Friday Science 3b: Eigenvectors

Fourth 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

So I continue working through chapter 3.

1. I think I have seen the term "eigenvectors" and "eigenvalues" since high school. These are more of the words I have heard for decades... they're familiar... but I haven't had any sense of what they meant (other such words include Fourier analysis, which I know a little now, the "Hamiltonian," and more).

So an eigenvector is one that you can put into a matrix machine and it's just like multiplying the matrix by a number.

So if the following matrix

  

is multiplied by the following vector

The answer for the top row is (1 x 1) + (2 x -1) = 1 - 2 = -1 .

The answer for the bottom row is (2 x 1) + (1 x -1) = 1 .

2. Now what is fun is that multiplying the matrix above by the vector is the same as multiplying the top and bottom of the vector by -1. So we say that the 1, -1 vector is an "eigenvector" of the first matrix and that -1 is the "eigenvalue" that goes along with the eigenvector.

In formal language, an eigenvector of M is a vector  ∣λ〉 such that

M∣λ〉 = λ∣λ〉 

and λ is the eigenvalue.

Not much, but it will have to do this week.