Name: Norman Anthony Aguero
Currently a student at FIU. My major is chemistry and my minor is physics. My goal is to hopefully earn a Ph.D. in physical organic chemistry.
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Amazing how the human mind interprets hidden cues.
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Bush Quotes
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Valence band (O.K. I yell "uncle" Pooklekufr 
, he was right and I was wrong.) Bands not bonds.
In solids, the valence band is the highest range of electron energies where electrons are normally present at absolute zero. In semiconductors and insulators, there is a bandgap above the valence band, followed by a conduction band above that. In metals, the conduction band has no energy gap separating it from the valence band. (The rest of this article refers to the valence band in semiconductors and insulators.)
Semiconductor band structure
See electrical conduction and semiconductor for a more detailed description of band structure.
Semiconductors and insulators owe their low conductivity to the properties of the valence band in those materials. It just so happens that the number of electrons is precisely equal to the number of states available up to the top of the valence band. There are no available states in the bandgap. This means that when an electric field is applied, the electrons cannot increase their energy (i.e. accelerate) because there are no states available to the electrons where they would be moving faster than they are already going.
There is some conductivity in insulators, however. This is due to thermal excitation - some of the electrons get enough energy to jump the bandgap in one go. Once they are in the conduction band, they can conduct electricity, as can the hole they left behind in the valence band. The hole is an empty state which allows electrons in the valence band some degree of freedom.
It is a common misconception to refer to electrons in insulators as "bound" - as if they were somehow attached to the nucleus and couldn't move. Electrons in insulators are quite free to move - in fact they move at a speed on the order of 100 km (60 mi) per second! They are also delocalised, having no well defined position within the sample.
What is life?
To a good approximation, one of the most oft-quoted references in thermodynamics is Erwin Schrödinger’s 1944 postulate that an organism keeps itself alive or aloof by feeding on negative entropy from its environment.[10][11][12] From the famous chapter six “Order, Disorder and Entropy” of his book What is Life?, Schrödinger asks: “what is the characteristic feature of life? and “when is a piece of matter said to be alive?” To answer these questions, Schrödinger turns to thermodynamics. Life, according to Schrödinger, avoids a decay to maximum entropy, or thermodynamic equilibrium, which Schrödinger equates with death, by feeding on negative entropy. Specifically, according to Schrödinger, an organism avoids decay by eating, drinking, breathing, and in the case of plants assimilating, a process called metabolism.
In the past, Schrödinger states, this process would have been considered an exchange of matter or energy, such that organisms stay alive by exchanging energy. He uses the example of how caloric values are printed in certain menus in the United States or Germany, but states that these caloric energy exchange values are useless in trying to quantify life. He then asks “what then is that precious something contained in our food which keeps us from death?” The answer, according to Schrödinger, is that because according the second law of thermodynamics an organism continually produces “positive entropy” it must continually draw in “negative entropy” from its environment to stay alive. Or, specifically “the essential thing in metabolism is that the organism succeeds in freeing itself from all the entropy it cannot help producing while alive.”
These suppositions, because they were intended for a lay audience, however, met with great opposition in the physics community. In later editions of his book, Schrödinger attached a note to chapter six explaining his use of the term “negative entropy”. He states “the remarks on negative entropy have met with doubt and opposition from physicist colleagues. Let me say first, that if I had been catering for them alone I should have let the discussion turn on free energy instead. It is the more familiar notion in this context. But this highly technical term seemed too linguistically near to energy for making the average reader alive to the contrast between the two things.”
Some Tools of Thermodynamics (The stuff I use on a daily Basis)
Some Miscellaneous Relationships
Recall that the combined first and second laws give the relationship (U = energy of the system, S is the entropy, p is the pressure and V is the volume.)
I'm sure Pooklekufr is gonna like this one.
(1)
This implies that U is a function of S and V. Sometimes we call S and V the "natural variables" of U. Regarding U = U(S,V) we can write
(2)
Comparing Equations 1 and 2 it is clear that
(3a)
and
(3b)
These two equations can be regarded as thermodynamic definitions of T and p.
