# Course Description

This one-semester course is designed to provide students with an understanding of the formalism, techniques, and important example problems of quantum mechanics. With this background, students will be prepared for subsequent in-depth studies in optical physics, quantum optics, quantum information, relativistic quantum mechanics and other advanced quantum mechanics topics, condensed matter physics, laser physics, and semiconductor physics. The course emphasizes a formal mathematical treatment of quantum mechanics, and is therefore **intended for students who have either (1) successfully (grade of A or B) completed at least one semester-long introductory course in quantum mechanics where the basic concepts, symbols, and mathematical approaches have been discussed**, or **(2) are strong in mathematical formalism, especially with linear algebra, eigenvalue equations, basis expansions (including Fourier transforms and plane-wave basis expansions), and abstract reasoning, and who are willing to do some extra reading of quantum mechanics concepts when needed. **

## Instructor Information

Instructor: Professor Brian P. Anderson

Department/College: Wyant College of Optical Sciences

Email: bpa@optics.arizona.edu

Campus phone: 520-626-5825

Office: Room 622, Optical Sciences (Meinel Building)

Mailing address: Wyant College of Optical Sciences, 1630 E University Blvd, Tucson, AZ 85721

Teaching Assistant (TA): TA information is announced at the beginning of each semester

## Course Format

OPTI 570 involves required reading assignments (approximately one reading assignment per class period) in which students will be challenged to learn many of the intricate details and derivations of quantum mechanics from the primary required textbook (which will be closely followed throughout the semester). Challenging homework problems will be assigned approximately weekly. Main-campus students will be expected to fully participate in classroom discussions. **Class time will be focused on**

Â· discussing and explaining the topics covered in the textbook

Â· discussion of concepts that are unclear or difficult to understand

Â· brief examples

Â· discussion of material and applications not covered in the textbook.

## Prerequisites

It is expected that students enrolling in this course are either particularly strong in math (esp. linear algebra) or have already studied the following topics in an introductory quantum mechanics course:

Â· deBroglie wavelength of a particle

Â· Schrodingerâs equation

Â· energy eigenstates of example potential wells (free particle, particle in a box, harmonic oscillator, hydrogen atom)

Â· Dirac notation (preferred but not essential)

Â· operator algebra and commutators

Â· angular momentum in quantum mechanics (spin, electron orbital angular momentum)

**It is not necessary that the topics above be fully understood, as they will be covered in detail in OPTI 570. A grade of A or B in an introductory Quantum Mechanics course such as OPTI 345 or OPTI 511R or equivalent is sufficient preparation. Students who have obtained a C in either of these classes or in their prior quantum mechanics course should obtain instructor permission before enrolling. Typically, undergraduate-level “Modern Physics” courses are not introductory Quantum Mechanics courses. **

**Students who have not completed an introductory quantum mechanics with a grade of A or B, including OPTI 511R or an undergraduate course, should consult with me before beginning this course**.

It is also **essential** that students are familiar with and can utilize the following mathematical concepts:

Â· matrix and vector multiplication

Â· finding the eigenvalues and eigenvectors of simple matrices

Â· working with complex numbers

Â· basic formalism of Fourier transform integrals

Â· interpreting and using differential equations (being able to solve challenging differential equations however is not expected or needed in this course)

Students will be expected to review on their own (as needed) these or other introductory or background topics that will be used in OPTI 570 but not covered in detail. Some students who are strong in mathematical formalism and methods, but without previous quantum mechanics experience, may benefit from occasional extra reading from an introductory QM book such as Introduction to Quantum Mechanics by Griffiths.

## Permissions

**The instructorâs permission/consultation is needed to take this class for students who do not have a previous introductory quantum mechanics course with a grade of A or B on a college/university course transcript, for non-degree-seeking students, and for undergraduate students.** **However: sStudents with strong mathematics experiences (such as Math/Applied Math graduate students) or students who have strengths in linear systems and fourier transforms do not need permission to register for this course but are encouraged to discuss their prior academic coursework and expectations with me prior to taking the course. **

## Recitation Sections

On most Wednesday afternoons or evenings, the TA for the course will hold an * optional* help session for about 1.5 hours. These are offered for your benefit, and are a chance for you to ask questions about material that is not making sense to you, and especially to discuss difficulties that you might be having with that weekâs problem set. Problem sets will usually be due on Thursdays for Main Campus students. Although recitation attendance is optional, recitation section attendance is strongly recommended: most students usually end up attending most help sessions. Additionally, 3 Wednesday afternoon sessions from 5-7pm during the semester are scheduled to be utilized for exams.

