It’s actually obvious and might fall under the “Duh!” heading for some, but it only recently sunk in on me that the Born Rule is really just another case of the Pythagorean theorem. The connection is in the way the coefficients of a quantum superposition, when squared, must sum to unity (one).
For that matter, Special Relativity, which is entirely geometric, is yet another example of the Pythagorean theorem, but that’s another story. (One I’ve already told. See: SR #X6: Moving at Light Speed)
The obvious connection is the geometry behind how a quantum state projects onto the basis eigenvectors axes.
In The Road to Reality (2004), Roger Penrose writes about a great analogy for symmetry breaking. Apparently, this analogy is rather common in the literature. (No, it’s not the thing about the pencil — this one involves an iron ball.) Once again, I find myself agreeing with Penrose about something; it is a great analogy.
Symmetry breaking (which can be explicit or spontaneous) is critical in many areas of physics. For instance, it’s instrumental in the Higgs mechanism that’s responsible for the mass of some particles.
The short post is for those interested in physics who (like I) have struggled to understand exactly what symmetry breaking is and why it matters.
Last time I began exploring the similarity between the Schrödinger equation and a classical heat diffusion equation. In both cases, valid solutions push the high curvature parts of their respective functions towards flatness. The effect is generally an averaging out in whatever space the function occupies.
Both equations involve partial derivatives, and I ignored that in our simple one-dimensional case. Regular derivatives were sufficient. But math in two dimensions, let alone in three, requires partial derivatives.
Which were yet another hill I faced trying to understand physics math. If they are as opaque to you as they were to me, read on…
This is the first of a series of posts exploring the mysterious Schrödinger Equation — a central player of quantum mechanics. Previous QM-101 posts have covered important foundational topics. Now it’s time to begin exploring that infamous, and perhaps intimidating, equation.
We’ll start with something similar, a classical equation that, among other things, governs how heat diffuses through a material. For simplicity, we’ll first consider a one-dimensional example — a thin metal rod. (Not truly one-dimensional, but reasonably close.)
Traveller’s Advisory: Math and graphs ahead!
A single line from a blog post I read got me wondering if maybe (just maybe) the answer to a key quantum question has been figuratively lurking under our noses all along.
Put as simply as possible, the question is this: Why is the realm of the very tiny so different from the larger world? (There’s a cosmological question on the other end involving gravity and the realm of the very vast, but that’s another post.)
Here, the answer just might involve the wavelength of matter.
I’ve been working my way through The Principles of Quantum Mechanics (1930), by Paul Dirac. (It’s available as a Kindle eBook for only 6.49 USD.) It’s perhaps best known for being where he defines and describes his 〈bra|ket〉 notation (which I posted about in QM 101: Bra-Ket Notation). More significantly, Dirac shows how to build a mathematical quantum theory from the ground up.
This is not a pop-science book. Common wisdom is that including even a single equation in a science book greatly reduces reader interest. Dirac’s book, in its 82 chapters, has 785 equations! (And no diagrams, which is a pity. I like diagrams.)
What I wanted to post about is something he mentioned about qubits.
In the last four posts (Quantum Measurement, Wavefunction Collapse, Quantum Decoherence, and Measurement Specifics), I’ve explored the conundrum of measurement in quantum mechanics. As always, you should read those before you read this.
Those posts covered a lot of ground, so here I want to summarize and wrap things up. The bottom line is that we use objects with classical properties to observe objects with quantum properties. Our (classical) detectors are like mousetraps with hair-triggers, using stored energy to amplify a quantum interaction to classical levels.
Also, I never got around to objective collapse. Or spin experiments.
In the last three posts (Quantum Measurement, Wavefunction Collapse, and Quantum Decoherence), I’ve explored one of the key conundrums of quantum mechanics, the problem of measurement. If you haven’t read those posts, I recommend doing so now.
I’ve found that, when trying to understand something, it’s very useful to think about concrete real-world examples. Much of my puzzling over measurement involves trying to figure out specific situations and here I’d like to explore some of those.
Starting with Mr. Schrödinger’s infamous cat.
In the last two posts (Quantum Measurement and Wavefunction Collapse), I’ve been exploring the notorious problem of measurement in quantum mechanics. This post picks up where I left off, so if you missed those first two, you should go read them now.
Here I’m going to venture into what we mean by quantum coherence and the Yin to its Yang, quantum decoherence. I’ll start by trying to explain what they are and then what the latter has to do with the measurement problem.
The punchline: Not very much. (But not exactly nothing, either.)
The previous post began an exploration of a key conundrum in quantum physics, the question of measurement and the deeper mystery of the divide between quantum and classical mechanics. This post continues the journey, so if you missed that post, you should go read it now.
Last time, I introduced the notion that “measurement” of a quantum system causes “wavefunction collapse”. In this post I’ll dig more deeply into what that is and why it’s perceived as so disturbing to the theory.
Caveat lector: This post contains a tiny bit of simple geometry.