2252022 PHY 742 Spring 2022 -- Lecture 17 1 PHY 742 Quantum Mechanics 12-12:50 PM MWF Olin 103 Plan for Lecture 17: The Dirac equation Chap. 16 in Professor Carlsons textbook: Dirac equation for a free particle Dirac equation for a
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Plan for Lecture 17:
The Dirac equation
Chap. 16 in Professor Carlson’s textbook:
Dirac equation for a free particle
Dirac equation for a hydrogen atom<br>
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2/25/2022 PHY 742 Spring 2022 -- Lecture 17 2<br>
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2/25/2022 PHY 742 Spring 2022 -- Lecture 17 3 P. A. M Dirac 1902-1984
Won Nobel Prize in Physics in 1933
together with Erwin Schrödinger<br>
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2/25/2022 PHY 742 Spring 2022 -- Lecture 17 4 Energy Momentum relationships in classical and quantum mechanics Non-relativistic mechanics Relativistic mechanics Focusing on treatment of Fermi particles Classical Quantum<br>
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2/25/2022 PHY 742 Spring 2022 -- Lecture 17 5 Relativistic relationships – continued
Ref: J. J. Sakurai, Advanced Quantum Mechanics<br>
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2/25/2022 PHY 742 Spring 2022 -- Lecture 17 6 Relativistic relationships – continued
Ref: J. J. Sakurai, Advanced Quantum Mechanics<br>
2/25/2022 PHY 742 Spring 2022 -- Lecture 17 8 Four component wavefunction of free Fermi particle<br>
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2/25/2022 PHY 742 Spring 2022 -- Lecture 17 9<br>
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2/25/2022 PHY 742 Spring 2022 -- Lecture 17 10<br>
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2/25/2022 PHY 742 Spring 2022 -- Lecture 17 11 What does this all mean?<br>
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2/25/2022 PHY 742 Spring 2022 -- Lecture 17 12 Dirac equation for free particles – slightly more convenient notation mc2 -mc2 For free particle, E>0 and E<0 solutions represent distinct physical situations.<br>
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2/25/2022 PHY 742 Spring 2022 -- Lecture 17 13 Dirac equation for a free particle Note: This notation is not entirely consistent with your textbook and is problematic because of other common usages of a and b.<br>
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2/25/2022 PHY 742 Spring 2022 -- Lecture 17 14 Dirac equation for Fermi particle in a scalar potential field<br>
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2/25/2022 PHY 742 Spring 2022 -- Lecture 17 15 Dirac equation for electron in the field of a H-like ion<br>
Eigenstate quantum numbers |nlml>; independent of electron spin
Relativistic spherical atom
Eigenstate quantum numbers |nk jmj>; electron spin included<br>
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2/25/2022 PHY 742 Spring 2022 -- Lecture 17 17 Dirac equation for electron in the field of a H-like ion -- continued<br>
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2/25/2022 PHY 742 Spring 2022 -- Lecture 17 18 Digression – useful identity for operator vectors A and B Note: You might like to verify these relationships!<br>
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2/25/2022 PHY 742 Spring 2022 -- Lecture 17 19 Digression – useful identity for operator vectors A and B -- continued<br>
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2/25/2022 PHY 742 Spring 2022 -- Lecture 17 20 Dirac equation for electron in the field of a H-like ion -- continued<br>
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2/25/2022 PHY 742 Spring 2022 -- Lecture 17 21<br>
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2/25/2022 PHY 742 Spring 2022 -- Lecture 17 22 Dirac equation for electron in the field of a H-like ion -- continued<br>
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2/25/2022 PHY 742 Spring 2022 -- Lecture 17 23 Dirac equation for electron in the field of a H-like ion -- continued<br>
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2/25/2022 PHY 742 Spring 2022 -- Lecture 17 24 More relationships between the operators<br>
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2/25/2022 PHY 742 Spring 2022 -- Lecture 17 25<br>
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2/25/2022 PHY 742 Spring 2022 -- Lecture 17 26 Dirac equation for electron in the field of a H-like ion -- continued<br>
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2/25/2022 PHY 742 Spring 2022 -- Lecture 17 27 Summary of allowed combinations of eigenvalues Alternatively<br>
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2/25/2022 PHY 742 Spring 2022 -- Lecture 17 28 Addition of angular momentum - orbital angular momentum and spin angular momentum<br>
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2/25/2022 PHY 742 Spring 2022 -- Lecture 17 29 Another representation of the spin-angular function in terms of spherical harmonic functions and eigenvectors of sz<br>
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2/25/2022 PHY 742 Spring 2022 -- Lecture 17 30<br>
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2/25/2022 PHY 742 Spring 2022 -- Lecture 17 31 Questions:
How do these results reduce to the Schrödinger equation in the non-relativistic limit?
What new physics is contained more generally?<br>
