Perturbation theory, as mentioned in an earlier post, is a very important part of the study of many fields but a recurring problem is the issue of summing divergent sequences which sometimes arise in a solution. Even some convergent solutions are very hard to sum because we can only calculate the first two or three termsContinue reading “Some Resummation Theory”
General Relativity is perhaps one of the most enlightening theories in all of physics that reconciles many fundamental ideas about gravity, mass, and spacetime. The understanding of such a theory requires, in some sense, a certain conceptual jump which in the past has been explained through a “stretchy fabric” where mass acts as wells andContinue reading “General Relativity”
This is just a quick fun article that may be trivial to some readers but will probably be very interesting to others. Imagine you jump through a hole that goes through the center of the earth like the little man below.
Integrals are a great point of interest in many areas of mathematics and, when learned about, are often overlooked on the fundamental level. The ideas of Riemann integration, which is what many learn about, are very vast and complex and can provide powerful results but there exists, in some sense, a better and more generalContinue reading “Lebesgue Integration”
There exists a certain class of “hard” problems that can’t be solved with exact form. Examples include solutions to certain differential equation or higher order polynomials like quintics which can’t be solved with a simple cubic formula or quadratic formula. Perturbation theory is a tool commonly used in mathematical physics and can easily provide solutions toContinue reading “Perturbation Theory”
The Heisenberg uncertainty principle seems like a principle that is so fundamentally experimental but it can actually be derived through theory. This requires the consideration of three concepts: the Cauchy-Shwarz inequality, measurement operators, and commutators.
Quantum teleportation is the idea that entangling states can cause very fast information transfer. The name however is misleading as this information transfer is not instantaneous but simply travels at the speed of light. The basis of this idea is based on a very interesting trick that arises out of some of the math ofContinue reading “Quantum Teleportation”
The speed of light may seem like an arbitrary constant of nature but, in some sense, it is actually set by other properties of the world. These other properties are the strengths of the electric and magnetic fields which are defined by the constants that are used in determination of them, otherwise known as theContinue reading “Speed of Light Derivation”
There is a certain luxury of circuit calculations for systems contain direct current that alternating current systems really do not have. It is the idea that voltage and current are “synced.” An increase in voltage will create a corresponding increase in current seemingly instantaneously. However, an alternating current that experiences voltage oscillations experiences a delay.Continue reading “Complex Impedance”
Complex exponentials are used immensely in math and as a result, in many fields of science. It is also used in abundance throughout this site so it is important to understand what they are for future reference. They show the relationship between exponentials and trigonometry on a fundamental level. The following is the relationship.