PH451 Capstone Quantum Mechanics

Course Name: Capstone: Quantum Mechanics

Course Number: PH451

Course Credits: This course combines approximately 90 hours of instruction, and assignments for 3 credits.

Prerequisites: (PH 424 or PH 524) and (PH 425 or PH 525) and (PH 426 or PH 526).

Course Description: Quantum mechanics is one of the two cornerstones upon which rests the modern theoretical conception of the rules by which the physical universe operates (the other stone being relativity.) Quantum mechanics is also beautiful. It is impossible to overstate the importance to a modern physics education of a sound understanding of, and facility with, quantum physics.

In this course, we will build on the foundation of understanding of the basic formalism of quantum mechanics and its applications laid in the Paradigms in Physics courses, applying it to a range of important physical systems, including quantum mechanical oscillating systems, the spectral structure of the hydrogen atom, and many-body systems. We will also explore techniques for solving problems too difficult to analyze exactly by means of approximation techniques such as perturbation theory. If time allows, we will finish the course by exploring some of the more unsettling implications of the quantum picture of the world as embodied in the EPR “paradox” and Bell inequalities, as well as the enormous practical utility of such quantum strangeness as implemented in quantum computers..

Course Content: Wave mechanics, Schrödinger equation, operators, harmonic oscillator, identical particles, atomic fine structure, approximation methods and applications.

Course Specific Measurable Student Learning Outcomes:

The emphasis in this course will be on learning by doing. The only way to really learn to understand the behaviour of quantum mechanical systems quantitatively is to analytically solve a lot of them by hand. We will solve many problems together, some of them using the power of computer to visualize the behaviour of quantum systems, and to explore computationally systems which are not easily solved with pencil and paper.

By the end of the course, students in PHY 451 should be familiar with and display some practical mastery of the concepts, definitions, and problem solving methods associated with the following topic areas: Mathematical tools, including linear algebra (vector spaces, orthonormal bases, inner products, matrices, linear operators, eigenvalue problems, projection operators and resolutions of the identity); Fourier series and Fourier transforms;

Dirac delta function; solution of ordinary and partial differential equations.

Computational tools such as Mathematica, Matlab, Python, or similar tools or software, for assistance with difficult calculations, numerical computations, and visualization.

Properties of solutions to the eigenvalue problem for the simple harmonic oscillator and associated mathematical methods (including raising and lowering operators). Coherent states. Applications to molecular and other vibrating quantum systems.

Properties of angular momentum and addition of angular momenta.

Approximation methods in quantum theory, including time-independent perturbation theory for approximate solutions to the energy eigenvalue problem and variational methods.

Quantum tunneling and quantum devices.

The structure of Hydrogenic atoms and perturbations thereof.

Time-dependent perturbation theory for quantum transition probabilities. Interaction of atomic systems with electromagnetic fields.

Properties of many-particle systems including systems of identical particles. Composite systems. Quantum Bose-Einstein and Fermi-Dirac statistics. Introduction to quantum statistical mechanics.

Basic appreciation of some of the ways in which quantum mechanics challenges our classical intuition about how

physical systems behave, including a quantitative understanding of Bell’s inequalities. (Call this one a “stretch goal”.)

This course is not a Bacc Core Course.

Evaluation of Student Performance:

The final score is calculated based on homework (30%), project paper (15%), midterm exam (25%) and final exam (30%).

Homework assignments are distributed through Canvas, and are expected before midnight of the due dates uploading to Canvas. No late homework will be accepted without prior consent.

Homework are best hand in pdf format so we can comment electronically. Using Latex is high recommended. Other editing programs, such as MS Word or Mathematica can export pdf as well. If you prefer handwriting, make sure it is clearly written and if possible, scan or take a picture then convert to pdf files.

(very) Tentative Schedule