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Quantum Entanglement ^

Posted on 02/23/2006 7:15:29 PM PST by strategofr


These pages explain quantum entanglement by way of colourful pictures, helpful analogies, and absolutely no math.

To understand quantum entanglement, several ideas and words must be explained, especially the idea of a photon. The photon is a key concept in physics, and so critical to entanglement that its behaviours must be fully understood. But before delving into the details of photons, let's take a look at the world of the very tiny, beginning with waves and atoms.

What is a Wave?

Tossing a pebble into a pond creates ripples that travel from where the pebble landed to the edge of the pond. The ripples are also called waves. Another way to make waves is with a long rope. If you swing your arm up and down while holding the rope (as in Figure 2.1), you can see waves start from your hand and disappear off the end; moving the rope faster makes more waves.

Waves in rope

Figure 2.1. Making Vertical Waves. Image copyright © National Science Foundation, 2000.

Waves have three important properties: frequency, amplitude, and wavelength. Frequency

This is the number of waves that happen over time. Figure 2.2 compares two frequencies:

Waves frequency Figure 2.2. Wave Frequency. Amplitude

This is the height of a wave. Figure 2.3 compares two amplitudes:

Wave amplitude Figure 2.3. Wave Amplitude. Wavelength

This is the distance between the top of one wave to the top of the next. Figure 2.4 compares two wavelengths:

Wave length Figure 2.4. Wave Length. Experiment

You can play with frequency and amplitude using your voice. When you talk, your throat vibrates which causes waves of sound to travel through the air around you. (This is just like how waves travel across the pond, except instead of the wave travelling across water, it goes through air.) Try this:

1. Place two fingers against your throat. 2. Start humming a single note. 3. Change the pitch of your hums (from ohhhh to eeeee). 4. Change how loud you hum.

You can feel the different vibrations: slower (lower, less frequent) waves, and faster (higher, more frequent) waves. The louder you hum, the larger the amplitude; likewise quiet hums have smaller amplitudes. A loud hum gives off more energy (and needs more energy), which causes a stronger vibration in your arm. Interference

When two waves interact with one another at the same space and time, they create an interference pattern. Figure 2.5 shows two waves interacting. Where they meet, the regular pattern of circles is disrupted because waves have constructive and destructive behaviour.

Wave interference

Figure 2.5. Interference Pattern. Image copyright © Museum Victoria, 2003. Constructive Waves

When the top (crest) of one wave meets the crest of another, they make a higher wave. Similarly, when the bottom (trough) of a wave meets the trough of another, they make a deeper trough. Figure 2.6 shows how constructive waves can merge together.

Constructive waves

Figure 2.6. Constructive Waves. Destructive Waves

When the trough of one wave meets the crest of another, or vice-versa, they cancel each other out. That is, the troughs are made more shallow and the crests are lowered. If the crests are as tall as the troughs are as deep, then the waves can cancel each other out completely. Figure 2.7 shows how destructive waves can merge together.

Destructive waves

Figure 2.7. Destructive Waves. Waves and Light

Before 1801, scientists thought that light travelled through space as tiny particles. In 1801, Thomas Young elegantly demonstrated his double slit experiment. The experiment showed that light also behaves like waves, because it interferes with itself. The role light plays with other particles is crucial to understanding how quantum entanglement works. And those particles are found within the realm of the atom ...

What is an Atom?

The atom was once thought to be the basic building block from which all else was created. It is the smallest particle of an element that still has the characteristics of that element. Helium atoms, for example, are used to fill balloons because they are lighter than air. One helium atom will rise up through the air, but because it is so tiny, many are needed to conquer the Earth's gravity. Today, physicists know that atoms are made from even smaller parts called elementary particles. Figure 3.1 represents a typical atom:

Atomic Structure

Figure 3.1. Model of an Atom. Image copyright © 2003 Protons, Neutrons, and Electrons

The nucleus, at the centre of an atom, is made up of elementary particles called protons and neutrons. Travelling around the nucleus, in a variety of ways, are electrons. An Electron Orbital Path is the space through which electrons travel as they tour the nucleus. Each Electron Orbital Path has a limit to the number of electrons allowed on it. Figure 3.1 shows blurry electrons because both their position and precise momentum can never be known at the same time (this is the Heisenberg Uncertainty Principle). These paths are known as shells.

