20. Angular Momentum States/Values From Commutators | Weinberg’s Lectures on Quantum Mechanics

#QuantumMechanics #StevenWeinberg #AngularMomentum

0:00 - Introduction
3:27 - Ladder Operators : A Reminder
5:57 - Angular Momentum Ladder Operators
8:56 - Some Commutators of Angular Momentum
11:00 - Eigenvalues of J^2 are positive
12:32 - Eigenvalue of J^2 ≥ (J_3)^2
13:15 - Solving f^2 in terms of j
17:29 - Showing that the Highest Weight State is unique
19:24 - Proving j’ = -j (j’ is min. m)
22:21 - Counting states in j-multiplet/the allowable values of “j”
24:38 - Normalisation of jm-states : phase convention
30:52 - Solving the matrix elements of J / Eg. Pauli matrices
36:03 - An Important Theorem on Inner Products in j-Multiplet
39:01 - Ending

This is lecture 20 of the series (part 2 of Chapter 4), where we discuss and explain the book, “Weinberg’s Lectures on Quantum Mechanics”.

In this video we shall give a derivation of all the eigenstates and eigenvalues of angular momentum; using just the algebra of rotations. The results are general and apply to all representations of rotation. It is shown that the angular momentum quantum number, j, can either be integer or half-integer values. This is a well-known and important result in quantum mechanics.

All these are accomplished, by using the angular momentum analog of the ladder operators, introduced in lecture 17 for the harmonic oscillator…

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►Ladder Operators For Harmonic Oscillator(Lecture 17)
https://youtu.be/Ly0GqiabuBE

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https://youtu.be/B_XN0Yfv_qQ
►Previous (Lecture 19)
https://youtu.be/aaTs6bafrHI
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