Over the past few weeks, I've tried to tunnel through the potential barriers to actual science blogging, with mixed success. One of my bigger oversights thus far has been to omit any sort of a "big picture" from my posts, leaving them as little islands in a vast sea of scientific ideas. Today, I'd like to correct that.
My most immediate goal in science blogging has been to explain how physical states function in the beautiful formalism of quantum mechanics. The language of quantum mechanics, however, is one of probabilities, of complex numbers and of linear algebra. Arguably the most fundamental part of the language of quantum mechanics, linear algebra may be roughly thought of as the study of vectors, and how they transform. As I shall discuss in a future post, by using the idea of a basis, we can represent a special kind of vector transformation by an object called a matrix (more generally, an operator). Thus, what we have discussed thus far is not a set of disparate islands so much as a set of stepping stones. If you prefer a more concrete metaphor, we have poured a foundation for future discussions, including a discussion of the quantum state itself.
Once we have the idea of a quantum state, the horizon opens wide for exploration. The quantum state gives us a language in which we can understand seemingly arcane consequences of a world described by quantum mechanics, such as entanglement or superposition. With the formal tools of mathematics at our disposal, we can overcome the limitations of our intuition, so that we can understand even such tricky concepts as these.
One downside to my stepping-stone approach, however, is to seemingly put the concept of a quantum state on a pedestal, inaccessible without a high degree of mathematical maturity. Little could be further from the case. Indeed, the mathematics with which we understand quantum states are not so difficult as they are esoteric. It is my own opinion that these areas of math need not be esoteric, save for that it has been arbitrarily decided upon (at least in my home, the United States) that Math Is Hard, and that concepts such as those discussed here Should Be Left To the Professionals. Bollocks. We live in a probabilistic world, and one in which statistics guide nearly every aspect of society, so why should understanding probability be so inaccessible? While complex numbers are not so manifestly real, even to the point that i is called the imaginary unit, it takes but a small amount of study to see that the complex numbers form an integral part of how we describe reality. Similarly, the concept of a vector may seem too far removed from reality for the layman to pursue, but in many ways, vectors formalize and encode much of our intuition about geometry, and are just as accessible as the sort of geometry that is taught in many grade schools.
No, quantum states are there for those who want them. My goal is to bring the concepts just a little bit closer, and to let the mathematical beauty underlying them shine through just a little bit brighter. In doing so, I won't always go from point to point in most straightforward way, but I ask your patience, for I am going somewhere. With a bit of looking back at what is, I hope you'll agree that we're going somewhere interesting.