Why are SPIN OPERATORS in the form of MATRICES and not CONTINUOUS?- Tutorial series on Spin [Part 3]
In this video I develop an intuition for spin as a quantum degree of freedom by establishing the form of spin operators in a general sense. The video discusses the physics of multicomponent wavefunctions in quantum mechanics. I also try to give a visualization of multicomponent wavefunctions, as opposed to the scalar or single component wavefunctions, by considering the more simple case of two dimensional space.
Through this visualization, I try to explain the nature of extra freedom in rotation facilitated by these multicomponent wavefunctions, and how this brings spin into the picture. This discussion leads to a conclusion that spin operators are n x n matrices acting on the discrete space in contrast to the orbital angular momentum operator (which are differential operators and act on continuous spaces).
The video also gives a very nice explanation for the commutator relation [L_i, S_j]=0.
Pre-requisites: Basic knowledge of Quantum Physics and the previous lectures of this series.
Previous video [Part 2] : • CONSEQUENCES of QUANTUM SPIN having NO CLA...
Next video [Part 4] : • EIGENSTATES and EIGENVALUES of SPIN OPERAT...
Timestamps:
00:00 Introduction/Recap
03:03 Action of angular momentum operator on state
07:03 State representation: Wave function (continuous) vs Column vector (discrete)
10:04 Rotation of Hybrid state (multicomponent) [Orbital + Spin operator]
15:29 Difference b/w Orbital and Spin angular momentum operators
16:48 Spin implies multicomponent wave function
21:52 Conclusion (Commutator of L and S operators)
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