The shell an electron follows depends on its energy. All the electrons in a specific shell have exactly the same amount of energy. To move from one shell to another, an electron must either gain or release a fixed amount of energy. A fancy way to say this is: electrons are restricted to quantized orbits. These shells are not necessarily fixed in size, but defined in terms of probability. Although there is a chance that an electron could be thousands of kilometres away, it is more likely to be close to the nucleus around which it travels.

Although useful to picture electrons orbiting a nucleus as planets encircle the Sun, it is not entirely accurate.

The next section illustrates these concepts with a race analogy. Electron Orbital Race

Imagine a race with the following rules:

1. runners on their track must run single file, without passing; 2. runners must remain on their track while they have the same physical fitness; 3. a runner that gets a boost of energy must jump to an outer track; 4. a runner that loses energy must jump to an inner track; and 5. there are many tracks, shown simplified in Figure 3.2.

Atom Analogy

Figure 3.2. Atom Analogy.

In this analogy, a runner gains energy by eating fruit. When this happens, the runner instantly obtains a new level of fitness, and must jump to a new track according to the rules. Runners can never be moderately fit, or somewhat lazy, but always lose or gain specific amounts (discrete units) of fitness. Quantum Leap

The word quanta means discrete units. It is analagous to the difference between a ladder and a slide; you can stand anywhere on a slide, but only on the rungs of a ladder. Now substitute the words track and runner with shell and electron. The word quantized describes the small, discrete, leaps that electrons make from shell to shell, as though ascending or descending a ladder.

The act of an electron jumping between two shells is called a quantum leap. Force Carrier

Electrons do not eat apples to make a quantum leap (their mouths are too small). They do, however, revolve around the nucleus as close as possible because it uses less energy; this is known as their ground state. If an electron gets extra energy, it must leap to a higher shell. After a short period of time, it will spontaneously release the extra energy, and leap back to its former shell. Just as a runner eats fruit for more energy, electrons get energy from a force carrier to make quantum leaps.

What is a Photon?

A photon is a force carrier particle. Elementary particles (electrons, protons, neutrons, and such) can only interact using force carriers. In the race analogy, photons are like fruit; they are responsible for electromagnetic interactions. Electromagnetic Interaction

Electromagnetic means having both electrical and magnetic properties. When physicists talk about interaction, it typically refers to the exchange of energy via photons. Keep in mind that visible light is only a small range of frequencies where photons exist.

Electromagnetic interaction between particles has three forms: forces within atoms, forces between atoms, and electromagnetic fields and waves. Forces Within Atoms.

This force causes the electrons to bind with an atom's nucleus (the positive nucleus attracts negative electrons, similar to the attraction between north and south poles of magnets). Forces Between Atoms.

The friction when tires roll, the pressure of squishing your thumb and forefinger together, and a chair holding you up occur because of changes in energy. The energy changes because electrons, or atoms, reposition themselves as material is deformed upon contacting itself or another material. Electromagnetic Fields and Waves.

This interaction is responsible for electric or magnetic fields, and electromagnetic waves that travel (light, x-rays, microwaves, and such). All these forms of waves are basically the same, differing only by wavelength. Quantized Behaviour

An electromagnetic wave must carry one or more whole units of energy (never arbitrary amounts). The units of energy carried are called quanta. Thus a single photon is an electromagnetic wave carrying one quantum of energy. Since it is a wave, it has a frequency; a higher frequency means a more energetic photon. Let's revisit the rope analogy to explore this concept.

Imagine making waves in a long strand of rope by swinging your arm up and down. The faster you move your arm, the more waves move along the rope. You have to work harder to make more waves, which means you have to put more energy into the rope. Since a photon behaves like a wave, when it has a high frequency it must have lots of energy! However, unlike the rope, a photon can only take on certain fixed frequencies. This is like being allowed to make 20 waves in the rope each second, or 30 waves, or 40 waves, but never 25, 37, or 42. The energy you put into the rope must be in quantized units.

Electrons will only absorb or emit photons of specific frequencies to perform a quantum leap. From the race analogy, this is similar to saying that runners are so picky that they will only eat fruit of a precise, fixed weight (for example, an apple weighing 140 grams is edible, but 142 grams is not).