Likewise, from the definition of enthalpy we wrote before that
(4)
Equation 4 implies that enthalpy is a natural function of S and p. Regarding H = H(S,p) we can write
(5)
From Equations 4 and 5 it is clear that
(6a)
and
(6b)
Equations 6a and 6b give us another thermodynamic definition of T and a thermodynamic definition of V (which is curious since we have always regarded V as a purely mechanical variable).
Helmholtz and Gibbs Free Energy
When we made the transformation from U to H by the Legendre transformation,
(7)
we remarked that V was not the most convenient independent variable. In the laboratory it is usually much easier to control pressure than it is to control V. Since both U and H are natural functions of entropy, it is fair to ask how convenient it is to have S as a variable. The answer is that it is not at all convenient to control entropy or to have entropy as an independent variable. We do not have a meter that reads entropy and we do not know how to hold entropy constant as we change some other variable. (Recall that we can control temperature, pressure, and volume.) So we make some more Legendre transformations.
(We will not give an extensive discussion of Legendre transformations here, but we should point out that they are not arbitrary. You can't just pick any two variables you wish and put them together to make a Legendre transformation. We could make the Legendre transformation from U to H by adding the pV term to U only because V is related to p and U through Equation 3b. Using this as a guide it would seem reasonable to use Equation 3a to change the variable S to T in the function U, and use Equation 6a to change the variable S to T in the function H. Let's try it.)
Define the Helmholtz free energy, A, as
(8)
Then,
(9a, b, c)
Equation 9c tells us that the Helmholtz free energy is a natural function of T and V. That is, A = A(T,V). T and V are much more convenient variables than S and V. Regarding A = A(T,V) we see that
(10)
Now compare Equation 10 with Equation 9c to see that,
(11a)
and
(11b)
Equation 11a gives us a thermodynamic definition of entropy and 11b gives another thermodynamic definition of pressure.
We now have a function of T and V, but we didn't much like V as an independent variable before so why should we like it any better now? Let's use the relationship 6a to define the Gibbs free energy, G, as,
(12)
Actually, we can use any one of the three equivalent definitions,
(13a, b, c)
Any one of Equations 13a, b, or c will give us the correct natural variables of G. Use Equation 12.
(14a, c, c)
From Equation 14c we see that the natural variables of G are temperature and pressure. Write
(15)
Comparing Equation 15 with Equation 14c we find that.
(16a)
and
(16b)
Equation 16a gives us another thermodynamic definition of entropy and 16b another definition of volume.
Meaning of A and G
What do A and G mean and what are they good for? We said after the introduction of the first law (which introduced the internal energy, U) that we would be introducing three more functions that have units of energy. We now know that these functions are H, A, and G. At the time we said that only U has a simple physical meaning - the sum of all the kinetic and potential energies of all the particles. There is no simple physical explanation for enthalpy and the two free energies. The best we can do is tell how they are used.
1) Most simple-minded.
Set
(17)
Then, using the definition of Helmholtz free energy as we have done above we find that,
(18)
For any process at constant temperature we have,
(19)
That is, for a constant temperature process the Helmholtz free energy gives all the reversible work. For this reason the Helmholtz free energy is sometimes called the "work function." When a physicist says "free energy" without indicating Helmholtz or Gibbs, he usually means Helmholtz free energy.
Similarly, we can write,
(20)
For a process at constant temperature and pressure we get,
(21)
That is, for a process at constant temperature and pressure the change in Gibbs free energy gives all the reversible work except the pV work. This work might include electrical work, work creating surface area, and so on. Chemists do most of their reactions on the bench top at constant pressure. When a chemist says "free energy" she almost always means Gibbs free energy unless she specifically states otherwise.
