Physics 3220: Links

Course Goals

Contents: (Click, or scroll down)

What we cover, and why
Expectations
Detailed chapter level content goals are at the bottom!

What we cover, and why:

Physics 3220 is likely to be your second course in quantum mechanics (QM) (following Physics 2170 or 2130), but really the first course in the fundamentals of the theory. We will cover roughly the first half of McIntyre (Phys 4410 continues with more advanced topics)

QM forms the basis for essentially all of modern physics, including nuclear and particle physics, condensed matter and atomic physics. It is also at the heart of a huge variety of modern technological innovations. It was originally developed at the start of the 20th century, a highly collaborative product of some of the greatest minds of physics, to describe the behavior of electrons in atoms. "Electronic physics" is still today the largest area of application (with quantum computing up and coming, perhaps!), but as far as we know quantum mechanics applies to everything. Quantum mechanics is more analogous to Newton's laws than to a specific theory such as Maxwell's in that it is a general framework rather than a description of a particular physical system. It can be thought of as the generalization of Newton's laws that must be used whenever the wave-like properties of matter are important. There are plenty of things that are not understood in physics, but as far as we know no system behaves in a way that is outside the scope of quantum mechanics. Even the speculative theories of physics at ultra high energies (such as string theory) are forms of quantum mechanics.

QM is one of the most practical areas of physics. Until the advent of computers it was hard to apply to most realistic systems, but today it is commonplace to calculate quantum properties of atoms, molecules, and solids. Chemists use QM on a daily basis to predict properties of molecules. The interaction of light with matter plays a role in many areas where QM is essential. Examples in biology include photosynthesis and vision, in astronomy the properties of stars are understood through spectroscopy, and engineers working with lasers, photodetectors, and semiconductor devices used quantum mechanics regularly. In academic labs, many aspects of quantum physics are under active development. Current topics include Bose-Einstein condensation of atomic vapors, superconductivity, quantum cryptography, the quantum Hall effect, quantum computing, and macroscopic quantum coherence. Increasing emphasis on nanotechnology research has brought many applied scientists into the quantum domain.

Since the very first papers of Schroedinger, QM has raised difficult interpretational and philosophical problems. Although some progress has been made, there are still issues that are far from settled, especially those related to the interface between the quantum description and classical reality. Many ideas in QM may seem counter-intuitive, abstract, and /or just plain "weird" to you, but you will find that doing quantum mechanics is perfectly within your grasp. At least at the moment, to understand the physical world, we absolutely need to understand quantum mechanics!

COURSE SCALE LEARNING GOALS

These are broad (course scale) goals - content goals are something different - see below!

Math/physics connection: Students should be able to translate a physical description of a junior-level quantum mechanics problem to a mathematical equation necessary to solve it. Students should be able to explain the physical meaning of the formal and/or mathematical formulation of and/or solution to a junior-level QM problem. Students should be able to achieve physical insight through the mathematics of a problem.
Visualize the problem: Students should be able to sketch the physical parameters of a problem (e.g., psi, potential, probability distribution), as appropriate for a particular problem.

Organized knowledge: Students should be able to articulate the big ideas from each chapter, section, and/or lecture, thus indicating that they have organized their content knowledge. They should be able to filter this knowledge to access the information that they need to apply to a particular physical problem.

Communication. Students should be able to justify and explain their thinking and/or approach to a problem or physical situation, in either written or oral form.

Problem-solving techniques: SStudents should be able to choose and apply the problem-solving technique that is appropriate to a particular problem. This indicates that they have learned the essential features of different problem-solving techniques (eg., separation of variables, power series solutions, operator methods). They should be able to apply these problem-solving approaches to novel contexts (i.e., to solve problems which do not map directly to those in the book), indicating that they understand the essential features of the technique rather than just the mechanics of its application. They should be able to justify their approach for solving a particular problem.

…a. Approximations: Students should be able to recognize when approximations are useful, and use them effectively (eg., when the energy is very high, or barrier width very wide,...). Students should be able to indicate how many terms of a series solution must be retained to obtain a solution of a given order.

…b. Symmetries: Students should be able to recognize symmetries and be able to take advantage of them in order to choose the appropriate method for solving a problem (eg., when parity allows you to eliminate certain solutions).

…c. Matrix methods and Dirac notation: Given a physical situation, students should be able to interpret and compute using Dirac notation, write down the required matrix equation for Schrodinger's equation, and correctly calculate the answer.

…d. differential equation methods: Given a physical situation, students should be able to write down the required (Schrodinger) differential equation, and correctly calculate the answer.

