Physics Discussion Thread

brihaspati · Post by **brihaspati** » 29 Jun 2011 05:12

Amber G. wrote:
vina wrote: Probably some sort of vortex /circulation gets created at the foot of the waterfall, that tends to draw swimmers underwater ?

Slightly different thought was from Brihaspatyiji regarding buoyancy being reduced by air/gas bubbles... What people think may be the main point?

I mentioned bubbles/air/gas because "foam" was mentioned. Recent rains could have also affected density - but that would require assumption of significant salinity/dissolved salts in "dry" phase. Even then the difference might not be that significant for floatation. But yes, strong vortex due to heavy water flow could also be responsible.

vina · Post by **vina** » 29 Jun 2011 05:33

kasthuri wrote:And mathematicians want to be decently paid!

And hence become bankers!

. Courant to Wall St is not even a subway ride away, but a short brisk walk away! Droves of them end up there.

And AmberG said, that lots of PhD and Post Doc Physics students she knew ended up in Wall St.

Well, if "Complex Analysis" = Physics is to be believed, that is really not a loss at all, but those folks are actually pushing the frontiers of Physics!

Think of it. Biology was something that scared me as a high school kid. I ran from it at the first opportunity. Much of that had to do with the way it was though in India about rote memorizations of plants and animals and strange sounding latin names and little "logic" (math,phy,chem was easy, you have a few basic axioms/ "knowns", you can arrive at the rest pretty decently at high school level without too much trouble). I regret that running away now.

And what about human psyche and intellect? Can Physics EVER model that ? You atleast need the abstraction of pure math for that. Now, if you want to include "Game Theory" , "Behavioral Effects" etc into Physics, be my guest. But before that , atleast get your arms around what the human psyche and mind is about before even attempting. How can Game Theory exist at all without the human mind ? It is a mind game after all. How about all that behavioral effects. Some of those aren't even "rational" ! Einstein is said to have been perturbed by the new fangled quantum theories and said "God Does Not Play Dice" . I dont know about God, but humans do , and play lot more than a nice "predictable" dice.

So net-net, cut out that "Physics can Describe the Universe " part. It cant . Ever. At best it can define the PHYSICAL part (or rather parts of it). There are stuff that it simply cant. Well ,let me back up a bit here. If you start including "Meta- Physics" as "Physics"

, sure you can.

GuruPrabhu · Post by **GuruPrabhu** » 29 Jun 2011 07:36

For a moment I thought "Physics envy" was about folks looking in from other fields who wished they could have the kind of fun physicists have.

One day banking will explain the inflation problem in cosmology. After all it is just a "bubble".

When was the last time that a field that ruined the world sought credit for doing so?

There is an index for "productivity" of various sectors of any economy - Banking scores approximately ZERO. People may think that there is some "output" in moving money around and skimming a profit - but, serious estimates rate that activity as parasitical.

vina · Post by **vina** » 29 Jun 2011 08:06

GuruPrabhu wrote:There is an index for "productivity" of various sectors of any economy - Banking scores approximately ZERO. People may think that there is some "output" in moving money around and skimming a profit - but, serious estimates rate that activity as parasitical.

While very very OT here in a Physics thread, I am just struck at that entire ridiculousness of that proposition you just stated. It is as ridiculous as stating "Physics created the atom bomb, so all of Physics is just a nihilistic, savage and amoral activity!".

In fact the "Soviet Union" did some incredibly great Physics and "Science" by any account. However it doesn't exist today despite all that "productivity" you quote. And a part of the reason is that it didn't have the "banking system" that you deride which is a vital part of the economy! In fact, in the history of the world, part of the reason why Western Europe could see sustained growth and prosperity for the last 400/500 years and country after country from that small part of the world could get the resources to explore , invest, populate and yes conquer the entire world, while no other part of the world could was in large part thanks to the "banking system". In fact, without it, the age of enlightenment and science could not have been sustained and invested in and grow to what it is today.

GuruPrabhu · Post by **GuruPrabhu** » 29 Jun 2011 08:10

Cognitive dissonance!

The new benefit of banking is colonial torture!!

vina · Post by **vina** » 29 Jun 2011 08:49

GuruPrabhu wrote:Cognitive dissonance!

The new benefit of banking is colonial torture!!

Well, if you think your asinine trolling makes you come across as very smart, suit yourself.

I could retort saying on the lines of "You could at least survive 'colonial torture'. Getting nuked will makes you atomized forever!" , but that is dragging down to your level. I have no desire to do it . So indulge yourself in your own make believe fantasies about "describing the universe" and the rest of it and all the rest of it.