2) More useful - two new criteria for equilibrium
Recall that the second law of thermodynamics,
(22)
gives us the fundamental criterion for equilibrium. That is, in a closed isolated system entropy seeks a maximum,
(23)
Although this is the fundamental definition of equilibrium it is not the most useful definition because we do not often work with closed isolated systems. More often we work with systems at constant temperature and either constant volume or constant pressure. We can use the second law, Equation 22, and our new functions A and G to find criteria for equilibrium under these conditions.
Rewrite the second law, Equation 22 as follows:
(24)
or
(25)
Now, going back to the original form of the first law with only pV work,
(26)
and making the transformation to Helmholtz free energy we get,
(27a, b, c)
For a process at constant temperature and volume we have,
(28)
We conclude that for a process at constant temperature and volume the Helmholtz free energy seeks a minimum. Any spontaneous process in a system at constant T and V must decrease the Helmholtz free energy (if the system is away from equilibrium) or leave the Helmholtz free energy unchanged (if the system is at equilibrium).
By the same token, we can use the Gibbs free energy to discuss processes at constant temperature and pressure,
(29a, b, c)
from which we conclude that for a process at constant T and p,
(30)
That is, at constant T and p the Gibbs free energy seeks a minimum. Any spontaneous process in a system at constant T and p must decrease the Gibbs free energy (if the system is away from equilibrium) or leave the Gibbs free energy unchanged (if the system is at equilibrium).
Maxwell's Equations
We now have the tools to derive some very useful relationships between thermodynamics variables. Maxwell's equations are based on the same principle as was Euler's test for exact differentials, namely that mixed second derivatives of "nice" functions must be equal. Applying this principle to our two new free energy functions we find,
(31)
but we already know the first derivatives of A from Equations 11a and 11b. So,
(32a, b, c)
We obtain another similar equation from the Gibbs free energy,
(33)
which becomes, using Equations 16a and 16b,
(34a, b, c)
There are two more Maxwell's equations from dU and dH, but these are not as useful as the ones just derived. We will leave it to the reader to find the Maxwell's equations from dU and dH.
First Application of a Maxwell's Equation
As our first application of a Maxwell's equation we will derive the so-called thermodynamic equation of state which we stated without proof earlier. Write the combined first and second laws,
Divide Equation 1 by dV and hold T constant to get,
(35)
Using the Maxwell's Equation, Equation 32c, to substitute for the entropy derivative we obtain,
(36)
Equation 36 is the equation that was written down without proof at the time we were discussing the Joule expansion.
The other version of thermodynamic equation of state, based on H instead of U, will be left as an exercise for the reader.
Summary
We now have four interesting and useful derivatives of entropy,
There are two other derivatives of entropy which might prove useful,
(37)
The other derivative, 
will be left to the reader.
WRS
From here you can:
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Copyright 2004, W. R. Salzman
Permission is granted for individual, noncommercial use of this file.
salzman@arizona.edu
Last updated 09 Jul 04
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A Lazy Layman's Guide to Quantum Physics
James Higgo 1999
What is Quantum Physics?
That's an easy one: it's the science of things so small that the quantum nature of reality has an effect. Quantum means 'discrete amount' or 'portion'. Max Planck discovered in 1900 that you couldn't get smaller than a certain minimum amount of anything. This minimum amount is now called the Planck unit.
Why is it weird?
Niels Bohr, the father of the orthodox 'Copenhagen Interpretation' of quantum physics once said, "Anyone who is not shocked by quantum theory has not understood it".
To understand the weirdness completely, you just need to know about three experiments: Light Bulb, Two Slits, Schroedinger's Cat.
Two Slits
The simplest experiment to demonstrate quantum weirdness involves shining a light through two parallel slits and looking at the screen. It can be shown that a single photon (particle of light) can interfere with itself, as if it travelled through both slits at once.
Light Bulb
Imagine a light bulb filament gives out a photon, seemingly in a random direction. Erwin Schroedinger came up with a nine-letter-long equation that correctly predicts the chances of finding that photon at any given point. He envisaged a kind of wave, like a ripple from a pebble dropped into a pond, spreading out from the filament. Once you look at the photon, this 'wavefunction' collapses into the single point at which the photon really is.