Now that we know the role photons play, and how selective they are about their energy levels, the two remaining items we need to understand are spin and polarization. Both are tricky properties that play a role in quantum entanglement. Spin

There are two types of spin covered in this section: particle spin and photon spin. Particles

All particles have a property known as spin. The following describes the spin of typical particles, excluding photons. Once the notion of spin has been highlighted, the property of spin as it relates to photons will be addressed.

Although there is no exact classical analogy, it helps to picture spin in terms of a globe. Figure 4.1 shows a tiny globe that you can imagine rotating around its axis as it travels through space. (But note that particles do not actually rotate like a globe.)

Spin analogy using a planet

Figure 4.1. Particle Spin Metaphor. Image copyright © NASA, 1972 from Apollo 17 - The Blue Marble.

The concept of spin can be clarified by showing an experiment that detects this property. Figure 4.2 shows the behaviour of a particle when passing between two magnets.

Detecting particle spin. Figure 4.2. Particle Spin Detection.

In the above figure, a particle is shot from the Emitter like an arrow from a crossbow. When the particle passes through the influence of the two magnets (the S and N rectangles), it will change course depending on its spin. If the particle is spinning up, it will deflect along the high path to the detector; if spinning down, it will travel the low path. This behaviour is caused by the particle's intrinsic angular momentum, which is a fancy term for spin.

The spin of a photon has slightly different behaviour than other particles. Photons

A photon's axis and its direction of motion are directly linked, making it impossible to change one without changing the other, much like a gyroscope. In all cases, a photon's axis must be 90 degrees to its motion. Since photons travel at light speed, their spin is limited to two states: clockwise or counter-clockwise. These states correspond to left-handed and right-handed photons.

The last property of photons that we need to explore, polarization, is related to spin. Polarization

Polarization is the direction that photons oscillate. In physics, oscillate means to vary between alternate extremes, often within a set time limit. For a photon, its polarization is the orientation of its axis, which in turn is linked to its direction of motion. The following sections explore the idea of how photons and waves are related to polarization. Polarized Waves

In Figure 4.3, a rope is being swung haphazardly, which makes not only horizontal and vertical waves, but all types in between. Note how the rope's length physically limits the size of waves that can be created.

Making rope waves.

Figure 4.3. Making Random Waves. Image copyright © National Science Foundation, 2000.

The size of light waves is different from rope in that it is restricted by the waves' frequency, and consequently their energy. The direction of light waves, like the rope, has no restriction; most light (from the sun or incandescent bulbs) travels randomly, similar to the rope movement in Figure 4.3. Passing light through a specially constructed material will filter out photons that do not have a specific polarization.

Figure 4.4 continues the rope analogy to illustrate the concept of filtering waves. When the same haphazard motion from Figure 4.3 is used to whip waves through a picket fence, only vertical waves will pass through the slit. Consequently, someone watching from the other side of the fence will only see vertical waves in the rope.

Making rope waves.

Figure 4.4. Making Polarized Waves. Image copyright © National Science Foundation, 2000.

Imagine placing a second fence after the first whose slats are horizontal. When waves try to pass through the first fence, only vertical ones succeed. The second fence, being horizontal, blocks all vertical waves. The result is that someone watching from the other side of the fences will see no waves in the rope. By using two fences with slits rotated 90 degrees to one another, all the rope waves are elimiated.

The same effect happens with photons, which can be demonstrated by a simple experiment. Experiment 1

To see the effect of polarization on light, try these simple steps:

1. Find two polarized sunglasses. 2. Hold one pair horizontally. 3. Hold one pair vertically. 4. Press a horizontal lens against a vertical lens. 5. Look through the pressed lenses.

As shown in Figure 4.5, the outer lens will block all vertically aligned photons, while the inside lens blocks the horizontal ones. The result is that no light completes the journey beyond the second lens.

In practice, some light may make it through. This could be due to flawed polarizing material in the lenses, or because the lenses are not exactly 90 degrees (perpendicular) to each other.

Two polarized sunglasses rotated at 90 degrees.

Figure 4.5. Polarized Lenses Blocking Light. Image copyright © Sierra Trading Post, 2004. Experiment 2

Polarization is not as simple as the previous experiment would lead you to believe. For example, place a third lens between the two lenses in the first experiment. Now rotate the third lens and watch what happens. Colorado University has an in depth explanation of polarization.

