Friday Science 3c. Hermitians and Fundamental Theorem of Quantum Mechanics

Fifth 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

More on chapter 3. I increasingly get the sense that this book should have been written somewhat in the reverse order that he did. Typical linear, building block thinking. Most human minds--especially those this book is allegedly written for, like me--work on a "need to know" basis. That's how this book should be written.

1. Made some progress this week in the book. Think I'm further than I've ever been in it, understanding more than I ever had. Probably could handle a re-read. Since I'm making good progress, I'll try just to jot down some notes.

A "Hermitian" conjugate is like the complex conjugate of a matrix. You do two things to a matrix to find its Hermitian conjugate:

• Interchange the rows and columns (so m₂₃ becomes m₃₂)
• Complex conjugate each matrix element.

A Hermitian conjugate is denoted by a dagger. So the Hermitian conjugage of M is M^† . The matrix might have a T in its upper right hand (for "transposed").

• So you might say that M^† = [M^T]*    (transposed and conjugates).
• So if M∣A〉 = B then 〈A∣M^† = 〈B∣

2. A Hermitian operator is one that is equal to its Hermitian conjugate: M = M^†

The eigenvalues of a Hermitian operator are all real.

3. We now get to what Susskind calls the fundamental theorem of quantum mechanics. It amounts to this: "Observable quantities in quantum mechanics are represented by Hermitian operators" (64). Another way to put it is that "The eigenvectors of a Hermitian operator form an orthonormal basis."

Here is my interpretation of how he unpacks it:

• The possible vectors for a Hermitian operator are all of its eigenvectors and their sums.
• The unequal eigenvalues of a Hermitian are orthogonal.
• Even equal eigenvalues can be analyzed as orthogonal. In other words, two eigenvectors can have the same eigenvalue. This is called "degeneracy."
• If a space is N-dimensional, there will be N orthonormal eigenvectors.

4. The Gram-Schmidt procedure is a procedure for teasing out orthonormal sets that relate to degenerated eigenvectors with the same eigenvalues. Here is the procedure:

• Divide vector one by its own length to get the first orthonormal basis of unit length.
• "Project" the second vector onto that unit vector by taking the inner product with it. 〈V₂∣v₁〉. 
• Subtract this from the second vector.
• Then divide the result by the length of the second vector to get an orthonormal basis for the second vector of unit length.

I don't entirely follow, but I'm making progress.