And get off your own high horse of pumped up self importance. If TIFR and BARC and all that "High Physics" fell off the coast of Mumbai into the sea, no one will bat an eyelid. If D Subba Rao ,so much as f*arts or SBI stops working, there will be turmoil roiling the country. Same if Ben Bernanke did in the US and Lawrence Livermore fell into the ground that opened up beneath it in an earthquake.

Countries and civlizations existed for millenia without "Physicists" (infact Somalia , Eritrea types do and still survive),but without a banking system and a functioning economy, will will not only sink into a Rwanda or worse, but in addition have no Physicists either.

So quit trolling and get a life. Bye.

GuruPrabhu · Post by **GuruPrabhu** » 29 Jun 2011 08:53

uh oh, someone who loves ROFL is offended by ROFL

on page 13 of dhaagaa:

vina wrote:

BENIS lingo has been replaced by indignant self-aggrandization and victim syndrome

negi · Post by **negi** » 29 Jun 2011 09:08

AmbarG is the said phenomenon related to rip currents ?

negi · Post by **negi** » 29 Jun 2011 09:12

One more question I hear that electron's mass and size are known so why/how does hisenberg's uncertainity principle hold true what prevents one from taking a snapshot of an electron in orbit in one of those 's','p' or 'd' orbitals and be able to measure it's position and momentum at any given instant say 't1' ?

GuruPrabhu · Post by **GuruPrabhu** » 29 Jun 2011 09:15

negi wrote:One more question I hear that electron's mass and size are known so why/how does hisenberg's uncertainity principle hold true what prevents one from taking a snapshot of an electron in orbit in one of those 's','p' or 'd' orbitals and be able to measure it's position and momentum at any given instant say 't1' ?

Mass is known. Size is an upper limit. In principle it is a "point particle".

However, the act of "snapshot" is what disturbs the electron [in a classical sense - snapshot involves scattering photons off the electron. This changes the state of the electron.]

In quantum sense, there are no "orbits" - there is only probability density.

In second quantized sense, there is no density either. There are only scattering cross sections.

negi · Post by **negi** » 29 Jun 2011 09:21

Ok just like scanning electron microscopy is used to resolve objects which are smaller than the wavelength of visible light why cannot one use smaller wavelengths to bombard an atom and image the electron by capturing and mapping the scattered/reflected/transmitted waves ?

I suppose the challenege is the electron has already chnaged location between the time period 'T' between transmit and rececption of the scanning beam ?

negi · Post by **negi** » 29 Jun 2011 09:25

in a classical sense - snapshot involves scattering photons off the electron. This changes the state of the electron.

Got my answer

; so for a layman a photon has enough kinetic energy to displace a electron ? (Is this what photoelectric effect is a consequence of ?)

GuruPrabhu · Post by **GuruPrabhu** » 29 Jun 2011 09:31

negi wrote: Got my answer ; so for a layman a photon has enough kinetic energy to displace a electron ? (Is this what photoelectric effect is a consequence of ?)

Photoelectric effect is when photon commits soosai and sends the electron to its 72.

For scattering off an electron (also known as Compton effect), a photon with very little energy can have an effect.

However, this is a classical concept regarding "uncertainty" and hence completely BOGUS. In QM, uncertainty is couched in non-commutation of operators, such as position and momentum.

But then, in QM, an electron sits in an "eigenstate" and can not change its energy (except when electron goes to jannat, i.e., "ionized", either via photoelectric effect or compton scattering]

so ultimately, one is forced to view this in that dreaded language called math.

Amber G. · Post by **Amber G.** » 30 Jun 2011 01:44

Negiji - You may already have seen that .. and probably any good resource would be helpful but I would recommend the following ...
Feynman's Lectures..
or Gamow's Mr Tompkins (or One Two Three Infinity)
or One of Online courses... (I have to look for the link but if interested, I can give one)

One way to look at it: If you want to "see" the electron aka nice microscope ..If you use visible light frequency the electrons are much smaller than the wavelength (few hundred nm about 10 millions time the size of electron).. and there will be quite a bit of uncertain of seeing it... If you use shorter wavelength (thus each photon will have much higher energy - h (nu)).. you will "disturb" the electron because now you have higher energy photon...(Shorter wavelength (more precision on position)==> higher frequency (energy of photon to disturb the electron)

But .. a good book would give much better treatment than I can give here...

hth

GP - Did you see the current Scientific American's review of a book ..
How the Hippies Saved Physics
or When flower power met quantum theory
Enjoy!

Amber G. · Post by **Amber G.** » 30 Jun 2011 01:48

From Harvard Physics Dept ..
.

You will seel traveling waves, standing waves, beating, and random motion... choreography of the dance of the pendulums is stunning! Aliasing and quantum revival can also be shown.

