Friday Science 3f: Spin Polarization Principle

Seventh installment summarizing Susskind's, Quantum Mechanics: The Theoretical Minimum.

Chapter 1: Dirac was much smarter than I (introducing linear algebra).
Chapter 2: Quantum States (a.k.a., more linear algebra)
Chapter 3a: Linear Operators
Chapter 3b: Eigenvectors
Chapter 3c: Hermitians and Fundamental Theorem of QM
Chapter 3d: Principles of Quantum Mechanics
Chapter 3e: 3-Vector Operators

Finishing up notes on chapter 3.

3.8 The Spin-Polarization Principle
Any state of a single spin is an eigenvector of some component of the spin.

I wish I more fully understood this principle, but I will do my best. It seems to me it is saying that you're going to get a +1 somewhere.

〈σ_x〉² + 〈σ_y〉² + 〈σ_z〉² = 1

This type of bracket indicates what is called an "expectation value" or the average value of a measurement. The square of the expectation value is the probability of finding a 1 there. So there has to be a 1 somewhere or the probability has to total one.

So, given any state ∣A〉 = α_u∣u〉 + α_d∣d〉

There is some direction 𝜎 ⃗∙𝑛 ̂ ∣A〉 = ∣A〉

3.7 An example
If I had fully followed the matrix analysis of the previous sections, I'm sure this section would be delightful. I get the general sense that he is playing out probabilities in a spherical framework. I generally understand spherical coordinates and the chart on p.89. But I think I'll skip summarizing this section and call chapter 3 concluded.