August 13th, 2020
Category Physics of Solar Energy
With a load of matched impedance, the output power of a solar cell can be maximized. As shown in Section 9.1.3, the maximum power is related to the nominal power by a form factor nf approximatelyRead More
In the microscopic treatment of matter, the particles, for example, molecules and electrons, are distributed in a system of energy levels. How the particles distribute among the energy levels determines the behavior of the system in a significant way. Regarding to applications with solar energy, an understanding of Maxwell-Boltzmann statistics and Fermi-Dirac statistics is essential. Maxwell-Boltzmann statistics is valid for a system of distinguishable particles in a system of energy levels allowing unlimited occupancy; whereas Fermi-Dirac statistics is valid for a system of indistinguishable particles in a system of energy levels allowing limited occupancy, that is, systems satisfying the Pauli exclusion principle.
The starting point of the derivation is the Boltzmann expression of en...Read More
The Hamiltonian of a one-dimensional harmonic oscillator is
– 1 a2 mj2
ff = 2mp + ~q •
where the momentum p and coordinate q satisfy the commutation relation
[p, q] = pq — qp = ih.
We introduce a pair of operators, the annihilation operator,
Physics of Solar Energy C. Julian Chen Copyright © 2011 John Wiley & Sons, Inc.
and the creation operator
„ + 1 „ . [шш
a тшш”- V-2ira (C4)
the meanings of these terms well be clarified soon. Using the commutation relation C.2, we have
H = НшаР a +— Нш,
and the commutation relation
[a, af] = aaP — aPa = 1. (C.6)
To find the eigenstates and energy levels of the Hamiltonian C.1,
Hn) = Enn), (C.7)
it is sufficient to find the eigenstates and eigenvalues of the operator apa,
ap an) = unn), ...Read More
Usually, introductory quantum mechanics starts with Schrodinger’s equation and using partial differential equations as the mathematical tools. For example, the hydrogen atom problem is resolved with spherical harmonics and Laguerre polynomials. A possible shortcoming with this approach is that the readers become submerged in pages and pages of mathematical formulas and lose the conceptual understanding of the physics. Historically, before Erwin Schrodinger discovered the partial differential equation format, Heisenberg and Pauli developed the algebraic approach of quantum mechanics, and resolved several basic problems in quantum mechanics, including harmonic oscillator, angular momentum, and the hydrogen atom...Read More
In planar trigonometry, there is the sine formula
sin A sin B sin C
In spherical trigonometry, a similar formula exists,
sin a sin b sin c
sin A sin B sin C
Obviously, for small arcs, Eq. B.15 reduces to Eq. B.14.
To prove Eq. B.15, we rewrite Eq. B.2 as
sin b sin c cos A = cos a — cos b cos c. (B.16)
Squaring, we obtain
sin2 b sin2 c cos2 A = cos2 a — 2 cos a cos b cos c + cos2 b cos2 c. (B.17)
The left-hand side can be written as
sin2 b sin2 c — sin2 b sin2 c sin2 A, (B.18)
By definition, in a spherical triangle, the sides and the vertical angles are always smaller than 180°. Therefore, in Eq. B.22, only a positive sign is admissible. Because Z is symmetric to A, B and C, we obtain
sin a sin b sin c sin A sin B sin C
We rewrite the cosine formula B...Read More
In planar trigonometry, there is a cosine formula
a2 = b2 + c2 – 2bc cos A. (B.1)
In spherical trigonometry, a similar formula exists:
cos a = cos b cos c + sin b sin c cos A. (B.2)
When the arcs are short and the spherical triangle approaches a planar triangle, Eq. B.2 reduces to Eq. B.1. In fact, for small arcs,
cos b & 1 – 1 b2, (B.3)
sin b & b, (B.4)
and so on. Substituting Eqs B.3 and B.4 into Eq. B.2 reduces it to Eq. B.1.
Here we give a simple proof of the cosine formula in spherical trigonometry by an analogy to that in planar trigonometry. To simplify notation, we set the radius of the sphere OA = OB = OC = 1. By extending line OB to intersect a line tangential to AB at a point D, we have
AD = tan c; OD = sec c.
Similarly, by extending line OC to intersect a line tengential ...Read More
A plane passing through the center of a sphere O cuts the surface in a circle, which is called a great circle. For any two points A and B on the sphere, if the line AB does not pass the center O, there is one and only one great circle which passes both points. The angle AOB, chosen as the one smaller than 180° or n in radians, is defined as the length of the arc AB. Given three points A, B, and C on the sphere, three great circles can be defined. The three arcs AB, BC, and CA, each less than 180° or n in radians, form a spherical triangle; see Fig. B.1.
Following standard notation, we denote the sides BC, CA, and AB by c, b, and a, respectively...Read More
When we look into the sky, it seems that the Sun and all the stars are located on a sphere of a large but unknown radius. In other words, the location of the Sun is defined by a point on the celestial sphere. On the other hand, the surface of Earth is, to a good approximation, a sphere. Any location on Earth can be defined by a point on the terrestrial sphere; namely by the latitude and the longitude. In both cases, we are dealing with the geometry of spheres.
To study the location of the Sun with respect to a specific location on Earth, we will correlate the coordinates of the location on the terrestrial sphere of Earth with the location of the Sun on the celestial sphere. The mathematical tool of this study is spherical trigonometry...Read More