The period of one complete cycle of the dance is 60 seconds. The length of the longest pendulum has been adjusted so that it executes 51 oscillations in this 60 second period. The length of each successive shorter pendulum is carefully adjusted so that it executes one additional oscillation in this period. Thus, the 15th pendulum (shortest) undergoes 65 oscillations. When all 15 pendulums are started together, they quickly fall out of sync—their relative phases continuously change because of their different periods of oscillation. However, after 60 seconds they will all have executed an integral number of oscillations and be back in sync again at that instant, ready to repeat the dance.

Amber G. · Post by **Amber G.** » 30 Jun 2011 02:09

Here is a classic (from Harvard Library) lecture from Prof Coleman.. Audience is supposed to be a mix of Physics Grad Students, Undergrads, Chemists, Philosophers, Bankers... etc..IOW perfect for this audience...

Quantum Mechanics in Your Face

GuruPrabhu · Post by **GuruPrabhu** » 30 Jun 2011 07:58

Amber G. wrote: If you want to "see" the electron aka nice microscope ..If you use visible light frequency the electrons are much smaller than the wavelength (few hundred nm about 10 millions time the size of electron)..

It is much worse than that. Wavelength of light is order 10^-7 m. The upper limit on the size of the electron is 10^-19 m. So 1 Trillion times smaller.

In Dirac/Feynman/QED picture, the electron is a point particle. Which is why these classical explanations are not complete. QFT is the only viable approach.

GuruPrabhu · Post by **GuruPrabhu** » 30 Jun 2011 08:08

Amber G. wrote: GP - Did you see the current Scientific American's review of a book ..
How the Hippies Saved Physics
or When flower power met quantum theory
Enjoy!

Thanks. I will try to find time to read this book. But I agree with this comment on the website:

I hate to nit-pick, but this article is titled "How the Hippies Saved Physics", and if you read the whole article, what it ends up saying is that, the hippies didn't save physics at all. So I feel sort of cheated.

I also spent the 70s reading Tao of Physics and Dancing Wu Li Masters while smoking the good stuff. Soviet invasion of Afghan increased the supply of good stuff. [Afghan refugees came to camps in Delhi. They brought the best commodity they had and sold it to make a living. Some bought guns and went back. This is before Charlie Wilson's War]

But these books did not increase my physics understanding. All they did was increase my appetite for illusions. Thankfully, it was corrected in a timely fashion

If I recall correctly, the author of Tao of Physics ends with how he "sat on a beach and watched the dance of Shiva in cosmic rays". Utter BS.

Amber G. · Post by **Amber G.** » 30 Jun 2011 08:53

GuruPrabhu wrote:
Amber G. wrote: If you want to "see" the electron aka nice microscope ..If you use visible light frequency the electrons are much smaller than the wavelength (few hundred nm about 10 millions time the size of electron)..
It is much worse than that. Wavelength of light is order 10^-7 m. The upper limit on the size of the electron is 10^-19 m. So 1 Trillion times smaller.

Well, I am a brf oldie.. about 10 Billion years old (trust me, when I was young, age of the universe was considered about 4 billion years... Now it is 14..

.) so electron was big then ... classical radius (Lorentz radius..) that was considered to be 10^(-13) (cm) in gcs. (but I just took it SI unit so missed a factor of 100) .s....

But it does not change the gist of what I was trying to say.. So let me quote my book:

they say..

An electron is sure hard to please.
When spread out, it sometimes will freeze.
Though agoraphobic,
It's still claustrophobic,
And runs off when put in a squeeze.

UBanerjee · Post by **UBanerjee** » 30 Jun 2011 11:19

Amber G. wrote:Negiji - You may already have seen that .. and probably any good resource would be helpful but I would recommend the following ...
Feynman's Lectures..
or Gamow's Mr Tompkins (or One Two Three Infinity)

I loved this book as a teenager and second the recommendation.

However this thread is completely over my head so back to lurking.

Vayutuvan · Post by **Vayutuvan** » 01 Jul 2011 00:15

I really liked Interactions by Sheldon Glashow (Nobel Laureate along with Salam and Weinberg). Planning to re-read. What do the Physics gurus here think?

negi · Post by **negi** » 01 Jul 2011 08:25

OK so I hear a lot about 'gravity bends the space time fabric' bhat does it mean ? What is space time fabric ? Is it an entity akin to 'lumiferous ether' which modern physicsts have conceived to explain the GTR ? On a related note it is now established that light rays appear to be bent by gravity; what happens to the speed of light ? Looks like concept of space time fabric has been introduced to somehow ensure that speed of light remains constant.

