The Neotropic Book of Physics

State vectors

phalacrocorax — Thu, 21 Jul 2011 17:39:35 +0000

A state vector is a vector in a Hilbert space that complete describes the state of a quantum system. Given an orthogonal basis , each one of its vectors representing a different possible outcome of a complete measurement of the system, any state vector can be written as a linear combination of the basis states with complex coefficients:

In case the measurement can yield continuous results, the summation can be replaced by an integral:

In the most general case, the basis may contain both continuous and discrete labels:

Filed under: Quantum Physics

Characteristic function of a Gaussian distribution

phalacrocorax — Wed, 20 Jul 2011 20:05:46 +0000

The characteristic function of a Gaussian probability distribution will simply be simply the result of the following integral:

The exponentials inside the integral can be turned into a single Gaussian by completing squares:

The remaining integral in can be easily solved as the integral of a Gaussian:

This is simply another Gaussian function:

Filed under: Statistics

Exponential of a projector

phalacrocorax — Wed, 20 Jul 2011 18:17:19 +0000

Let be a scalar and be a projector, thus satisfying for every . An exponential of can be expanded in a power series:

Recognizing that the series in the last right-hand term can be interpreted as another exponential with subtracted, we find:

Filed under: Linear Algebra

Gaussian probability distribution

phalacrocorax — Wed, 20 Jul 2011 15:06:11 +0000

A Gaussian probability distribution will be proportional to a generic Gaussian function:

where 0' title='a > 0' class='latex' /> and .

According to the integral of the Gaussian over the real line, the normalization of this function must be:

Therefore, the constant must be :

Now, the first moment of this distribution can be calculated from the integral of a power multiplied by a Gaussian:

The second moment may be calculated using the same formula:

According to the definition of variance, its value will be:

In this way, we can write this distribution in terms of its average value and its variance :

Filed under: Statistics

Conservative forces

phalacrocorax — Sun, 17 Jul 2011 17:04:57 +0000

A conservative force is a position-dependent force that can be expressed as the negative gradient of a scalar potential :

The work exerted by this force depends only on the value of the potential at the initial point and at the final position , as we conclude by applying Gauss’s theorem:

As the work is equivalent to change in kinetic energy , we find the following law of conservation for the total energy of the system:

Therefore, a conservative force field keeps constant the total (kinetic plus potential) energy of the system.

Filed under: Classical Mechanics

Work exerted on a particle

phalacrocorax — Sun, 17 Jul 2011 14:31:32 +0000

The work exerted by a force on a particle over a path is simply its line integral over the path. Formally,

If is the total force exerted on the particle, Newton’s second law and the fact that give us:

where is the initial time and is the end time of the particle’s trajectory, and we are supposing that the particle’s mass is constant. As it is obvious that:

the work can be expressed as:

This means that the total work exerted on the particle is the same as the change in its kinetic energy.

Filed under: Classical Mechanics

Integral of a centered Gaussian multiplied by a power over the real line

phalacrocorax — Fri, 15 Jul 2011 20:48:50 +0000

We will call those Gaussian functions symmetric in respect with the origin the centered Gaussians. The most general form of the integrals of such centered functions multiplied by powers of is:

Using the result of the general case on the right-hand side, we find:

The only term in the summation that does not vanish is the one where . This is only possible if is an even number ():

which is the same as:

The odd case, where the terms in the summation vanish, is obviously null:

Filed under: Integral Zoo

Integral of a Gaussian multiplied by a power over the real line

phalacrocorax — Fri, 15 Jul 2011 20:20:01 +0000

The general form of the integral of a Gaussian multiplied by the th power of the variable over the whole real line is:

The first thing we must do is change variables to , arriving at:

Therefore, we just need to solve the integrals , which are:

These integrals vanish when is an odd number, as this would turn the integrand into an odd function. Therefore, we just need to be concerned about the functions where , with :

The trick now is to notice that we can express the integrand as a partial derivative of a simpler function:

The last term on the right-hand side is simply an integral of a Gaussian:

Replacing it in the original expression for the integral, we find:

Results for the first powers

:		(Integral of a Gaussian)
:
:
:

Filed under: Integral Zoo

Malthusian growth model

phalacrocorax — Wed, 13 Jul 2011 02:13:28 +0000

The Malthusian growth model for a population states that the number of individuals at instant , represented by the non-negative function , has a rate of change directly proportional to the current population:

where is a positive real number.

This differential equation is solved by an exponential function:

The result for this model is a rapidly growing population.

Filed under: Calculus

Linearity of an average value

phalacrocorax — Mon, 11 Jul 2011 18:40:18 +0000

The average value of a function of a random variable is given by the following integral of the function multiplied by the probability density :

As long as each average value of a series of functions exists, the average value of any linear combination of these functions will be linear:

which is the same as:

Filed under: Statistics