Schroedinger's Cat
In this experiment, we take your pet cat and put it in a box with a bottle of cyanide. We rig it up so that a detector looks at an isolated electron and determines whether it is 'spin up' or 'spin down' (it can have either characteristic, seemingly at random). If it is 'spin up', then the bottle is opened and the cat gets it. Ten minutes later we open the box and see if the cat is alive or dead. The question is: what state is the cat in between the detector being activated and you opening the box. Nobody has actually done this experiment (to my knowledge) but it does show up a paradox that arises in certain interpretations.
If you dare to think about it (you're not really supposed to), you have to believe one of the following things:
MENU
Your consciousness affects the behaviour of subatomic particles
- or -
Particles move backwards as well as forwards in time and appear in all possible places at once
- or -
The universe is splitting, every Planck-time (10 E-43 seconds) into billions of parallel universes
- or -
The universe is interconnected with faster-than-light transfers of information
----
Full English Breakfast
Coffee or Tea
These are the results of the different interpretations of quantum physics. The interpretations all compete with each other. Otherwise respectable physicists can get quite heated about how sensible their pet interpretation is and how crazy all the others are. At the moment, there's about one new interpretation every three months, but most of them fit into these categories.
What does it mean?
The meaning of quantum physics is a bit of a taboo subject, but everyone thinks about it. To make it all a bit more respectable, it is better to say 'ontology' than 'meaning' -- it's the same thing. There are several competing interpretations and the one thing they all have in common is that each of them explains all the facts and predicts every experiment's outcome correctly.
Copenhagen Interpretation (CI)
This is the granddaddy of interpretations, championed by the formidable Niels Bohr of Copenhagen university. He browbeat all dissenters into submission (with the notable exception of Einstein) at a Brussels conference sponsored by a man called Solvay in 1927. Bohr thereby stifled the debate for a generation or two.
The CI has a bit of a cheek calling itself an interpretation, because it essentially says "thou shalt not ask what happens before ye look". He pointed out that the Schroedinger equation worked as a tool for calculating where the particle would be, except that it 'collapsed' as soon as you took a peek. If anyone asked why this was, he would say, "shut up and calculate" (or he might as well have done).
When you do try to take Copenhagen seriously you come to the conclusion that consciousness and particle physics are inter-related, and you rush off to write a book called The Dancing Wu-Li Masters.
More recently, Henry Stapp at the University of California has written papers such as On Quantum Theories of the Mind (1997). Stapp's central thesis is that the synapses in your brain are so small that quantum effects are significant. This means that there is quantum uncertainty about whether a neuron will fire or not - and this degree of freedom that nature has allows for the interaction of mind and matter.
What happens to the cat? You're not allowed to ask.
Many Worlds Interpretation (MWI)
The various paradoxes that the Copenhagen Interpretation gave rise to (famously Schroedinger's cat, and Einstein's dislike of "spooky action at a distance") led others to keep on trying to find a better interpretation.
The simplest was put forward by a student, Hugh Everett, in 1957. He simply said that the Schroedinger equation does not collapse. Of course, everyone laughed at him, because they could see that the photon, for example, was in just one place when they looked, not in all possible places. But after a couple of decades, this issue was resolved with the concept of decoherence - the idea that different universes can very quickly branch apart, so that there is very little relationship between them after a tiny fraction of a second.
This has led to what should strictly be called the 'post-Everett' Interpretation, but is still usually called MWI. It is now one of the most popular interpretations and has won some impromptu beauty contests at physics conferences. Unfortunately it means that billions of you are splitting off every fraction of a second into discrete universes and it implies that everything possible exists in one universe or another. This comes up with its own set of hard-to-digest concepts, such as the fact that a 500-year-old you exists in some universes, whereas in others you died at birth.
