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Chap. 20 in Shankar: The Dirac equation

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1 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 Lecture 32

2 11/17/2017 PHY 741 Fall Lecture 32

3 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 Lecture 32

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

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

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

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

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

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

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

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

12 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 Lecture 32

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

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

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

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

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

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

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

20 11/17/2017 PHY 741 Fall Lecture 32

21 11/17/2017 PHY 741 Fall Lecture 32

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


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