Quantum Entanglement

Whatever happened to one particle would thus immediately affect the other particle, wherever in the universe it may be. Einstein called this "Spooky action at a distance."

Amir D. Aczel, Entanglement, The Greatest Mystery In Physics. The Theory

When a photon (usually polarized laser light) passes through matter, it will be absorbed by an electron. Eventually, and spontaneously, the electron will return to its ground state by emitting the photon. Certain crystal structures increase the likelihood that the photon will split into two photons, both of them with longer wavelengths than the original. Keep in mind that a longer wavelength means a lower frequency, and thus less energy. The total energy of the two photons must equal the energy of the photon originally fired from the laser (conservation of energy).

When the original photon splits into two photons, the resulting photon pair is considered entangled.

The process of using certain crystals to split incoming photons into pairs of photons is called parametric down-conversion.

Normally the photons exit the crystal such that one is aligned in a horizontally polarized light cone, the other aligned vertically. By adjusting the experiment, the horizontal and vertical light cones can be made to overlap. Even though the polarization of the individual photons is unknown, the nature of quantum mechanics predicts they differ.

To illustrate, if an entangled photon meets a vertical polarizing filter (analagous to the fence in Figure 4.4), the photon may or may not pass through. If it does, then its entangled partner will not because the instant that the first photon's polarization is known, the second photon's polarization will be the exact opposite.

It is this instant communication between the entangled photons to indicate each other's polarization that lies at the very heart of quantum entanglement. This is the "spooky action at a distance" that Einstein believed was theoretically implausible. The Practice

Experiments have shown that Einstein may have been wrong: entangled photons seem to communicate instantaneously. Figure 5.1 illustrates how to create entangled photons.

Parametric down-conversion. Figure 5.1. Photon Entangler Device. Image copyright © European Space Agency

1. An ultraviolet laser sends a single photon through Beta Barium Borate. 2. As the photon travels through the crystal, there is a chance it will split. 3. If it splits, the photon will exit from the Beta Barium Borate as two photons. 4. The resulting photon pair are entangled.

An ultraviolet laser is used because the laser light has a high frequency. A high frequency implies a greater chance of splitting into two entangled photons. The Result

Figure 5.2 is an enhanced photograph of a photon that has split into an entangled pair.

Entangled photons. Figure 5.2. Entangled Photons.

Spooky Behaviour

This section describes some of the strange behaviours seen in experiments. Double-slit Experiments

Figure 6.1 illustrates what happens when a source of light shines through two tiny slits onto a screen. The detector screen illuminates a wave-like pattern caused by light interfering with itself. This is how waves are expected to behave.

Light Wave Experiment. Figure 6.1. Experiment 1 - Light Waves.

Since photons are also particles, we can transmit them one at a time. Figure 6.2 shows the result of many solitary photons being fired at the detector.

Light Particle Experiment. Figure 6.2. Experiment 2 - Light Particles.

The interference pattern still appears; but if photons are fired alone, then with what do they interfere? Quantum theory tells us that each photon interferes with itself. If true, then it implies that we cannot know through which slit the photon travels; the photon seems to have travelled both slits simultaneously!

Trying to detect which slit the photons travel puts this quantum weirdness in the spotlight, so to speak.

For example, we can polarize the light before it goes through the slits. Like rippling a rope through a picket fence to permit only vertical waves (see Figure 4.4), polarizing allows us to limit the type of light waves that make it through the slits to the detector.

When we put a polarizing filter around one of the two slits, the interference pattern disappears. The result is shown in Figure 6.3.

Polarized Light Experiment. Figure 6.3. Experiment 3 - Polarized Light Waves.

Whenever we can detect, or deduce, through which slit a photon has travelled, the interference pattern instantly disappears. An interference pattern only appears when the photon's path is unknown.

It gets weirder.

Even if we examine the photon's trail after it passes the double-slits (but before it reaches the detector), the interference pattern disappears. And it disappears regardless of whether the examination uses a direct or indirect measurement of the photon.

But what if we used two photons that are inextricably linked (through entanglement), to perform the experiment? Entanglement Experiments

I was born not knowing and have had only a little time to change that here and there.