*(One of these exams might instead be given as a take-home exam). Each exam will be two hours. Exam dates and formats will be announced near the beginning of each semester. Recitation session recordings, extra office hours, and a delayed homework and exam due-date structure will be offered for remote students.*

**These particular sessions will not be optional for main-campus students; attendance at these times is required****unless otherwise arranged prior to the start of the semester.**For students taking this course remotely, recitation sections will either be recorded and distributed, or extra recitation sections or office hours for remote students will be given. Homework due dates for remote students will be delayed by a few days, with homeworks typically due the Tuesday after the Thursday due dates for Main Campus students. Exam dates will be similarly extended.

## Learning Outcomes

Upon completion of this course, it is expected that students will be able to:

Â· interpret and assign physical meaning to the notation and symbols of quantum mechanics when encountering new quantum mechanics problems or reading quantum mechanics literature;

Â· construct equations using the formalism of quantum mechanics that directly correspond with a wide variety of scenarios and processes of the physical world;

Â· solve a wide variety of equations and systems of equations using the formalism of quantum mechanics in order to reach conclusions about physical problems and make predictions about the outcomes of possible measurements;

Â· be prepared to take more advanced courses involving quantum mechanics and quantum physics, such as UA courses OPTI 544, OPTI 646, PHYS 570B, and OPTI 561.

## Required Texts and Materials

**Required:** The primary required course textbook is a hardcopy or electronic copy of Volumes 1 and 2 of *Quantum Mechanics* by C. Cohen-Tannoudji, B. Diu, & F. Laloe, published by Wiley & Sons. The 3^{rd} edition of this classic textbook (printed in 2019) has three volumes. The earlier 1^{st} and 2^{nd} editions each have two volumes, and are no longer in print. New or used copies of Volumes 1 and 2 of *any* edition are required. **PLEASE NOTE: The 3 ^{rd} volume of the 3^{rd} edition is not used and does not need to be purchased.**

**Required:** *Field Guide to Quantum Mechanics*, by Brian P. Anderson, published by SPIE Press, 2019. UA students can obtain a free PDF copy from the SPIE Digital Library website via a subscription held by the UA Libraries. From the UA Libraries main website (library.arizona.edu) perform a **Library Search** with the **title** âField Guide to Quantum Mechanicsâ (including quotes makes the search results easier to navigate). The book should be the only result. Select the book, then select the link for **SPIE Digital Library eBooks**. If you are logged in with your UA NetID, you will then be able to download the entire PDF. You may also purchase a hardcopy of the book through SPIE press.

**Required:** OPTI 570 handwritten class notes. A free PDF copy will be available to all students through this website. Bound copies may be borrowed from a student who previously took OPTI 570 (printed versions of these notes used to be required). These class notes contain content presented in lectures, expanded discussions of the content of the *Field Guide to Quantum Mechanics*, and other **required reading**.

## Schedule of Topics and Activities

OPTI 570 aims to cover the following topics (numbers in parentheses indicate **approximate** number of 75-minute lectures for each topic):

1. Mathematical formalism I. State space and state vectors, scalar product, Dirac notation. Linear operators, Hermitian operators. Representations and bases. Eigenvalue equations, observables, commuting observables. Unitary operators and unitary transformations. (4)

2. Postulates of quantum mechanics. Physical implications, interpretations. Time dependence. Time translation (evolution) operator, SchrĂ¶dinger and Heisenberg pictures. (2)

3. Wavepackets: example to illustrate representations, transformations, translations, other concepts. (1)

4. The harmonic oscillator. Creation and annihilation operators, operator algebra. Solution of the eigenvalue problem. Stationary states in position and momentum representations. Quasi-classical (coherent) states, time evolution of expectation values, comparison to classical harmonic oscillator. (7)