...e. Metacognition: Students should be able to justify their choices in problem solving methods (see LG #4 above) verbally or in writing, and explicitly engage in discussion about their thinking and what helped them learn. (See also LG #6 below) .
Problem-solving strategy: Students should be able to draw upon an organized set of content knowledge (LG#3), and apply problem-solving techniques (LG#4) to that knowledge in order to organize and carry out long analyses of physical problems. They should be able to connect the pieces of a problem to reach the final solution. They should recognize that wrong turns are valuable in learning the material, be able to recover from their mistakes, and persist in working to the solution even though they don’t necessarily see the path to the solution when they begin the problem. Students should be able to articulate what it is that needs to be solved in a particular problem and know when they have solved it.

Expecting and checking solution: When appropriate for a given problem, students should be able to articulate their expectations for the solution to a problem, such as general shape of the wave function, dependence on coordinate choice, and behavior at large distances. For all problems, students should be able to justify the reasonableness of a solution they have reached, by methods such as checking the symmetry of the solution, looking at limits, relating to cases with known solutions, checking units, dimensional analysis, and/or checking the scale/order of magnitude of the answer.

Intellectual maturity: Students should accept responsibility for their own learning. They should be aware of what they do and don’t understand about physical phenomena and classes of problem. This is evidenced by asking sophisticated, specific questions; being able to articulate where in a problem they experienced difficulty; and take action to move beyond that difficulty.

Build on Earlier Material. Students should deepen their understanding of Phys 2170 material. I.e., the course should build on earlier material.

Important comment on preparation:

Physics 3220 is a challenging, upper-division physics course. Unlike earlier courses, you are fully responsible for your own learning. Physics 3220 covers much material you have not seen before, at a higher level of conceptual and mathematical sophistication than you may have encountered in a physics class so far.
Therefore you should expect:

a large amount of material covered quickly.
no recitations, and few examples covered in lecture. Most homework problems are not similar to examples from class.
long, hard homework problems that usually cannot be completed by one individual alone.
challenging exams.

YOU control the pace of the course by asking questions in class. We tend to speak quickly, and questions are important to slow down the lecture. This means that if you don’t understand something, it is your responsibility to ask questions. Attending class and the homework help sessions gives you an opportunity to ask questions. We are here to help you as much as possible, but we need your questions to know what you don’t understand.

Physics 3220 covers some of the most fundamental physics and mathematical methods in the field. Your reward for the hard work and effort will be learning important and elegant material that you will use over and over as a physics major (and beyond!) Here is what we have experienced:

most students reported spending a minimum of 10 hours per week on the homework (!!)
students who didn’t attend the homework help sessions often did poorly in the class.
students reported learning a tremendous amount in this class.

Specific content goals, organized by chapters:

McIntyre Ch 1: (If you see something from Ch 1 you think is MISSING from this list, or that does not BELONG in this list, let me know! )

You should be able to...

...Describe and derive the basic classical physics of the Stern-Gerlach device, including predicting qualitative motion of uncharged classical objects in a given Stern-Gerlach field
…predict the outcome probabilities and outcome "states" along any given paths of Chained-Stern-Gerlach devices of any arbitrary configuration, including ones oriented in X, Y, Z, or n (tilted by theta and phi) directions.
... work backwards along Chained-S-G devices, e.g. given outcomes, decide what S-G orientations were needed or what starting states we began with
...work with basic Dirac notation, including manipulating basic formal expressions (e.g. knowing when term order matters, or which expressions are meaningless), or how to distribute terms involving sums in the bra or ket.
... be able to convert kets to and from bras, normalize kets, find orthogonal kets, compute brackets, and handle the basic complex number manipulations associated with such problems.
... Use Postulate 4 to predict experimental outcomes. This includes working with the formal notation of Postulate 4 and being able to move back and forth between sections of Chained S-G's and the Postulate 4 probability formula.
... generalize to quantum systems beyond spin 1/2 (e.g. doing normalization or use Postulate 4 for spin-1 systems)
... Be able to use matrix notation to compute brackets and express bras and kets

McIntyre Ch 2: You should be able to...
(If you see something you think is MISSING from this list, or that does not BELONG in this list, let me know! )