Richard Feynman, Letter to Armando Garcia. Parametric Down-conversion

We have already seen how to create entangled photons through a process called spontaneous parametric down-conversion:

Parametric down-conversion. Figure 6.4. Photon Entangler Device. Image copyright © European Space Agency

To review, the laser in Figure 6.4 fires a high-energy beam into a special type of crystal. Every once in a while one of the photons from the beam will split into two less energetic photons. These two entangled photons will have opposite polarizations and travel in two different directions, resulting in two streams of polarized light.

The previous double-slit experiments detected interference patterns by shining a single light source through two slits (Figure 6.1 and Figure 6.2). The next experiment uses two streams of entangled photons. Quantum Eraser

Figure 6.5 shows a Bell-state quantum eraser, named after John Bell. It illustrates the application of the following steps:

1. a laser fires photons into a Beta Barium Borate (BBO) crystal; 2. the crystal entangles some of the photons; and then 3. entangled photons travel to two different detectors: A and B.

Placed between the crystal and detector B is a double-slit, like in the previous experiments. Immediately in front of detector A is a polarizing filter that can be rotated. Figure 4.5 showed an experiment using sunglasses to see the effects of rotating a polarizer. Those same effects apply here.

Quantum Eraser. Figure 6.5. Experiment 4 - Bell-state Quantum Eraser.

The Bell-state quantum eraser has one more feature: each slit is covered by a substance that changes the polarization of a photon. Consequently, the left-hand slit will receive photons with a counter-clockwise polarization, and the right-hand slit will pass photons with a clockwise polarization.

Note: Polarization does not affect interference patterns.

Initially, neither detector shows an interference pattern. Since we control the polarization of photons passing through the slits and we know the polarization accepted by each slit, we can deduce which way the photons travelled (counter-clockwise through the left; clockwise through the right). Thus no interference patterns are detected.

However, if we rotate the polarizing filter in front of detector A so that the polarizations of the photons that hit the detector are the same (that is, we can no longer distinguish between clockwise and counter-clockwise polarizations), then the interference pattern appears at both detectors!

How do the photons arriving at detector B know that the polarizations have been "erased" at detector A? Conclusion

Quantum Theory is continually being challenged and tested; physicists are finding new ways to explain the world of the tiny. Each passing year brilliant minds add to, or subtract from, the pool of knowledge about quantum behaviour.

Unlike the static nature of the web pages presented here, quantum physics is ever changing. Physicists are confronted with problems that will take many iterations, many years, to solve. Scores of theories will be presented, some of them merely tweaking, while others radically alter, our perceptions of quantum nature.

Whatever we observe in the future, it promises to be exciting!

TOPICS: Technical
KEYWORDS: quantummechanics
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Got a little bogged down in this.
1 posted on 02/23/2006 7:15:30 PM PST by strategofr
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To: strategofr

(Actually, I think this is pretty good. Go to website for illustrations.)

2 posted on 02/23/2006 7:21:10 PM PST by strategofr ( Davidson: "...50 or more [like Foster]..murdered [by Clintons]." Hillary's Secret War, Poe, p. 100)
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To: strategofr
Thus a single photon is an electromagnetic wave carrying one quantum of energy. Since it is a wave, it has a frequency;

Here's where the "explanation "lost" me.
The discussion went from "behaves like a wave" to it is a wave. Not buying it. If it can be both, there needs to me an explanation for that; analogous or otherwise...

3 posted on 02/23/2006 7:35:52 PM PST by Publius6961 (Multiculturalism is the white flag of a dying country)
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To: Publius6961; strategofr; NormsRevenge; Grampa Dave; SierraWasp; Marine_Uncle; RadioAstronomer; ...
Perhaps this will help explain this:

A non-running computer produces fewer errors

4 posted on 02/23/2006 7:46:32 PM PST by Ernest_at_the_Beach (History is soon Forgotten,)
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To: Ernest_at_the_Beach


5 posted on 02/23/2006 7:51:47 PM PST by A CA Guy (God Bless America, God bless and keep safe our fighting men and women.)
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To: strategofr

I will read thru again tomorrow....going to go read about Cantor's levels of infinity...for awhile now....thanks.

6 posted on 02/23/2006 7:59:39 PM PST by Ernest_at_the_Beach (History is soon Forgotten,)
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To: Ernest_at_the_Beach

Rather heady stuff to say the least. I'll stick with the cat in the hat.

7 posted on 02/23/2006 8:11:40 PM PST by Marine_Uncle (Honor must be earned)
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To: strategofr
How do the photons arriving at detector B know that the polarizations have been "erased" at detector A?

