The Theoretical Minimum: Quantum Mechanics

Always been a bit of a science geek, but my last real quantum encounter was with that Black Mirror episode, Joan is Awful. Then, out of nowhere, I found myself deep in a book on quantum mechanics, a book that used to be just confusing jargon to me. Thanks to some college Linear Algebra, it’s started making sense. It’s weirdly thrilling, to realize that quantum mechanics is not just sci-fi stuff — it’s real, and it’s so weird.

What’s exactly “weird”?

Everyone talks about Quantum Mechanics as this weird world where a cat can be both dead and alive, and particles are playing dice games. But when you really get down to it, most people can’t explain how it's weird beyond these buzzwords like entanglement, tunneling etc.

Leonard Susskind, a renowned theoretical physicist, and Art Friedman, a data consultant who also happens to be one of Leonard’s students and, crucially, a fiddle player—teamed up to write ‘The Theoretical Minimum Vol.2: Quantum Mechanics’ back in 2014. It’s been a few years, but don’t dismiss it as outdated. This book is still one of the best resources to understand Quantum Mechanics — without the hand-waving.

Curious about what freaked Einstein out about ‘spooky action’? Want to get your head around the Uncertainty Principle? This book breaks it down.

The Book Overview

It’s not exactly Pop-Sci. It looks like one, especially with that cover and size, but it’s quite different from any pop-sci book I’ve read. The pages are filled with math. No, don't run away now — as it turns out, that’s how you really start doing Quantum Mechanics.

The book integrates the necessary math in separate sections, allowing readers to choose their level of engagement. It starts with state vectors and operators, using bra-ket notation, and then covers measurements, entanglement, and time-evolution, with spin-states as practical examples. The discussion on Bell’s theorem is more conceptual than detailed. Readers who complete the book should have a good grasp of Bell’s theorem. The later chapters introduce the general wave function for particles, the Schrödinger equation, the derivation of the uncertainty principle, and a brief look at path integrals. The book concludes with a section on the harmonic oscillator, hinting at quantum field theory.

Want to taste a little weirdness? Keep Reading (Grab a Pencil, I’ve freestyled a bit.)

Systems and Experiments

The spin of an electron is about as quantum mechanical as a system can be, and any attempt to visualize it classically will badly missed the point.

The isolated quantum spin exemplifies a broader class of simple systems known as qubits. Susskind discusses an experiment where the first interaction with the apparatus prepares the system in one of two states. Subsequent experiments confirm this state. During the experiment, changes in sigma suggest that sigma represents a degree of freedom associated with a directional sense in space. Were we to conceive of a spin vector, it would logically be described by three components (sigma_x, sigma_y, sigma_z).

For example, if we prepare the state sigma_z as +1 and then measure sigma_x, classical expectations would suggest a result around 0. However, the actual measurement of sigma_x consistently yields either +1 or -1. Only the average of these repeated measurements behaves classically — as the expectation value is 0.

Each experiment is invasive; measuring one spin component obliterates information about another.

Then, Susskind talks about classical and quantum propositions. Say, proposition A is the z component of the spin is +1, and proposition B is the x component of the spin is +1. Let’s say the unknown agent has set things up. Then we discover that sigma_z equals +1, and now we know that (A or B) is true. But, now let’s reverse the order of measurement, starting with measuring sigma_x; if it turns out that sigma_x equals +1 are finished, (B or A) is true. Suppose we find the opposite result, sigma_x equals -1. So now, with a probability of 1/4, we find that (B or A) is false. This occurs despite the fact that the hidden agent had originally made sure that sigma_z equals +1. Inclusive OR is not symmetric in quantum mechanics. This defies classical logic, as the truth of propositions (A or B) may depend on the order of their confirmation. Worse, (A and B) is not even confirmable. In quantum physics, the second part of an experiment can interfere with confirming the first part. Thus, propositions like ‘the particle has position X’ or ‘the particle’s momentum is Y’ become meaningless.

Susskind then introduces mathematical concepts, starting with dual number systems and the phase factor as a complex number whose real component is 1. He describes entities with magnitude and direction in space as 3-vectors. Vector spaces used to define quantum mechanical states are called Hilbert spaces, which may have a finite or infinite number of dimensions. 3-vectors form a real vector space, while bras and kets form a complex vector space. Susskind also discusses inner products and orthonormal basis. The dimension of a space is defined as the maximum number of mutually orthogonal vectors it can contain.

Quantum States

Knowing a quantum state means knowing as much as can be known about how the system was prepared.

If the apparatus is oriented along the Z axis, the two possible states that can be prepared correspond to sigma_z equals +1 or -1. Let's call them up and down and denote them by ket-vectors u and d. If the apparatus is oriented along the X-axis and registers -1, the state |l> has been prepared.

The idea that there are no hidden variables has a very simple mathematical representation: the space of states for a single spin has only two dimensions.

To describe any quantum state of the spin, we choose |u> and |d> as the two basis vectors. Any state can then be written as a linear combination of these two. The components of such a state vector are probability amplitudes, which, when squared, give the probability of finding the electron in that state upon measurement. It’s important to note that the state vector represents the potential outcomes of a measurement, not the actual values obtained from the measurement. Furthermore, the terms ‘up’ and ‘down’ in this context should not be mistaken for literal directions in space; they are simply labels associated with the state vectors in the abstract space of quantum states.

The Perfect Transition to Real Science

So, I’m going to go ahead and assume that you breezed through the above paragraphs, and everything made total sense. But hey, if it didn’t, that’s totally okay. Just grab a copy of this book and just let it hang out on your shelf. You never know; one day, you might be as bored as I was and decide to give it a spin (pun intended.)

But, when you finally close this book, you’ll realize you’re not just flipping through Pop-Sci stuff anymore. You’ve stepped up to the real-deal science. It’s like signing off from your Pop-Sci Era — a fitting end, a perfect transition. Making it through this book isn’t easy. It’s like a hike up a steep hill — tough for sure, but the view at the top? This is what you were depriving yourself of by reading only Pop-Sci.

The Theoretical Minimum
_The Theoretical Minimum is a series of Stanford Continuing Studies courses taught by world renowned physicist Leonard…_theoreticalminimum.com

I am leaving a link to the website to accompany the lectures in the book! Happy Learning.