....represent an operator in the Sz basis, given its eigenvalue equations and eigenvectors.
…diagonalize a 2x2 matrix (find the eigenvalues, and/or given an eigenvalue, find an eigenvector)
... Use the Sx, Sy, and Sz matrices and eigenvectors (which will be given on MY crib sheet, Eq 2.38 on page 41) in problems involving S-G problems, probability predictions, or other typical situations.
... Use the Sn matrix, and eigenvectors (which will be given on MY crib sheet, Eq 2.41 and 2.42 on pp 41-42) to handle chained S-G problems involving N-hat oriented S-G's. This includes understanding how to relate "theta" in the n-hat picture to "theta/2" which appears in the |+/->n kets.
... Use postulate 4 to predict probabilities of measurements given any combination of spin 1/2 states and detectors, or work backwards (given the results, to draw conclusions about the measurement device's orientation)
... identify when a matrix is Hermitian, and form a hermitian conjugate of a 2x2 or 3x3 matrix.
... Know the definition of a "projection operator", Pn = |+n><+n| , and be able to use it to project components of kets, as well as engage in basic formal dirac notation manipulations ( such as what appears in Eq 2.54, or throoughout section 2.2.4)
... Interpret both numerator and denominator of postulate 5, including being able to comput either of those separately given a state and a measurement outcome, and interpret the meaning of Postulate 5 to predict outcome states from chained S-G setups.
... compute "expectation values" of any given operator, interpret this as a sum of possible results*probabilities (as given e.g. in Eq 2.74, but for any operator).
... compute the RMS deviation (Delta) for an operator, given a state.
...construct or interpret probability distributions such as McIntyre's figure 2.8 for arbitrary states and measurements.
... compute the commutator of 2 operators,
... connect commutation (or non-commutation) to measurements ("compatibility") , uncertainties, and whether 2 operators share common eigen-bases.
... Know the definition of S.S (the S^2 operator), and evaluate its effect on spin 1/2 states.
... apply the rules and postulates of quantum mechanics described above to spin 1 systems in simple cases. (e.g. normalizing spin 1 states, finding brackets, determining orthogonality, computing probilities using Postulate 4, predicting output states using Postulate 5, or sketching probability distributions as in e.g. Fig 2.13)
... Generalize to arbitrary system of spin. We will not go too far computationally here - E.g. no diagonalizing of nasty matrices bigger than 2x2 on a test. But e.g. I would expect you to interpret formulas written in n dimensions, "projection operators" in spin-n systems, or compute probabilities in the usual way with Postulate 4 no matter what the dimensionality, at the level shown in e.g McIntyre Chapter 2.8)

That was it for Exam #1, just Ch 1-2.
... Apply the Uncertainty principle (Eq 2.98, which will be on MY crib sheet) to arbitrary spin 1/2 operators and states.
It can be helpful to know (or quickly rederive) the commutation relations in McIntyre Eq 2.96

_____

McIntyre Ch 3 : You should be able to...

....construct a Hamiltonian operator (2x2 or 3x3 matrix) for a spin 1/2 (or 1) particle in an arbitrary B-field
... solve for the "cn" coefficients in the initial state in the |En> basis.
…write down the time dependent solution given initial conditions, and use this to compute probabilities or measurement outcomes or expectation values of any operators of interest at later times. (You should be able to do this even if the field points in a direction other than z-hat, and be able to handle spin-1 as well as spin 1/2)
... qualitatively describe "spin precession", for an *arbitary* magnetic field direction.
... apply Rabi's formula to find "flip probability".
... extend the time-dependent formalism above to "non-spin" systems (neutrino oscillations being one good example, but any 2-state system would work the same)
Note that we did not cover 3.4 in any detail, I will not be testing you on the formulas or formalism in that section.

McIntyre Ch 4 : You should be able to...

....Use, interpret, and generate 2-particle states in the form used in McIntyre, e.g. his Eq 4.1
... Do standard Quantum calculations with 2-state systems including computing brackets and acting single-particle operators on a state (as e.g. done in Eq 4.2 and 4.3) The key here is to recognize that "1" and "2" states and operators ignore states and operators of the opposite number, and otherwise behave as they aways have.
... Given an EPR gedanken experiment setup (like in Fig 4.1) predict how observations at A will correlate with observations at B.
... distinguish "hidden variable" from "quantum" predictions - e.g. be able to decide if a setup will or will not obey the Bell inequalities.

McIntyre Ch 5 : You should be able to...

