Friday, July 27, 2018

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


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 δij, 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.

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