Chap. 20 in Shankar: The Dirac equation

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Chap. 20 in Shankar: The Dirac equation PHY 741 Quantum Mechanics 12-12:50 AM MWF Olin 103 Plan for Lecture 32: Chap. 20 in Shankar: The Dirac equation Some simple concepts of special theory of relativity Energy and momentum relationships Dirac equation for a free particle 11/17/2017 PHY 741 Fall 2017 -- Lecture 32

11/17/2017 PHY 741 Fall 2017 -- Lecture 32

Notions of special relativity The basic laws of physics are the same in all frames of reference (at rest or moving at constant velocity). The speed of light in vacuum c is the same in all frames of reference. y’ y v x x’ y’ x’ y x 11/17/2017 PHY 741 Fall 2017 -- Lecture 32

Lorentz transformations y’ y v x x’ y’ x’ y x 11/17/2017 PHY 741 Fall 2017 -- Lecture 32

Lorentz transformations -- continued 11/17/2017 PHY 741 Fall 2017 -- Lecture 32

Examples of other 4-vectors applicable to the Lorentz transformation: 11/17/2017 PHY 741 Fall 2017 -- Lecture 32

Lorentz transformation of the velocity For an infinitesimal increment: 11/17/2017 PHY 741 Fall 2017 -- Lecture 32

Lorentz transformation of the velocity -- continued 11/17/2017 PHY 741 Fall 2017 -- Lecture 32

Example of velocity variation with b: ux/c uy/c 11/17/2017 PHY 741 Fall 2017 -- Lecture 32

Velocity transformations continued: Velocity 4-vector: 11/17/2017 PHY 741 Fall 2017 -- Lecture 32

Some details: 11/17/2017 PHY 741 Fall 2017 -- Lecture 32

Significance of 4-velocity vector: Introduce the “rest” mass m of particle characterized by velocity u: Properties of energy-moment 4-vector: 11/17/2017 PHY 741 Fall 2017 -- Lecture 32

Properties of Energy-momentum 4-vector -- continued 11/17/2017 PHY 741 Fall 2017 -- Lecture 32

Summary of relativistic energy relationships 11/17/2017 PHY 741 Fall 2017 -- Lecture 32

How do these relationships effect quantum mechanics? Focusing on treatment of Fermi particles Non-relativistic mechanics Relativistic mechanics 11/17/2017 PHY 741 Fall 2017 -- Lecture 32

Relativistic relationships – continued Ref: J. J. Sakurai, Advanced Quantum Mechanics 11/17/2017 PHY 741 Fall 2017 -- Lecture 32

Relativistic relationships – continued Ref: J. J. Sakurai, Advanced Quantum Mechanics 11/17/2017 PHY 741 Fall 2017 -- Lecture 32

Relativistic relationships – continued 11/17/2017 PHY 741 Fall 2017 -- Lecture 32

Four component wavefunction of free Fermi particle 11/17/2017 PHY 741 Fall 2017 -- Lecture 32

11/17/2017 PHY 741 Fall 2017 -- Lecture 32

11/17/2017 PHY 741 Fall 2017 -- Lecture 32

What does this all mean? 11/17/2017 PHY 741 Fall 2017 -- Lecture 32