....Interpret "probability histograms" (like in Fig 5.2), be able to go back and forth between that representation or an expansion of the form |psi> = Sum c_n |En> (where you should be able to compute c_n by evaluating <En|psi> and interpret |c_n|^2 as the robability of finding the energy En, as shown in that histogram.
... know what the x and p operators are in the position representation
... Compute probability of finding a particle within dx of position x using the wave function (i.e. you should know that Prob density = |psi(x)|^2, and you should know the difference between "probability density" and "probability"
...be able to form brackets of position-space wavefunctions by forming an integral, and interpreting the bracket as we have in earlier chapters to compute probabilities, or normalize wavefunctions.
... Compute expectation values of given operators in the position space representation by integrating
... write down the time-independent Schrodinger equation as a 2nd order ODE for a given potential function.
... Solve the infinite square well (even if I make small changes, e.g. the location of the starting/ending points of the well, the width of the well, the value of the base of the well)
... use the rules of QM we have developed to compute the time dependence of superposition states of "infinite well" |En> states, find the probabilities of measuring energy for such time dependent superposition states, and find the position probability density for such states.
... know the standard definitions of wavelength, wave number,
... know and appy the "boundary conditions" - namely, that wave functions are continuous everywhere, d/dx(wave function) is continuous unless V=infinity, in which case the wave function must vanish)
... Be able to write the "general solution" to Schrodinger's equation in regions where V(x)=constant
... solve the "finite square well" problem (even if I make small changes, e.g. the location of the starting/ending points of the well, the width of the well, the value of the base or top of the well)
... Sketch qualitatively correct wave functions even when V(x) is not simple to solve exactly - including getting the sign of the curvature right, the behaviour in "allowed" or "forbidden" regions, application of boundary conditions, getting the relative wavelength right based on KE, and getting the relative amplitudes right based on classical arguments about where the particle spends the most time...
(Note that I am not going to cover computational solutions to Schrodinger's equation on exams)

McIntyre Ch 6 : You should be able to...

... Solve the Schrodinger equation in position space for unbound particles (when V=constant): both the time-independent (exp[i kx]) solutions , knowing what "k" is..., and also the time-dependent solutions.
... Know and use the deBroglie relation for plane waves to relate momentum, wavenumber k, and wavelength.
... know the eigenfunctions of momentum in position space (using the Dirac normalization)
... Be able to use the definitions to compute Fourier transform and Inverse fourier transform (which will be on the crib sheet). I will remind you of any "Gaussian" formulas if you need them, but I would expect you to be able to integrate delta functions or constant functions on your own...
... Have a qualitative understanding of wave packets, including familiarity with terminology like "envelope", "phase velocity" and "group velocity", and "momentum representation"
... Be able to relate the width of wave packets in position and momentum space.
... Use the position-momentum Heisenberg uncertainty relation (which will be on the crib sheet) to make rough estimates of of energy or momentum, like we did in HW or CH 6.3.1
... (As above in item 1) solve the Schrodinger Equation in regions of space where V=constant , using "boundary conditions" on the wave function (continuity of psi and psi'(x)) to piece together solutions of Schrodinger equation for unbound particles hitting a potential well or barrier
...Interpret the coefficients of plane waves as a measure of "flux", thus deciding which (if any) vanish
... Use formulas for T or R (I will give them on my crib sheet if you need them) to make qualitative and quantitative predictions of reflection and transmission, in scattering and tunneling situations. This means you know and can interpret/evaluate the "k" values in the different regions, including forbidden ones.
Section 6.6 will not be on the exam.

McIntyre Ch 7 : You should be able to...

... Recognize situations where you can use the general method of "separation of variables" to simplify partial differential equations, and do so.
... Separate the Center of mass from relative motion coordinates in central potential problems. (You should be able to simply write down the CM part of the wave function, which is a 3-D plane wave)
... know and use spherical coordinates including volume integrals
... be able to compute basic commutation relations involving angular momentum operators, including knowing which ones do and do not commute.
...be able to compute or predict experimental outcomes of measurements (including probabilities, and also the "final state") of L^2 or components of L (Lz, or Ly, or Lx) for states given in the usual form |l, m>
... solve "particle on a ring" problems, including knowing the energy formula, and being able to predict experimental outcomes of energy as well as Lz. You should also be able to add in "time dependence" and comput spatial probability densities for such states.
... solve "particle on a sphere" problems, including knowing the energy formula, and being able to predict experimental outcomes of energy as well as Lz or L^2. You should also be able to add in "time dependence" and comput spatial probability densities for such states.
... Know the solution for the Phi(phi) separated solution of wavefunctions (plane waves in phi), including normalization
... Know and interpret the standard notation of Legendre functions and Spherical harmonics
... Use the postulates of quantum mechanics to compute probabilities of measurements on angular momentum eigenstates (either in position representation, Y_lm(theta,phi) or dirac notation |l,m>

McIntyre Ch 8 : There will not be a whole lot of emphasis on this. But, you shouldstill be able to...

... know and interpret the standard notation of general solutions to the hydrogen atom, psi_n,l,m,
...use the postulates of quantum mechanics to compute probabilities of measurements of energy or angular momentum on hydrogen atom states
... use the standard "time evolution" methods from Ch 3 on hydrogen atom wave functions.
... Answer very basic questions about the spectrum of Hydrogen.

As always, let me know if you think I'm missing some key ideas, I'll add them! If you don't understand what I'm after, or are wondering about some particular topical area and what I might expect from you on an exam - don't hesitate to ask!

Physics 3220 (Fa16)

Course Goals