In 1997, Max Tegmark at Princeton University proposed an experiment to prove that MWI was correct. It involved pointing a loaded gun at your head and pulling the trigger. Of course, you will only survive in those universes where the gun, for whatever reason, fails to go off. If you get a misfire every time, you can satisfy yourself -- with an arbitrarily high level of confidence -- that MWI is true. Of course, in most universes your family will be weeping at your funeral (or possibly just shaking their heads and muttering).
What happens to the cat? It's dead in half the subsequent universes and alive in the other half.
Pilot Waves, Hidden Variables and the Implicate Order
David Bohm (1917-1992) was a very brilliant physicist and that's why people went along with him when he came up with an elegant but more complicated theory to explain the same set of phenomena (normally, more complicated theories are disqualified by the principle known as Ockham's Razor).
Bohm's theory follows on some original insights by Prince Louis de Broglie (1892-1987), who first studied the wave-like properties of the behaviour of particles in 1924. De Broglie suggested that, in addition to the normal wavefunction of the Copenhagen Interpretation, there is a second wave that determines a precise position for the particle at any particular time. In this theory, there is some 'hidden variable' that determines the precise position of the photon.
Sadly, John von Neumann (1903-1957) wrote a paper in 1932 proving that this theory was impossible. Von Neumann was such a great mathematician that nobody bothered to check his maths until 1966, when John Bell (1928-1990) proved he'd bodged it and there could be hidden variables after all -- but only if particles could communicate faster than light (this is called 'nonlocality'). In 1982 Alain Aspect demonstrated that this superluminal signaling did appear to exist, although David Mermin then showed that you could not actually signal anything. There is still some argument about whether this means very much.
Bohm's theory was that the second wave was indeed faster than light, and moreover it did not get weaker with distance but instantly permeated the entire universe, acting as a guide for the movement of the photon. This is why it is called a 'pilot wave'.
This theory explains the paradoxes of quantum physics perfectly. But it introduces a new faster-than-light wave and some hidden mechanism for deciding where it goes -- to create an 'implicate order'. That's quite a lot of extra baggage, and scientists like to travel light. Worse still, Bohm went on to become a mystic, identifying his 'implicate order' with Eastern spirituality and spawning books like Fritjof Capra's The Tao of Physics . That's heretical behaviour in the eyes of any decent physicist.
What happens to the cat? It's either dead or alive, of course!
Consistent Histories
The Consistent Histories interpretation, put forward by Robert Griffiths in 1984, works backwards from the result of an experiment, arguing that only a few possible histories are consistent with the rules of quantum mechanics. It's an interesting idea but not very popular because it still doesn't explain how a particle can go through two slits and interfere with itself. Roland Omns, in The Interpretation of Quantum Mechanics (1994) wrote down 80 equations in a single chapter and came to the conclusion that the 'consistent histories' interpretation was pretty much the same as Copenhagen, with a few knobs on.
What happens to the Cat? Again, you're not supposed to ask.
Alternate Histories
The Alternate Histories Interpretation is quite different, being similar to the Many-Worlds Interpretation, but with the insistence that only the actual outcome is the real world and the ones we're not in don't actually exist. Unfortunately this gets us right back to their being some kind of 'collapse'.
What happens to the cat? Again, you're not supposed to ask.
Time Reversibility
Richard Feynman (1918-1988) was a genius who developed a new approach to quantum mechanics. He formalised its crowning achievement, Quantum Electrodynamics, which is the most accurate scientific theory ever devised. He also developed the Feynman Diagram, which represents the interaction of two particles as the exchange of a third particle. This diagram has time on one axis and space on the other and the interaction can be viewed as happening both in forward and in reverse time.
An electron, on its way from point A to point B, can bump into a photon. In the diagram this can be drawn as sending it backwards not just in space, but also in time. Then it bumps into another photon, which sends it forward in time again, but in a different direction in space. In this way, it can be in two places at once.
