Friday Science 3e: Three-Vector Operators

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

Much is still not clicking but let me finish what I can of chapter three.

3.5 A Common Misconception
1. Measurements in quantum mechanics correlate with operators. However, the two are not exactly the same. Measurements come up with definite answers. For example, if you are measuring a particular spin vector, it will either be 1 or -1.

By contrast, the operator has to do more with the probability of a certain outcome. Operators are mathematical rather than actual. They are the tools used to calculate eigenvalues and eigenvectors. We use them on state vectors like "up" "down" "right" "left" "in" and "out." And the result of his operation is another state vector combination that may involve square roots and imaginary numbers--things you will never get in an actual measurement. The actual measurement is always either 1 or -1.

3.6 3-Vector Operators Revisited
2. So Susskind distinguishes three types of vector in this section. The first is a 3-vector space like we use in ordinary directions in life (two miles south, then a mile east, on the sixth floor).

Then he's been talking about state vectors like up, down, right, left, in, out. These are metaphors, I think.

Now he speaks of spin components x, y, and z. He calls these operators, written as matrices. They are the three measurable components of spin. He calls them a new kind of 3-vector, a 3-vector operator. I don't seem to understand, but I'm going with it.

3. Now what if we want to measure spin in any direction sigma n, where n is the direction? Then we can break down the spin in this direction to

σ_n = σ_xn_x + σ_yn_y + σ_zn_z

So if we use the Pauli matrices from the previous post for the components of sigma, we can express the spin in that direction as:

Susskind does some matrix voodoo to combine all these into one big matrix.

Apparently, if we know the eigenvectors and eigenvalues of this particular σ_n, we can use this matrix to calculate all the probabilities for all the outcomes of our measurements of the spin.