Friday Science: 3a. Linear Operators

Third installment reviewing 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)

1.  So on to chapter 3 (a.k.a., even more linear algebra). This is really as far as I've gotten in the several times I've started plodding through this book. This week I want to summarize a first few pages from chapter 3. It will probably take two more weeks to finish the chapter.

The first half of the chapter is a mathematical interlude. Once again, I think these interludes are most effective after you have introduced a problem you need to solve. Then the math makes sense as a way to solve the problem. Oh well.

2. I think I'm beginning to get a better sense of what bras and kets are. That it is so simple to say is part of my frustration with Susskind's pedagogy. In laypeople's language, a ket like ∣A〉 is a collection of complex numbers (I could explain complex numbers). By convention, they are written in an up and down matrix like this:

We still don't really know why we would want such a collection, but we are calling this a vector.

Bras are written as horizontal collections of complex numbers and are the complex conjugates of the bra equivalent (I could explain complex conjugates). We call them vectors too. They are written like this:

The "inner product" of a bra and a ket is simply the matrix multiplication of the two (I could explain matrix multiplication).

3. Machines and Matrices
So linear operators are basically matrices that bras and kets are multiplied by using matrix multiplication. John Wheeler, a famous twentieth century physicist, called them "machines." Again, Susskind hasn't really given any sense of why we would need these or when we would use them.

But you basically use them to "operate" on bras and kets. For example, here's a linear operator that you might multiply a bra or ket by:

"Operating" this on a bra or ket is like plugging a number into an equation, except we are multiplying a bra or ket by this matrix.

In notation, we might say M∣A〉 = ∣B〉 . The operator M takes the input and spits out the output. ∣A〉 and ∣B〉 are kets.

4. Linear operators 1) relate to observable features in quantum mechanics (=real not imaginary stuff), 2) are like functions--you need to get an output for every imput, 3) multiplying the input by something needs to get the output multiplied by that something, and 4) whether you do the machine on the sum of vectors to begin with or do it on the sum of the outputs, the result should be the same.

Some of the "observables" you use these in relation to include: position of a particle, its energy, its momentum, its angular momentum, or an electric field at a point in space.

I read more than the six pages this post covers (51-56). If I get a chance before next Friday, I may blog some more, but gotta fly.