#f304

[physicsandchemistryrevision]

Quantum Physics

Expectation value general form

Cont'd from "Expectation value in quantum mechanics"

In turn, we can express this in a general form. If a general observable operator Ô has an eigenfunction | ψ ⟩ which produces an eigenvalue o, given the by the eigen-equation

then its projection onto the basis of | ψ ⟩, given by ⟨ ψ |, produces the expectation value for an observation of | ψ ⟩

which in turn produces the expression

often written as

in conformity with the expected value in statistical mathematics. Note that the eigenvalue o may not necessarily be independent of x, which is why we’ve left it in the integral.

This definition can be extended to consider all 3-dimensional space, where the expectation value of the measurement of an observable Ô of a free particle is found by

In the next post, we'll look at some examples of using this.

[physicsandchemistryrevision]

Electromagnetism & Optics

Electrostatics Continuous charge distribution Cont'd from "Electric fields"

Thus far we have been working with discreet charge distributions (See “Coulombs law" and "Principle of superposition" ), where point charges are considered in their individuality. A continuous charge distribution is made from considering a continuous 'volume' of charge.

E =  1/4πε0 · ∫ 1/|r|² dq r

where ε0 is the permittivity of free space, dq is an infinitesimal charge element within a continuous distribution of charge, r is the position vector of the charge element from the test charge and r is the unitary position vector.

[physicsandchemistryrevision]

Mathematics

Stokes' theorem

Cont'd from "Level surfaces", see "Curl of a vector" 

Simply put, Stokes' theorem states that the circulation of a vector field V around a surface is equal to the integral of the component of the curl of V which is in a plane perpendicular to the surface over the total area A of the surface. Therefore, a vector V will have circulation around a surface in the x-y plane equal to

circulation = ∬ (∇×V)z dA                      A                      

where, in this case, dA = dydx since the surface is in the x-y plane. (∇×V)z represents the z-component of curl(V).

—

For example, a vector field A = (x, x²y, xy²z) has a curl

∇×A = 2xyz i - y²z j + 2xy k

Now, to find the circulation C of A around a 2 by 1 rectangle in the x-y plane centred at the orgin, we take the z-component (i.e., the k part) of A and integrate as above, substituting dA = dxdy

C = ∬ 2xy dxdy       A                    

The limits are the area over which the integration happens. This will be the surface we're finding the circulation around. In this case, the rectangle is centred at the origin and has 2 units in the y-direction and 1 unit in the x-direction, implying that the limits are

0 ≤ x ≤ 1

0 ≤ y ≤ 2

 We now have the definite integral

      2  1             C = ∫  ∫ 2xy dxdy       0 0                   

which can be solved as follows!

      2        x = 1     C = ∫ [x²y]   ·   dy       0        x = 0   

      2       C = ∫ y dy       0          

                  y = 2 C = [1/2 · y²]                   y = 0

C = 2

—

Interestingly, Stokes' theorem was not initially presented as a formal publication, nor was it entirely his own work. Instead, the idea was suggested to him by Lord Kelvin. Stokes included it in his 1854 Smith's Prize exam as a question for his students to prove and the theorem quickly became attributed to Stokes after the potential of the question was realised. As a result, Stokes' theorem is less commonly known as the Kelvin-Stokes theorem.

[physicsandchemistryrevision]

Physics

Quantum physics: Introduction

Although I've covered the basics of quantum mechanics before on this blog, I hope the next series of posts will be more in-depth than those from Unit G482.

Black body radiation & empirical deductions

When a black body absorbs energy it re-emits the energy with equal intensity across all wavelengths, following the Stephan-Boltzmann law

I = σT4

where I is intensity of emission, σ is the Stephen-Boltzmann constant (σ = 5.67×10-8 W m-2 K-4) and T is absolute temperature of the body.

However, when a non-ideal body re-emits energy the intesity varies with each wavelength, following a bell-curve shape with a peak wavelength. This can be represented using the spectral emittance (or emissivity) of the body

I =  ∫0∞ I(λ) dλ

Rayleigh predicted the spectral emittance of a black body can be expressed as

I(λ) = 2πckT/λ4

where c is the speed of light in a vacuum and k is Boltzmann's constant. Empirically this was true for wavelengths λ  ≫ 0. However, since it held no fundamental basis the equation no longer applied for higher frequency, shorter wavelength waves. This became known as the UV catastrophe.

Wien's displacement law was experimentally derived to predict at which wavelength the distribution would peak, λmax.

λmax· T = const. = 2.9×10-3 m·K

Interestingly, Wien's law held for short wavelengths but failed at longer wavelengths.

Clearly there were no fundamental rules determining why the waves behaved in this way. Planck proposed a different model for the way a black body would radiate. His theory could combine the two laws and went on to change the scope of physics for years to come.

[arq911eng-blog]

The Mexico city Government and the College of Architects gave the 3rd. Place to arq 911sc + at103 + F304 and Caterina Pregazzi for the project "4->7->85" of the Integral Merced Rescue contest organized last December. 

[arq911-blog]

El Gobierno del Distrito Federal y el Colegio de Arquitectos de la Cd. de México otorgaron el 3er. Lugar a arq 911sc + at103 + f304 y Caterina Pregazzi en el concurso "Rescate Integral de la Merced" el pasado 27 de diciembre.