19. Introducing Spin, Rotations and Group Representations | Weinberg’s Lectures on Quantum Mechanics

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0:00 - Introduction
3:02 - Angular Momentum dependence of Energies in Atoms
4:16 - Energies independent of “m”; due to Rotation Invariance
6:47 - Unexplained doublets of Sodium D lines(Fine Structure)
9:03 - Introducing 4th Quantum Number : Spin of Electron
11:08 - Problem with Electron Spinning on its own axis
12:19 - Calculating Electron “radius”
14:23 - Spin as Generator of Rotations
15:54 - Properties of Rotation matrix “R”
17:09 - N-dim Rotations as SO(N)
18:55 - Definition of a Group
21:11 - Parameters of Rotations, ω
22:33 - Counting Rotation parameters
23:41 - Constructing Representations of Rotations
25:28 - Generator of Rotations, “J”, is anti-symmetric
26:21 - A Representation that is equal to “R”(defining rep.)
29:08 - Faithful vs Unfaithful Representations
30:45 - Proof : Generators of Rotations form an Algebra
37:49 - Lie Algebras can be used to build Representations
40:25 - 2-index Generator, transforms like tensor, in any Rep.
42:04 - Commutators between “J” and Vector Operators
44:14 - Relations between 1-index and 2-index “J”
46:46 - Invariance of Levi-Civita symbol under Rotations
47:39 - 1-index “J” Rotates like a Vector
50:28 - Deriving Commutators of 1-index “J”
51:50 - Differences between Total and Orbital Angular Momentum
52:49 - Deriving the Commutators of Spin
55:52 - Connecting Actual Rotations with Representations
58:44 - N-dimensional Representation of SO(3) vs SO(N)
59:35 - Proof : Generators of Lie groups are Traceless
1:00:31 - Ending

This is lecture 19 of the series (part 1 of Chapter 4), where we discuss and explain the book, “Weinberg’s Lectures on Quantum Mechanics”.

Today we move on to chapter 4; on the general theory of rotations and angular momenta, where a new quantum number, spin, is introduced. Spin is recognised to be the intrinsic angular momentum of an electron, and is introduced to explain the doubling of spectral lines(fine structure), first observed in the sodium D lines.

The representation theory of the rotation group is developed. This allows us to define the angular momenta, in a general way, and also tells us how quantum states are rotated. Alone the way, we define the concept of a symmetry group in a rigorous way, focusing on Lie groups; which describe continuous symmetries. The Lie algebra of the rotation group is also derived.

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