There is little doubt that a Feynman diagram offers the easiest way to predict the results of a subatomic experiment. Many physicists have seen the power of this tool and taken the next step, arguing that reverse time travel is what actually happens in reality. Victor Stenger of the University of Hawaii argues strongly for this ontology in his forthcoming book. Of course, for a layman, it is hard to understand why a photon bounces around in such a way that it appears in two slits at once.
What happens to the Cat? It is both dead and alive simultaneously. We don't see this because of the macroscopic 'measurement problem'.
Transactional Interpretation
Like Stenger's, John Cramer's Transactional Interpretation relies on the fundamental time-symmetry of the universe. He argues that particles perform a kind of 'handshake' in the course of interacting. One sends out a wave forward in time, and another sends one out backwards in time.
What happens to the Cat? Ermm...
Gremlins
A new interpretation, presented for the first time here, is that there are little green gremlins hovering around, going backwards and forwards in time, shaking hands and collapsing with mirth as they poke and prod subatomic particles in a way they calculate most likely to confuse us. This explains all of the observed experimental results, but it does introduce gremlins, and the need for a further theory about why they should want to confuse us. Using the principle of Ockham's razor, this interpretation will probably not find much popularity among the scientific community although it may be the basis for a new religion. Watch this space.
What happens to the Cat? Depends on what the gremlins think will confuse us most.
Dark Matter
How do we know that dark matter isn't just normal matter exhibiting strange gravity? A new observation of gravitationally magnified faint galaxies far in the distance behind a massive cluster of galaxies is shedding new dark on the subject. This image from the Hubble Space Telescope indicates that a huge ring of dark matter likely exists surrounding the center of CL0024+17 that has no normal matter counterpart.
What is visible in the above image, first and foremost, are many spectacular galaxies that are part of CL0024+17 itself, typically appearing tan in color. Next, a close inspection of the cluster center shows several unusual and repeated galaxy shapes, typically more blue. These are multiple images of a few distant galaxies, showing that the cluster is a strong gravitational lens. The relatively weak distortions of the many distant faint blue galaxies all over the image, however, indicates the existence of the dark matter ring. The computationally modeled dark matter ring spans about five million light years and has been digitally superimposed to the image in diffuse blue.
A hypothesis for the formation of the huge dark matter ring holds that it is a transient feature formed when galaxy cluster CL0024+17 collided with another cluster of galaxies about one billion years ago, leaving a ring similar to when a rock is thrown in a pond.
Image credit: NASA, ESA, M. J. Jee and H. Ford et al. (Johns Hopkins Univ.)
+ Full Resolution (720 Kb)
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Cat's Diary
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Fourth dimension
There are three conventional spatial dimensions: length (or depth), width, and height, often expressed as x, y and z. x and y axes appear on a plane Cartesian graph and z is found in functions such as a "z-buffer" in computer graphics, for processing "depth" in imagery. The fourth dimension is often identified with time in physics, and as such is used to explain the non-Euclidean space-time used in Einstein's theories of special relativity and general relativity.
When a reference is used to four-dimensional co-ordinates, it is likely that what is referred to is the three spatial dimensions plus a time-line. If 4 (or more) spatial dimensions are referred to, this should be stated at the outset, to avoid confusion with the more common notion that time is the Einsteinian fourth dimension.
The implications of another spatial dimension are now discussed. This would be orthogonal to the other three spatial dimensions. The cardinal directions in the three known dimensions may be referred to as up/down (altitude), north/south (latitude), and east/west (longitude). When speaking of the fourth spatial dimension, an additional pair of terms is needed. Attested terms include ana/kata (sometimes called spissitude or spassitude), vinn/vout (used by Rudy Rucker), and upsilon/delta.
If time is counted as the "fourth dimension", the additional fourth spatial dimension would be referred to as the fifth dimension.
Dimensional analogy
To make the leap from three spatial dimensions into four, a device called dimensional analogy is commonly employed. Dimensional analogy is studying how (n – 1) dimensions relate to n dimensions, and then inferring how n dimensions would relate to (n + 1) dimensions.