5. Angular momentum. Commutation relations. Angular momentum ladder operators, operator algebra. Solution of the eigenvalue problem (operator approach). Pauli matrices. Stern-Gerlach experiment. Spin 1/2 problem. Two-level systems. Bloch sphere. (4)

6. Eigenvalue problem for the central potential, separation in angular and radial equations. Hydrogen atom. Orbital angular momentum, spherical harmonics. Addition of angular momentum, Clebsch-Gordan coefficients. (4)

7. Stationary perturbation theory. Perturbation equations. Non-degenerate perturbation theory. Degenerate perturbation theory. (2)

8. Fine and hyperfine structure. Corrections to hydrogen atom problem: spin-orbit coupling, relativistic effects, Darwin correction. Fine structure of the n=2 shell in hydrogen. Hyperfine structure. Atomic Structure. (2)

9. Time-dependent perturbation theory. Interaction picture. Perturbation equations, solutions to first and second order, transitions between discrete states, limits of validity. (4)

## Grading Scale and Policies

Each studentâs final course grade will be based on the total points accumulated over the semester. A grade of âAâ will be given for 90-100 total points, âBâ for 80-89 points, âCâ for 70-79 points, etc. For this course, A is interpreted as âExcellent â has demonstrated a more than acceptable understanding of the material; exceptional performance; exceeds expectations,â B is interpreted as âGood â has demonstrated an acceptable understanding of the material; adequate performance; meets expectations,â C is interpreted as âBelow Average â has not demonstrated an acceptable understanding of the material; inadequate performance; does not meet expectations,â D is interpreted as âPoor â little to no demonstrated understanding of the material; exceptionally weak performance.â, and E is interpreted as âFailure â usually reserved for non-attendance.â

Extra credit points may be given for the completion of certain assignments or available on exams, but should not be expected. Students who obtain a low grade on an exam may be given a chance to bring their grade up by completing an extra assignment.

## Administrative Drop

I may make use of the administrative drop option for students who:

- neglect to turn in the first few homework assignments with a demonstrated reasonable attempt at completion
- are clearly (to me) struggling with the material (based on homework submission or grades on the first or second exam) and who are on track to receive a low course grade, if they do not seek out additional help from me or the TA, attend recitation sections, or complete extra assignments that I assign to help bring them up to speed
- do not respond to my requests to meet or discuss their progress in the course, particularly if it appears to me that they are struggling with the material
- are struggling with the course material and have not discussed their enrollment in the course with me, if they are in a category listed above on this page that needs my permission to enroll in the course
- miss an exam

I will notify any such students that they are facing an administrative drop from the course prior to dropping them. I have never had to use the administrative drop before in this course, and I hope that trend continues!

## Assessments

The following assessments and their percentages (weight of final grade) are used to calculate grades for this course.

**Assessments: Percentage of final grade**

Participation, including attendance, asking and attempting to answer questions occasionally in class, asking questions during office hours or recitations when struggling with a topic, etc : 5%

Completing and turning in homework assignments: 11%

First mid-term exam (written, tentatively scheduled to be a 2-hour exam): 21%

Second mid-term exam (written, tentatively scheduled to be a take-home exam): 21%

Third mid-term exam (written, tentatively scheduled to be a 2-hour exam): 21%

Final exam OR final project OR final homework set: 21%

## Attendance and Participation Policies

OPTI 570 follows the Class Attendance and Participation Policies described at https://www.optics.arizona.edu/osc-students/grading-attendance-policies and the University of Arizona academic policies available at https://academicaffairs.arizona.edu/syllabus-policies.

## Subject-To-Change Notice

Information contained in the course syllabus, other than the Grading Scale and Policies and Absence Policies, may be subject to change with reasonable advance notice, as deemed appropriate by the instructor of this course.

## Graduate Student Resources

The University of Arizonaâs Basic Needs Initiative is comprised of programs and resources that can be found at: http://basicneeds.arizona.edu/index.html