2022 Relativistic Quantum Mechanics

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Academic unit or major
Undergraduate major in Physics
Nishida Yusuke 
Class Format
Lecture    (Livestream)
Media-enhanced courses
Day/Period(Room No.)
Mon1-2(W321)  Thr1-2(W321)  
Course number
Academic year
Offered quarter
Syllabus updated
Lecture notes updated
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Course description and aims

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.

Student learning outcomes

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

Competencies that will be developed

Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

Class flow

Lectures by blackboard

Course schedule/Required learning

  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

Out-of-Class Study Time (Preparation and Review)

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 T2SCHOLA.

Reference books, course materials, etc.

K. Nishijima, Relativistic Quantum Mechanics, Baifukan (Japanese)
Y. Kawamura, Relativistic Quantum Mechanics, Shokabo (Japanese)
M. Oka, Quantum Mechanics II, Maruzen (Japanese)

Assessment criteria and methods

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.

Related courses

  • PHY.Q208 : Quantum Mechanics II
  • PHY.Q311 : Quantum Mechanics III
  • PHY.E212 : Electromagnetism II

Prerequisites (i.e., required knowledge, skills, courses, etc.)

No prerequisites are necessary, but enrollment in the related courses is desirable.

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