Pages: 24

Charge is an intrinsic property of the electron. There is no such thing as an electron without its charge. No free particles have ever been observed with fractional electron charge. The function Quantity[“ElementaryCharge”] is used to acquire the fundamental electric charge in Mathematica. The charge is elegantly displayed as e and has units assigned. At the heart of modern physics lies the fact that measured quantities such as energy and angular momentum are not continuous. Instead, they take on discrete values, a concept referred to as quantization. The fine structure constant appears frequently in atomic physics which is described by the electromagnetic force strength combined with the quantization in terms of Planck's constant.

Pages: 17

In its classical interpretation, radiation comes from atomic oscillators having a continuous probability distribution, which is distributed exponentially. The exponential distribution has a number of remarkale properties. To get the total radiation energy density, one needs to count the number of oscillators and then multiply by the average energy per oscillator. The average energy comes from an exponential probability distribution. The small-wavelength correction to the Rayleigh-Jeans formula was obtained by Planck who assumed that the atomic oscillators that produce the radiation do not have a continuous distribution of energy levels, but that they were quantized with energies. The peak wavelength value of the thermal spectrum is found by taking its derivative with respect to lamda and setting it equal to 0.

Pages: 27

Radioactive decay produced the mystery particles alpha, beta, and gamma (He nucleus, electron, photon) and played a central role at the beginning of the modern physics era. Study of these decays revealed the structure of the atom at a time when alpha, beta, and gamma were still unknown radiation phenomena. Decay always occur if they are not forbidden. Alpha decay is a common decay of a heavy nucleus. The exponential distribution has the special property that the rms deviation from the mean is equal to the mean. The charge-to-mass ratio of the electron was measured by J. J. Thompson who built the first mass spectrometer consisting of parallel plates having perpendicular electric and magnetic fields. The idea of the spectrometer is that an electron velocity is very difficult to measure but its path is relatively easy.

Pages: 13

The first postulate of special relativity states that the laws of physics are identical in all inertial (non-accelerated) frames of reference. The second postulate states that the speed of light in vacuum (c) is the same for all observers. Now consider the motion as viewed from a frame S' that moves with a speed v in the -x direction. In this frame both points A and B move with speed v in the x direction. In the time that light is traveling, point B is moving and the light has a longer distance to travel to get there. Its speed is fixed so it takes a longer time to get from A to B. Since the speed of light is fixed, the two frames are related by the velocity triangle.

Pages: 10

The Bohr model is historically important in the understanding of quantum physics. It gives many important results for the hydrogen atom including atomic size and ionization energy. It is sometimes referred to as a semi-classical calculation, because it includes the physics of angular momentum quantization without the full quantum mechanical description of an electron wave. It is tremendously useful for gaining physical intuition of the quantum nature of the atom. In the Bohr model of the hydrogen atom, an electron has a circular orbit with speed v and radius r about the proton. The Bohr quantization condition is that an integer number of electron (de Broglie) wavelengths must fit to the atomic circumference.

Pages: 20

The particle in a box, also referred to as “infinite square well” is an important example because it illustrates the many features of the solution to the Schrödinger equation in an example requiring minimal math. The problem will be worked first in one dimension. The box is defined by a potential that is zero in some region (0<x<L) and infinite at the boundaries. Thus, the particle is absolutely confined and the wave function is zero beyond the boundaries. There are multiple solutions (in this case, an infinite number) corresponding to the number of oscillations that fit inside the box. The energy levels increase with n, but the percentage of increase decreases with n, in such a manner that we arrive at the classical limit at large n.

Pages: 15

The harmonic oscillator is defined by a quadratic potential. Many important problems in physics reduce to that of a quantum harmonic oscillator because the leading term of a potential near its minimum is quadratic. It is a general and noteworthy feature of quantum systems that the lowest energy is not zero. Calculate the probability that the quantum oscillator is found beyond the classical limit. Agreement with the uncertainty principle may be checked by calculating the expectation values of x2 and p2 and then taking the square root of the product. For large quantum numbers, the solution must approach that of the classical harmonic oscillator.

Pages: 25

The Bohr model is highly successful in explaining many features of the hydrogen atom. The ground state wave function does not have any angular dependence. The normalized hydrogen atom wave functions are directly accessible in Mathematica and can be readily verified as solutions of the Schrodinger equation. Get the hydrogen atom ground state wave function in terms of the Bohr radius a. The probability of finding the electron per volume at a given location is given by the square of the wave function. The electron has only a very tiny probability of being found inside the proton, whose size is about 1 fm.

Pages: 21

Probability distributions are at the heart of modern physics and have been encountered in thermal radiation, radioactive decay as well as from the solution to the Schrodinger equation. In general, a probability distribution may be written as P(x) which represents some probability per unit of x. The function is usually normalized to unity, P(x)dx=1. Some distributions like flipping a coin are not continuous but have discreet outcomes. The Poisson distribution corresponds to large n and small x and is obtained from the binomial distribution with the approximations. A major-use case of the Poisson distribution comes about when one wants to know an upper limit on the average value (a) when a small number of occurrences are observed.

Pages: 15

The primary mechanism by which energy is produced in the sun is by the proton cycle. This is a multistep process in which the first step is the fusion of 2 protons. Two protons cannot make a stable bound state because of their electrical repulsion, but the bound state of a proton and a neutron, a deuteron, is stable. There are other variances that have the same net reaction. The parsec is a length unit defined by the distance at which the mean radius of the earth's orbit subtends an angle of one second of arc. The authors may use the solar luminosity together with the Stefan-Boltzmann law to calculate the surface (blackbody) temperature of the sun.