Monthly Archives: February 2011


Scientists say it’s high ‘NOON’ for microwave photons

An important milestone toward the realization of a large-scale quantum computer, and further demonstration of a new level of the quantum control of light, were accomplished by a team of scientists at UC Santa Barbara and in China and Japan.

More about the group that is working on this:

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Derivation of Simple Harmonic Motion w/ Dampening

Simple Harmonic Motion (SMH) is derived from Newton’s force law, F = ma and Hooke’s spring law, F = -kx. If we start with -kx = ma, and ask ourselves what function could represent x, we have a differential equation, which is solved by x = A cos ωt. This leads to ω=sqrt(k/m).

We can also use e^iωt as a solution, or at least the real part of it. But these equations do not accurately depict what is observed. When watching an oscillation, eventually it slows down. This is because of the friction due to force, expressed as F = -βv. Now we rewrite the sum of the forces as ma = -kx – βv, and try to find a solution for this differential equation. We guess a solution based on e^iωt, and then solve for ω using the quadratic equation, and plug into the e^iωt equation.

We end up with a dampening constant γ=β/m, and we use this to determine a halflife or 1/e which is about 1/3 the decay time.

An out of phase driver means a small driver can produce large oscillations, leading to Resonance!

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Properties of Matter Part II

Solid to Gas = Sublimination
Gas to Solid = Deposition

The primary difference between states of matter is the molecular structure.

A phase diagram shows how if pressure and temperature is varied, matter will be solid, liquid or gas.

Element + Element = mixture

If a chemical reaction takes place in the mixture it becomes a compound.

Physical properties can be observed without changing. For example: Density, luster, hardness, color, state

Chemial properties can only be observed when the matter is reacting with another substance.

Intensive property: scale invariant (color)
Extensive property: varies w/ amount (mass)

The ratio of 2 extensive quantities that scale in the same manner is an intensive property. ie density = mass/volume.

Measurement and Units:

Accuracy: How close to the true value
Precision: How well can be repeated

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Intro to ODQ’s

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How does a laser diode work?

I was looking at the Helium Neon laser at the LTC today, and wondering how a laser diode like the ones in a laser pointer works, since it must be so much smaller and I doubt it contains a tube of gas.

What makes laser light special is that the light is a coherent stream of photons. There are various types of devices that are used to produce laser light. Some types of lasers are: ruby laser, chemical laser, dye laser, gas laser, and rare-earth solid-state laser. All of these lasers work by exciting atoms to a higher state, so more atoms are in the higher state than the lower one. This is called an inverted population of atoms. The reason the population needs to be inverted is so that emission of photons will be dominant, instead of absorption.

The second important feature is that the higher state must be metastable, meaning the electrons remain excited for longer than usual so the transition to the lower state happens because of stimulated emission. Stimulated emission is the process of a photon of energy equal to an excited atom striking the excited atom, causing it to transition to the lower state and emit a second photon of the same frequency. This process gives two photons in exactly the same phase and moving in the same direction.

Cd players and laser pointers use semiconductor diode lasers. These lasers work by layering two materials, one on top of the other.  One of the materials is an n-type semiconductor. An example is an arsenic-doped silicon crystal, where negatively charged electrons carry the current. The other material is a p-type semiconductor, which  is populated with net positive gaps. Overall through both n and p type semiconductors have no net charge.

Because of the energy difference between the n and p layers there is an inverted population of atoms.  This means an electron can jump down causing a photon to be emitted, which in turn stimulates another electron to transition to a lower energy level and emit a photon, causing stimulated emission. The two materials form the laser diode. I’m not sure how the beam emerges, and will look for books on this at the LTC .

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Maxwells Equations

Maxwells equations are four laws used to understand electromagnetism.

Gauss’s law for electricity:
A generalized form of Coulomb’s law relating electric field to its sources, electric charges

Gauss’s law for magnetism: The same for the magnetic field, except that if there are no magnetic monopoles, magnetic fields likes are continuous, they do not begin or end.

Faraday’s Law: An electric field is produced by a changing magnetic field

Ampere’s Law:  A magnetic field is produced by an electric current or by a changing electric field.

Electromagnetic waves are produced by changing electric charges.

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Introduction to Differental Equations

Differential equations are equations involving derivatives. For example:

y”+xy’+2y = sin x

This is a second order, linear differential equation.

The order is the highest derivative, in the above case, it’s 2 because y”.

Linear can be thought of as a 1st degree polynomial.

Differential equations involving more than 1 independent variable are called partial differential equations. This course is mainly concerned with ordinary differential equations.

We learned two methods of solving ODQ’s today. The first method was Separation of Variables. This means rearranging the equation so that one variable is on one side and the other variable is on the other side, then integrating and finally solving for the function variable.

The second method was not given a clear name, but I think it could be called using an integration factor. We used this method because the variables could not be easily separated, but we observed the structure of the eq was similar to the output of the product rule for derivatives. So we added an extra term, mu, to both sides and integrated using it.

Finally we learned the Theorem of 1st Order Linear Differential Equations. Which states y’+p(x)y=q(x) If p,q are continuous on a < x < b, containing x, there is a unique solution satisfying the equation, and an initial condition y(x) = y.

An initial condition is ie y(0)=1, it is used for setting the constant.

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Filed under MAT 303