators, Hermitian and unitary operators.
Readings: Shankar 1.7-1.8
And more math: active and passive transformations, eigenvalues and eigenvectors, diagonalization. There are some topics that I won't cover that you should read up on in Shankar: degeneracy and simultaneous diagonalization of Hermitian matrices.
Readings: Shankar 1.8-1.10
We'll first go over in some detail a classical example which illustrates use of the linear algebra tools we've just seen; we'll be applying very similar machinery to quantum mechanical problems soon. Then we will begin to discuss infinite dimensional vector spaces of functions, which are the kind of vector spaces we'll be dealing with for most of the rest of the course. I will skip section 1.9 in lecture, but you should read it, and the problems will be on the homework. Be sure also to read all of the material on Dirac delta functions, which will also be needed for your homework.
Readings: Shankar 1.10
This will be the last lecture of math review (except for a tiny bit next class). We will continue with the discussion of infinite-dimensional vector spaces, Dirac delta functions, and representations of operators in such spaces.
Readings: Shankar Chapter 2, especially 2.1, 2.5, 2.7
After finishing up a little bit leftover from last class (K and X operators), I will review briefly some key concepts from classical mechanics: generalized coordinates, Lagrangian and Hamiltonian formulations. I will not cover this in as much detail as in Shankar. We may come back to some of the later material in Chapter 2 as needed.
Readings: Shankar 4.1-4.2
Now that we're tooled up, we will start covering quantum mechanics proper. The material in Chapter 4 is the most important material of the course; much of the subsequent course material will be applications of these ideas. In this lecture I will list the postulates of quantum mechanics, pointing out similarities and differences to their analogues in classical mechanics. We will then start to explore the consequences of these postulates: we'll discuss particle states, operators corresponding to observables, calculation of probabilities, normalization of wavefunctions, degeneracies and "collapse of the wavefunction".
Readings: Shankar 4.2 (skip density matrix section), but be sure to read the material on degeneracies and "complete sets of commuting observables", which will not be covered in lecture.
We'll continue to explore consequences of the postulates: I'll define expectation and uncertainty, look at what happens when you have simultaneous observables, and go through the example of a Gaussian wavepacket.
Readings: Shankar 4.3. The last few pages of this chapter, while worth reading, are less important.
We'll finish up a little bit leftover from last lecture (plane waves), and then I will discuss the Schrödinger equation, which governs evolution of wavevectors. We'll talk about the quantum Hamiltonian, and stationary states.
Readings: Shankar 5.1-5.2
We will discuss free particle solutions of the Schrödinger equation, and evolution in time of the Gaussian wavepacket. We'll then begin discussing the particle-in-a-box example.
Readings: Shankar 5.2-5.3
I'll go over the continuity (matching) conditions for wavefunctions, and then we'll go through the particle-in-a-box example in some detail. I'll then start introducing some more tools: we'll generalize to 3D and we'll start on the "continuity equation for probability" (to be finished next class).
Readings: Shankar 5.3-5.4. We won't cover 5.5-5.6 in class but you should read them.
I will discuss the continuity equation for probability, and the concept of probability current, by making an analogy to classical electromagnetism. Then we'll look at the example of scattering off a step potential barrier in some detail. I will skip much of the computational detail of Shankar, but instead will try to give you a guide to his steps for calculating the time propagation of the wavefunction.
Readings: Shankar Chapter 6, 7.1-7.3
We will consider the conditions under which we can treat a system as classical. As time permits, we will begin to discuss the quantum harmonic oscillator, an extremely important system.
Readings: Shankar 7.1-7.3.
Guest lecturer: Dr. Springer
Readings: Shankar 7.4-7.5
Guest lecturer: Dr. Springer
Readings: Shankar 7.4-7.5
Guest lecturer: Dr. Chandrasekharan
Readings: Shankar 10.1-10.2
Guest lecturer: Dr. Chandrasekharan
Readings: Shankar 10.3 (I will not cover in class the sections ``Determination of Particle Statistics" and ``When Can We Ignore Symmetrization and Antisymmetrization?", but you should read them.)
I will briefly review direct product and treatment of multiple particles and coordinates. Then I will discuss what happens in QM if particles are identical. We will talk about symmetric and antisymmetric states, Hilbert spaces of bosons and fermions, and normalization of multiparticle states.
Readings: Shankar 2.7-2.8 (skip last section on S and E), 11.1-11.2. Be sure to read the material on pp. 289-290 on finite translations.
First we'll review classical mechanics material on canonical transformations, and active and passive transformations. Then I'll introduce the idea of a symmetry (invariance under transformation), and we'll look at the consequences of symmetries in CM. The most important idea here is that if the Hamiltonian is invariant under some infinitesimal transformation "generated" by a dynamical variable, then that variable is a constant of the motion. We'll look in some detail at the example of translational invariance. Next, we'll look at analogues of these ideas in quantum mechanics, again taking the example of translational invariance. I will not be covering all of Chapter 11, but will highlight the most important ideas.
Readings: Shankar 11.3, 11.4. I recommend skipping the last part of 11.4 on the parity violation experiment (it's confusing, and not really quite right).
We will discuss consequences of invariance under time translation. We'll then consider parity, a discrete transformation, and talk about eigenstates of parity. I will discuss the famous experiment showing that the weak interaction is not parity invariant.
Readings: Shankar Chapter 12. The most important sections for this lecture are 12.2 up to p. 310, 12.3, 12.4, 12.5 after p. 333, beginning of 12.6.
I will cover some of the most important points of Chapter 12. We will look first at rotations in the x-y plane, see how infinitesimal rotations are generated by the z-component of angular momentum, and consider solutions of problems with rotational invariance. Then we will do the same for rotations in 3D, and will find that the solutions are spherical harmonics.
Readings: Shankar Chapter 12.6 (up to p. 345), 13.1
We will look at an important specific case of the rotationally invariant Hamiltonian-- the hydrogen atom, which can be treated as an electron in the attractive Coulomb potential of the proton. Here, we already know the angular part of the solution (since we found last class that the angular wavefunctions are spherical harmonics), and we are faced with solving the radial equation with the Coulomb potential. We'll do this by a power series method, and eventually obtain the famous hydrogenic wavefunctions and the quantized Bohr energies.
Readings: Shankar 12.5, pp. 321-329
I will go back to cover a few more things about angular momentum: in particular, we will look at how to use angular momentum ladder operators to find the eigenvalues of angular momentum. This will lead into a discussion of spin, to be continued next lecture.
Readings: Shankar 13.1, especially pp. 373-379, properties of the Pauli matrices (pp. 381-383)
After a brief review of the angular momentum discussion from last lecture, I will discuss experimental evidence for half-integer eigenvalues of the z component of angular momentum: this can be interpreted as ``spin", an intrinsic angular momentum. By looking at the matrix elements of the components of J, we can determine spin operator matrices. Finally I will discuss various notations for describing particles with spin.
The review session will be from Noon-1:30 pm in Room 154.