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The state of a quantum mechanical system is described by an element , called a ket, from a Hilbert space, i.e., a vector space that is complete and equipped with an inner product as well as a norm related to this inner product. Using bra-ket notation, the inner product of vectors and is represented asContinue reading “Bra-ket formalism of quantum mechanical measurement”

We consider a free particle restricted to a ring of length , with a complete lap around the ring taken to begin at position and end at position . The general TISE is where We take in the ring and we assume the periodic boundary condition . The TISE becomes which has general solution InContinue reading “Free particle in a ring with a periodic boundary condition”

A mass connected to a spring and executing simple harmonic motion will oscillate at a natural frequency which is independent of the initial position or velocity of the mass. The particular pattern of vibration at the natural frequency is referred to as the mode of vibration corresponding to that natural frequency. Obviously, there is onlyContinue reading “Some mathematics relating to phonons”

I was struggling to explain the concept of a solid angle to a student. I found that the following approach, by analogy with plane angles, succeeded. In the case of a plane angle (in radians) subtended by an arc length , the following relationship holds: On the left-hand side we have the ratio of toContinue reading “Quick way to explain the concept of a solid angle”

A Laplace transform of a function is defined by the equation The idea is that a Laplace transform takes as an input a function of , , and yields as the output a function of , . In many applications of this idea, for example when applying a Laplace transform to the solution of aContinue reading “Useful methods in Laplace transform theory”

In this note, I want to quickly record a couple of interesting results involving the beta function. First, consider the reciprocal beta function where It is easy to see by using Stirling’s formula in the numerator and denominator of that for with fixed. Therefore, the reciprocal beta function converges to a simple power law. Second,Continue reading “A couple of results involving the beta function”

I sometimes see things like or instead of the usual . Where do these strange notations come from? Suppose we have a random variable , considered as a function which maps events from a -field to an interval on the real line. Let the random variable take values in the range , which we willContinue reading “Strange notations in mathematical probability”

Suppose we divide a time interval of length into steps. Let be a random variable with initial value . At the first step, the value of the random variable can go up by or down by with probability and respectively. (This is a Bernoulli trial with probability of success ). Then at the first stepContinue reading “From random walk to Brownian motion with drift”

I recently needed to approximate a logarithm of the form where is some large number. It was not possible to use the usual Maclaurin series approximation for because this only holds for . However, the following is a useful trick. We have Therefore, suppose we replace with , where is any large number. Then TheContinue reading “Approximating logarithms of large numbers”

Zipf’s law refers to the phenomenon that many data sets in social and exact sciences are observed to obey a power law of the form with the exponent approximately equal to 2. In the present note I want to set out a simple Yule-Simon process (similar to one first discussed in Simon, H, 1955, OnContinue reading “A simple Yule-Simon process and Zipf’s law”