For example, in Edwin Abbott Abbott's book Flatland, he writes about a square that lives in a two-dimensional world, like the surface of a piece of paper. A three-dimensional being has seemingly god-like powers from the perspective of this square: such as being able to remove objects from a safe without breaking it open (by moving them across the third dimension), see everything that from the two-dimensional perspective is enclosed behind walls, and remaining completely invisible by standing a few inches away in the third dimension. By applying dimensional analogy, one can infer that a four-dimensional being would be capable of similar feats from our three-dimensional perspective. Rudy Rucker demonstrates this in his novel Spaceland, in which the protagonist encounters four-dimensional beings who demonstrate such powers.
A useful application of dimensional analogy in visualizing the fourth dimension is in projection. A projection is a way for representing an n-dimensional object in n − 1 dimensions. For instance, computer screens are two-dimensional, and all the photographs of three-dimensional people, places and things are represented in two dimensions by removing information about the third dimension. In this case, depth is removed and replaced with indirect information. The retina of the eye is a two-dimensional array of receptors but it can allow the brain to perceive the nature of three-dimensional objects using indirect information (such as shading, foreshortening, binocular vision etc.). Artists use perspective to give three-dimensional depth to two-dimensional pictures.
Similarly, objects in the fourth dimension can be mathematically projected to the familiar 3 dimensions, where they can then be more conveniently examined. In this case, the 'retina' of the four-dimensional eye is a three-dimensional array of receptors. A hypothetical being with such an eye would perceive the nature of four-dimensional objects using indirect information contained in the images it receives in its retina. Perspective projection from four dimensions produces similar effects as in the three-dimensional case, such as foreshortening. This adds four-dimensional depth to these three-dimensional pictures.
Dimensional analogy also helps in understanding such projections. For example, two-dimensional objects are bounded by one-dimensional boundaries: a square is bounded by four edges. Three-dimensional objects are bounded by two-dimensional surfaces: a cube is bounded by 6 squares. By applying dimensional analogy, one may infer that a four-dimensional cube, known as a tesseract, is bounded by three-dimensional volumes. And indeed, this is the case mathematically: the tesseract is bounded by 8 cubes. Knowing this is key to understanding how to interpret a three-dimensional projection of the tesseract. The boundaries of the tesseract project to volumes in the image, not merely two-dimensional surfaces. This helps in understanding features of such projections that may otherwise be very puzzling.
Likewise the concept of shadows can help us better understand the theory of four dimensions. If you were to shine a light on three dimensional object, it would cast a two dimensional shadow. Therefore light on a two-dimensional object would cast a one-dimensional shadow (in a two-dimensional world), and light on a one-dimensional object in a one-dimensional world would cast a zero-dimensional shadow, that is, a point of non-light. This idea can be used in the other direction; light on a four-dimensional object would cast a three-dimensional shadow.
As an example of this, imagine that light is shone down through a wireframe cube onto a flat surface. The shadow that results is that of a square within a square with each of the corners connected. Similarly, if a four-dimensional cube were lit "from above", its shadow would be that of a three-dimensional cube within another three-dimensional cube.
Being three-dimensional we are only able to see the world with our eyes in two dimensions; a four-dimensional being would see the world in three. Thus it would be able, for example, to see all six sides of an opaque box simultaneously. Not only so; it would also be able to see what was inside the box at the same time, just like in Flatland, where the sphere sees objects in the two-dimensional world and everything inside them simultaneously. Analogously, a four-dimensional viewer would see all points in our 3-dimensional space simultaneously, including the inner structure of solid objects and things obscured from our three-dimensional viewpoint.
Reasoning by analogy from familiar lower dimensions can be an excellent intuitive guide, but care must be exercised not to accept results that are not more rigorously tested. For example, consider the formulas for the circumference of a circle C = 2πr and the surface area of a sphere: A = 4πr2. One might be tempted to suppose that the surface volume of a hypersphere is V = 6πr3, or perhaps V = 8πr3, but either of these would be wrong. The correct formula is V = 2π2r3.
