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This is Part II of my blog post series giving an introduction to the basic principles of quantum mechanics. In this post, we will cover the importance of linearity in quantum mechanics and develop some intuition regarding the relationship between linear operators (which act on functions) and matrices (which act on vectors).
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Depending on the degree of experience you have with physics, you may or may not have encountered a few concepts from quantum mechanics in your college physics or chemistry class. Often, the way that quantum mechanics is first introduced to students is through some of the key experiments in which we observe some aspect of so-called “quantum weirdness”. Somehow, when we take a peek at the world of very tiny things, we tend to observe phenomena that we would never expect to see on the scale at which we experience the everyday world. In this multiple-part blog post, I will give a survey of the basic principles of quantum mechanics and show how it has changed our perspective on the macroscopic world.
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In this post, we will pick up from where we left off in Part 2 of this blog post and finish the proof that the Domino problem (introduced in Part 1) is undecidable:
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In this post, we will pick up from where we left off in Part 1. So far, we have introduced Turing Machines (TMs) and Universal Turing Machines (UTMs). Now let’s see how they help us in proving the undecidability of the domino problem we encountered earlier.
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In mathematics, it is often assumed that every well-posed yes or no question about some object or property has a definite answer. Despite this assumption, there still exist many important questions in mathematics that have failed to admit some kind of answer. Among these are some of the most challenging unsolved problems faced by modern mathematicians, such as the Riemann Hypothesis and the P vs. NP Problem. However, there are some questions we can pose in mathematics that we can actually prove are “unprovable”, meaning we can show with absolute certainty that no method for determining a yes or no answer exists. In the field of computer science, these are referred to as undecidable problems. In this two-part (or possibly three-part) blog post, we will be doing a deep dive into the world of undecidability and introduce some important concepts from mathematics and computer science along the way.
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Welcome to my blog! Here I will do my best to give occasional expositions on topics that I find interesting, profound, or meaningful. I’m not sure what kind of trajectory this blog will take, but my goal is that these posts will each serve as self-contained tutorials that provide a deep-dive into a technical topic that I find to be either relevant to my area of research or interesting in its own right. In these posts, I will do my best to make the content accessible to readers with a technical background, while not providing redundant information that would otherwise be a Google search away.