Friday Science: Susskind 4a: Unitarity

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
Chapter 3f: Spin Polarization Principle

Now that I'm done reviewing Hawking, I thought I would return to wending my way through Susskind's book. Here beginneth chapter 4.

4.1 Classical Reminder
A state is the way something is at a particular point in time. The main rule for how states change in classical mechanics is deterministic. If you know the formula, you know what the next state is going to be. The second rule is reversibility. If you know the state now and the formula for change, then you know what the previous state was too.

If two identical systems have the same state at some point in time, then their past and future is the same as well. We call this "unitarity."

4.2 Unitarity
So consider a closed system. Let's use the Greek letter psi to indicate the quantum state of something: |ψ⟩ . To say that "the state was |ψ⟩ at time t, we will use the notation |ψ(t)⟩ . In a sense, this notation |ψ(t)⟩ represents the entire history of the system.

Assuming a system has unitarity, we can use the operator U to say this too:

|ψ(t)⟩ = U(t)|ψ(0)⟩ 

The entire history of the states of a system is this "time-development operator" producing a series of states that start at time 0.

4.3 Determinism in Quantum Mechanics
The development of a state vector in quantum mechanics is deterministic just like in classical mechanics but with one very significant difference. In classical mechanics, determinism tells us the result of the next experiment with certainty. In quantum systems, it tells us the probabilities of the outcomes of later experiments.

4.4 Closer look at U(t)
1. This time-development operator in quantum mechanics must be linear. That means that for every time you put in, you get one quantum state out and the relationship between the two develops at a constant ratio.

2. The unitarity operator also implies that if two basis vectors are orthogonal (are distinguishable), then they will always be orthogonal. This is called "the conservation of distinctions." This means that, for example:

⟨ψ(t)|Φ(t)⟩ = 0

if these two functions are orthogonal.

3. Susskind then shows that for unitary operators

U^†U = I

where U^† is the Hermitian conjugation of U [1] and I is the "unit matrix." The unit matrix is one where, when multiplied by something, results in the same matrix. It is a matrix with all ones down its diagonal and zeros everywhere else.

It has the equivalent result to the Kronecker delta δ_ij, which yields the value 1 when two things with the same basis vector are multiplied but 0 when orthonormal basis vectors are multiplied.

4. This adds a fifth principle to quantum mechanics. The evolution of state-vectors with time is unitary.

[1] As a reminder, an operator is Hermitian if the matrix version and its transposed version (where you interchange the rows for the columns) yield an equivalent result.

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.

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.