What's even weirder is that if you put detector A far away such that the photon arrives at B first, you still get the interference pattern. How do the photons arriving at detector B know that the polarizations eventually will be "erased" at detector A?
8 posted on 02/23/2006 8:18:14 PM PST by non-anonymous
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To: Publius6961
It's been years since I've had a p-chem headache. Thanks, everyone!

LOOSELY: All matter consists of waves. Everything is quantized. Everything exists at a certain state/energy level. For very tiny things (electrons, protons, photons, etc.), the allowed energy levels are easily distinguished because these levels are clearly separated by energy zones that are forbidden (you can only stand on the run of a step in a staircase and not the rise). As we consider larger and larger objects, the permitted states run closer together (rise becomes smaller and smaller) to the point that they seem to run together into one smooth continuum of allowed states. This creates the illusion of no quantization for macroscopic objects, which is described by Newtonian Mechanics (which may be considered a subset of Quantum Mechanics).

Waves have wavelength and particles have mass. De Broglie related the two qualities in a simple equation. So, they are related and experiments using the relation do work!

de Broglie relation

A decent reference

9 posted on 02/23/2006 8:38:11 PM PST by MilesVeritatis (War is an ugly thing, but not the ugliest of things...." - John Stuart Mill)
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To: ImaGraftedBranch; Ultra Sonic 007

Bump for homeschool physics...

10 posted on 02/24/2006 6:43:39 AM PST by ImaGraftedBranch ("Toleration" has never been affiliated with the virtuous. Think about it.)
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To: Publius6961

It is neither a wave nor a particle; it is a photon. There are measurements that give "wave-like" results; when performing one of these, the photon has all the properties of a wave. There are other experiments that give "particle-like" resluts; when performing of these other expemints, the photon has all the prpoerties of a particle.

Photon is a photon is a photon. (If you prefer Rand to Stein, delete the first two words of the previous sentence.) Wave and particle are our descriptions of experimental results, not descriptions of photons.

11 posted on 02/24/2006 6:52:12 AM PST by Doctor Stochastic (Vegetabilisch = chaotisch ist der Charakter der Modernen. - Friedrich Schlegel)
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To: AntiGuv

FutureTech ping!

12 posted on 02/24/2006 7:01:50 AM PST by Ultra Sonic 007 (Hitler and Stalin have nothing on Abortion)
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To: PatrickHenry

Seems to me a prime candidate for the Science ping list!

13 posted on 02/24/2006 7:54:15 AM PST by AntiGuv
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To: AntiGuv
Very nice, but it's really just a tutorial. No ping list deployment.
14 posted on 02/24/2006 8:20:53 AM PST by PatrickHenry (Virtual Ignore for trolls, lunatics, dotards, scolds, & incurable ignoramuses.)
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To: PatrickHenry

I bow to your better judgment!

15 posted on 02/24/2006 8:38:32 AM PST by AntiGuv
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To: AntiGuv

Not "better." Just my judgment. I worry about "ping fatigue," so I try not to ping for everything that comes along.

16 posted on 02/24/2006 8:46:18 AM PST by PatrickHenry (Virtual Ignore for trolls, lunatics, dotards, scolds, & incurable ignoramuses.)
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To: ImaGraftedBranch

Agreed. My public school had a lame physics class for senior level students. We had to go elsewhere for an AP physics class.

17 posted on 02/24/2006 10:37:29 AM PST by MilesVeritatis (War is an ugly thing, but not the ugliest of things...." - John Stuart Mill)
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To: strategofr
Oh. I thought perhaps it was a thread about a beautiful physicist/beach babe...

18 posted on 02/24/2006 10:43:01 AM PST by r9etb
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To: r9etb
If there were physicists like that when I was at university, I would have switched departments...and would have developed the need for much tutoring during her office hours.

Hey, wait just a minute! She can't be a physicist--where's the pocket protector??? For that matter, I don't see space for a pocket in that suit.

19 posted on 02/24/2006 12:10:44 PM PST by MilesVeritatis (War is an ugly thing, but not the ugliest of things...." - John Stuart Mill)
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To: MilesVeritatis
at university

I'm guessing you're Canadian or English?

20 posted on 02/24/2006 12:18:22 PM PST by r9etb
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