Quantum Mechanics

PHYS 5115, Spring 2019

Instructor: Professor Jim Halverson

E-mail: j.halverson@northeastern.edu

Phone: 617-373-2957

Office Hours: MR 8:30-10:00 Dana 226

Grader: Anindita Maiti (maiti.a@husky.neu.edu) and Jiahua Tian (tian.jiah@husky.neu.edu)

"Quantum mechanics is the description of the behavior of matter and light in all its details and, in particular, of the happenings on an atomic scale. Things on a very small scale behave like nothing that you have any direct experience about. They do not behave like waves, they do not behave like particles, they do not behave like clouds, or billiard balls, or weights on springs, or like anything that you have ever seen." - Richard Feynman. See here for more.

Course Information

Course Schedule: MR 11:45-1:25

Course Location: Forsyth 202

Description: Lays out the basic axioms of quantum mechanics, and the complementarity between the wave function and matrix mechanics approach to quantum systems. Considers systems of spins in static and dynamical settings, deriving the algebra for addition of angular momenta. Introduces the notion of time evolution and coherent states/wave packets. Demonstrates the solution to numerous one and three-dimensional problems, including the harmonic oscillator and hydrogren atom. Extends these results to new settings via the use of perturbation theory. Some advanced topics will be covered, such as the path integral and GHZ systems.

Course Goals:

• Understand the relation between physical observables and quantum operators.
• To appreciate the need for complex quantum states, and how phases are related to physically meaningful quantities.
• To develop familiarity in solving physical eigenvalue problems.
• To be able to understand the time evolution of dynamical systems.
• To become familiar with the use of raising and lowering operators in various systems.
• To learn basic techniques in the solving of differential equations relevant to quantum systems in one and three dimensions.
• To become adept at the use of approximation methods such as perturbation theory.
• To expose students to topical advanced subjects such as groups and symmetries, entanglement, decoherence and quantum information theory.

Resources: The textbook we will use is A Modern Approach to Quantum Mechanics (2nd ed.) by John S. Townsend. Not all material covered in lectures will be found in the textbook, so attendance at lectures is strongly encouraged. It is assumed that you will have read the relevant textbook passages before attending the lectures.

I will also try to regularly post lecture notes before lecture. They won't be perfect, but small errors not caught beforehand will hopefully be caught during class, and I will update accordingly. See discussion below about why I'm posting notes.

Office Hours: Three hours per week are set aside for office hours, when I will meet with students outside of class. Office hours are primarily for discussion of homework problems or course concepts, though questions about broader areas of physics are welcomed if no one present is looking to discuss course material. Please, come, office hours are fun and students often find them helpful!

Homework: Problem sets will be assigned each week. They will be collected at the start of the lecture on the day they are due. New assignments will be posted to the homework folder. Late assignments will not be collected under any circumstance. Instead, your lowest homework score will be dropped; if it is a missed assignment it is your responsibility to learn the material, since it may appear on the midterm or final exam. You may discuss the assignments with your classmates, but the work you submit must be your own. Homework questions will not be answered via e-mail; please come to office hours. Questions regarding homework grading should be directed to both graders via e-mail.

Midterm: There will be an in-class midterm on February 14 that may test any of the material from homework or class up to that date.

Final Exam: There will be a take home final exam, distributed during the last class. All final exams must be returned to my office, Dana 226, by 9AM on the last day of finals. If I am not in my office, you may put it under my closed door, but you must send me an e-mail alerting me and telling me the time of delivery. The exam is your work only.

Grading: Your final grade will be determined based on your performance on the homework assignments and the final exam. The scoring breakdown is as follows: homework 50%, final 30%, midterm 20%.

Final Letter Grade: Your course grade will be determined both by your overall weighted total score, with the weights given above under “Grading”. The following table indicates ‘target’ overall score ranges corresponding to various final course grades

• A = 93-100%
• A- = 88-92%
• B+ = 84-88%
• B = 78-84%
• B- = 74-78%
• C+ = 70-74%
• C = 65-70%
• C- = 60-65%

Since exams and assignments can vary slightly in difficulty from semester to semester, the actual score ranges may be adjusted slightly downward from those given in the table. That is, if your final percentage falls in one of the above ranges, you will receive the associated letter grade or higher. Under extreme circumstances the ranges may be shifted in the opposite direction, but this will be announced explicitly in class before Mar 20.