9 g in a centrifuge
A pilot training in a centrifuge where the acceleration exceeds 9G.
A normal person faints between 4 and 6G. The figure in the bottom left corner represents the number of G.
Stellar Jewel Box
Thousands of sparkling young stars are nestled within the giant nebula NGC 3603, one of the most massive young star clusters in the Milky Way Galaxy.
NGC 3603, a prominent star-forming region in the Carina spiral arm of the Milky Way about 20,000 light-years away, image reveals stages in the life cycle of stars.
Powerful ultraviolet radiation and fast winds from the bluest and hottest stars have blown a big bubble around the cluster. Moving into the surrounding nebula, this torrent of radiation sculpted the tall, dark stalks of dense gas, which are embedded in the walls of the nebula. These gaseous monoliths are a few light-years tall and point to the central cluster. The stalks may be incubators for new stars.
On a smaller scale, a cluster of dark clouds called "Bok" globules resides at the top, right corner. These clouds are composed of dense dust and gas and are about 10 to 50 times more massive than the sun. Resembling an insect's cocoon, a Bok globule may be undergoing a gravitational collapse on its way to forming new stars.
The nebula was first discovered by Sir John Herschel in 1834.
Image Credit: NASA, ESA, and the Hubble Heritage (STScI/AURA)-ESA/Hubble Collaboration
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My other side, after my little Halloween stint:
There's a hidden message in "JIngle Bells" when played backwards. Press the button 'go' and listen carefully. Make sure to turn the volume up so you can really hear the hidden message.
Triple product rule
The triple product rule, known variously as the cyclic chain rule, cyclic relation, or Euler's chain rule, is a formula which relates partial derivatives of three interdependent variables. The rule finds application in thermodynamics, where frequently three variables can be related by a function of the form f(x, y, z) = 0, so each variable is given as an implicit function of the other two variables. For example, an equation of state for a fluid relates temperature, pressure, and volume in this manner. The triple product rule for such interrelated variables x, y, and z is given by
Here the subscripts indicate which variables are held constant when the partial derivative is taken. That is, to explicitly compute the partial derivative of x with respect to y with z held constant, one would write x as a function of y and z and take the partial derivative of this function with respect to y only.
The advantage of the triple product rule is that by rearranging terms, one can derive a number of substitution identities which allow one to replace partial derivatives which are difficult to analytically evaluate, experimentally measure, or integrate with quotients of partial derivatives which are easier to work with. For example,
Various other forms of the rule are present in the literature; these can be derived by permuting the variables {x, y, z}.
Derivation:
An informal derivation follows. Suppose that f(x, y, z) = 0. Write z as a function of x and y. Thus the total derivative dz is
Suppose that we move along a curve with dz = 0, where the curve is parameterized by x. Thus y can be written in terms of x, so on this curve
Therefore the equation for dz = 0 becomes
Dividing by dx and rearranging terms gives
Dividing by the derivatives on the right hand side gives the triple product rule
Note that this proof makes many implicit assumptions regarding the existence of partial derivatives, the existence of the total derivative dz, the ability to construct a curve in some neighborhood with dz = 0, and the nonzero value of partial derivatives and their reciprocals. A formal proof based on mathematical analysis would eliminate these potential ambiguities and grey zones.
Black Widow Nebula Hides in the Dust
In this Spitzer image, the two opposing bubbles are being formed in opposite directions by the powerful outflows from massive groups of forming stars. The baby stars can be seen as specks of yellow where the two bubbles overlap.
When individual stars form from molecular clouds of gas and dust they produce intense radiation and very strong particle winds. Both the radiation and the stellar winds blow the dust outward from the star creating a cavity or bubble.
In the case of the Black Widow Nebula, astronomers suspect that a large cloud of gas and dust condensed to create multiple clusters of massive star formation. The combined winds from these groups of large stars probably blew out bubbles into the direction of least resistance, forming a double bubble.
Image credit: NASA/JPL-Caltech/Univ. of Wisc.
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