Author Archives: Dr Christian P. H. Salas

The Legendre transform is a mechanism for converting a relationship like into a relationship like In other words, by going from a function to its Legendre transform and vice versa, the roles of and in (1) and (2) can be reversed. In moving from (1) to (2), the required Legendre transform is because differentiating bothContinue reading “The role of the Legendre transform in large deviation theory”

Markov’s inequality is a basic tool for putting upper bounds on tail probabilities. To derive it, suppose is a non-negative random variable. We want to put an upper bound on the tail probability where . We do this by observing that where is the indicator function which takes the value 1 when and the valueContinue reading “Markov’s inequality and the Cramér-Chernoff bounding method”

In relation to a calculation involving sums of random variables, I needed to differentiate with respect to a double integral of the form The parameter appears in both upper limits, but appears on its own in the outer upper limit, and as in the inner upper limit. I went through the motions of using Liebniz’sContinue reading “Differentiating a double integral with respect to its upper limits”

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”