Course Material

An approximate outline of the lectures is given below, in order of appearance of topics.

• Superposition
• Quantum states
• Rotations and basis change
• Matrix representation of operators
• Angular momentum
• Eigenvalue problems
• Generalized uncertainty principle
• Heisenberg uncertainty principle
• Time evolution and Schrodinger equation
• Example: ammonia molecule
• Two-particle systems: addition of angular momenta
• Wave functions and spatial translation
• Wave packets and 1D potentials
• Quantum harmonic oscillator (HO)
• Time dependence of HO and coherent states
• Schrodinger equation in three dimensions
• Angular solutions to central potentials
• Hydrogen atom
• Perturbation theory
• Stark effect
• Relativistic corrections to hydrogen atom
• Path integral approach to QM.
• GHZ systems and EPR.

Comments and Recommendations

Lessons from Previous Semesters: Happily, this is my third time teaching this beautiful subject, and based on previous student experience there are some things that you should know at the outset:

• Why post lecture notes? I cannot emphasize this enough: this course moves quickly. As your professor, I owe it to you to prepare you as much as possible for next stages, which means learning a lot of material; I'll explain this in more depth in class. We'll be able to cover a lot of material by stressing a deep understanding of concepts, which will be buttressed by concrete calculations in class, as well as on homeworks and exams. However, this means that information will be coming quickly on the board, and in previous semesters a few students have found it difficult to think about the concepts while also hearing the content of my words and copying down equations. To that end, I want to make your life easier: generally, I will attempt to post lecture notes before class. See description and caveats above.
• Eigenvectors and eigenvalues: they are a major part of the course, and I will review them and all of the other aspects of linear algebra below, but I will not teach you how to actually compute them (just as I will not teach you matrix multiplication), since that's what a linear algebra course is for. Be sure you know how to compute eigenvalues and eigenvectors for 2x2 and 3x3 matrices, because you might need to know it on the midterm, and I will often ask for an interpretation of the result of an eigenvalue or eigenvector calculation --- it is impossible to get the right interpretation without the calculation being done correctly. On homework or the take-home final, feel free to compute the eigenvectors or eigenvalues using your favorite software package (such as Wolfram Alpha) and print the result, because in this course we care about the physics of the eigenvectors, not precisely how they are computed (for which there are many algorithms).

Linear Algebra Background: Quantum mechanics relies heavily on linear algebra. While some of the basics will be covered in the course, I strongly encourage you to review the following concepts:

• Vectors and vector spaces.
• Matrices / linear maps. ("operators" in QM language).
• Inner product / norm.
• Bases.
• Basis change.
• Eigenvalues and eigenvectors.

Study Groups: You are strongly encouraged to form small groups to work together on the homework. This will aid in the learning process, but note that it is very important that you participate in finding the solutions, and you present your own write-up. Doing so is critical for learning the material and doing well on the final, since the latter must be entirely your own work.

Help: If you have trouble with the homework, seek help immediately; do not fall behind in the course. You have several places to go for help: your lecturer (after class or during office hours); the Physics Workshop in 300 Churchill near the physics labs (a schedule should be posted near 111 Dana by the second week of class). There is also peer tutoring available.

Academic Integrity and Misconduct: Be sure to review Northeastern Academic Integrity policies, which are here.

Appropriate disciplinary action, potentially including failing the student, will be taken in the event of cheating, plagiarism, dishonesty, or other academic misconduct. Since students in this course are often encouraged to work in teams, some specific remarks are in order. It is not considered cheating if you:

• Work together on homework assignments, as long as you each work out and submit your own final answers.
• Get help from professors, physics workshop, tutors, etc. on the homework assignments.
• Work together on preparing for exams.

It is considered cheating if you:

• Submit work done by others (without your participation) as your own.
• Copy work on exams.

In addition, please review the relevant College of Science Academic Course Policies