In this course I discuss relativistic quantum mechanics. After giving a review on special relativity, I introduce the Klein-Gordon equation as a relativistic generalization of the non-relativistic Schroedinger wave equation and discuss its problem. Then I introduce the Dirac equation, which is the relativistic wave equation for an electron, based on Dirac. Then I discuss the plane wave solution of the Dirac equation, interaction with electromagnetic fields, Lorentz covariance, non-relativistic approximations, an application to hydrogen atom, anti-particle, the Weyl equation and scattering problem. Finally I give an introduction to quantization of fields.
Special relativity theory and quantum mechanics are the most important subjects in modern physics. Learning main ideas unifying these theories and how this unification leads to quantum theory of fields are very important in deeply understanding quantum mechanics and to catch up advanced subjects of modern physics such as elementary particle physics.
You will be able to understand quantum mechanics describing relativistic phenomena, in particular, (1) basics and applications of relativistic quantum mechanics of spin 1/2 particle based on the Dirac equation and (2) an introductory part of quantization of fields.
Special relativity, Klein-Gordon equation, Dirac equation
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | ✔ Practical and/or problem-solving skills |
Lectures by blackboard
Course schedule | Required learning | |
---|---|---|
Class 1 | special relativity | Understand special relativity and Lorentz transformation |
Class 2 | Klein-Gordon equation | Understand neccesity of relativistic wave equation and the Klein-Gordon equation |
Class 3 | Dirac equation | Understand how to derive the Dirac equation |
Class 4 | solution of the Dirac equation for a free partilce | Understand how to find the plane wave solution to the Dirac equation |
Class 5 | interaction with electromagnetic fields and nonrelativistic limit of the Dirac equation | By coupling to the electromagnetic fields and taking the nonrelativistic limit, derive the well-kown non-relativistic Hamiltonian |
Class 6 | Lorentz covariance of the Dirac equation (1) infinitesimal Lorentz transformations and the Lorentz algebra | Understand how the infinitesimal Lorentz transformation forms an algbera |
Class 7 | Lorentz covariance of the Dirac equation (2) Lorentz algbera and Lorentz group | Understand the relation between the algbera of infinitesimal Lorentz transformation and the Lorentz transformation |
Class 8 | Lorentz covariance of the Dirac equation (3) Lorentz transformation of spinors and the plane wave solution constructed from the Lorentz transformation | Understand the properties of the Dirac wave functions under the Lorentz transformations |
Class 9 | non-relativistic approximation and the Foldy-Woutheuysen transformation | Understand the non-relativistic approximation of the Dirac Hamiltonian |
Class 10 | relativistic quantum mechanics | Understand the relativistic correction to the spectrum of the hydrogen atom |
Class 11 | anti-particle, charge conjugation, Weyl equation | Derive the anti-paricle solution from the Dirac equation and understand its physical meaning |
Class 12 | scattering problem (1) Feynman's propagator function | Understand the formalism of scattering problem using the propagator function |
Class 13 | scattering problem (2) Coulomb scattering of an electron | Apply to the scattering problem of electron in Coloumb field |
Class 14 | quantization of scalar and electricmagnetic fields | Understanding the canonical quantization of fields |
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Lecture notes will be distributed via OCW-i.
K. Nishijima, Relativistic Quantum Mechanics, Baifukan (Japanese)
Y. Kawamura, Relativistic Quantum Mechanics, Shokabo (Japanese)
M. Oka, Quantum Mechanics II, Maruzen (Japanese)
Students will be assessed on their understanding of basic ideas in relativistic quantum mechanics and their ability of solving problems.
The scores are based on reports.
No prerequisites are necessary, but enrollment in the related courses is desirable.