Prasad · Post by **Prasad** » 01 Jul 2011 08:30

edit: crap posted it in the wrong thread. sorry adminullah

Rahul M · Post by **Rahul M** » 01 Jul 2011 09:07

maulana prasad, did you mean to post it here ?

spacetime fabric of GTR is same as 4 dim space we use in special theory, 3 space and 1 time. if you think about it, we basically live in it, don't we ? that is independent of whether GTR is a valid theory or not.
it is not quite like ether theory because spacetime is real so to speak and not an artificial construct added by hand, which was the case with ether. even if in the future gravitational phenomena are explained by theories that do not require spacetime or 4-space, the construct of spacetime won't be affected.

I mean we do live in 3 dim space and move through time in one direction (towards future), so there's your 4-space !

of course, that is a simplistic way of looking at it, the other is from the POV of gravity and where its properties are inherently connected with matter. even then, it is not supposed to be a physical object like ether. a rough analogy would be the boundaries of a picture is the spacetime while the picture itself is matter.

p.s. oh yes, light slows down (or speeds up or remains same) in grav fields, depending on where you observe it from. it does not however slow down if you are in the same frame, or so einu baba tells us.

there's a very approachable (but not accurate since we can't represent 4-space) analogy of spacetime bending, the rubber sheet example is one name for it. a typical representation.

sanjeevpunj · Post by **sanjeevpunj** » 01 Jul 2011 09:25

Light can be slowed down to near 38 mph speeds!
http://news.harvard.edu/gazette/1999/02.18/light.html

negi · Post by **negi** » 01 Jul 2011 09:31

Rahul M wrote: p.s. oh yes, light slows down in strong grav fields.

Does it ? can you elaborate ? I thought that concept of gravitation bending space time was introduced to ensure that speed of light remain constant (i.e. by increasing the net distance traveled by the light). Secondly if what you say is correct then does it mean refractive index of a medium is influenced by gravity/mass ?

Rahul M · Post by **Rahul M** » 01 Jul 2011 09:33

sanjeevpunj wrote:Light can be slowed down to near 38 mph speeds!
http://news.harvard.edu/gazette/1999/02.18/light.html

that's a different thing. in a medium the photons still travel at c but collide, get absorbed and then emitted etc. we can think of it as the photons taking a detour inside the medium and hence getting late in coming out.

negi, that part was loosely worded originally. I think I have elaborated now.
check it now.

negi · Post by **negi** » 01 Jul 2011 09:42

Got it; another question from where the hell does photon get this 'energy' to travel with constant speed irrespective of reflection/refraction or even when bent by gravity ? Also if all the matter in the universe is supposed to comprise of a same fundamental particle then shouldn't there be a relationship (as in being made of same fundamental building block) between photon and quarks(or whatever is the latest particle discovered of late) ?

sanjeevpunj · Post by **sanjeevpunj** » 01 Jul 2011 09:46

Rahul M wrote:
sanjeevpunj wrote:Light can be slowed down to near 38 mph speeds!
http://news.harvard.edu/gazette/1999/02.18/light.html
that's a different thing. in a medium the photons still travel at c but collide, get absorbed and then emitted etc. we can think of it as the photons taking a detour inside the medium and hence getting late in coming out.

Right, it is a different situation in the lab.That slowing down of light is under really extreme conditions in a lab, with extreme conditions of temperature and vaccuum, suspended by strong magnetic fields, as described in the article. I just posted it as a useful reference.

Amber G. · Post by **Amber G.** » 01 Jul 2011 19:59

^^^That is of course interesting.. (This was also in top news item in many mainstream news media then.

Also were in the news ...
UC Berkeley had the speed slowed down to about 10Km/s in a semiconductor .. and IBM had announced microchips which can slow down light (switches using that will use million times less power, they say)..

U of R (Rochester) had a nice demo once, where letters "UR" projected were slowed down..

GuruPrabhu · Post by **GuruPrabhu** » 02 Jul 2011 01:48

Rahul M wrote: ... in a medium the photons still travel at c but collide, get absorbed and then emitted etc. we can think of it as the photons taking a detour inside the medium and hence getting late in coming out.

This is not correct. If the photon were to be absorbed and re-emitted etc, it would change wavelength. Clearly, this does not happen.

Actually, the particle picture of light does not have any way of explaining refractive index. Only the wave picture does. Refractive index is a bulk phenomenon, just like cerenkov radiation, transition radiation etc. Thus, they require coherent behavior from the medium - particle physics/QM does not have a formalism for that. It is good old classical physics of Maxwell and electronic model of matter that comes to the rescue.

This is why it is difficult to get rid of photon particle/wave duality.

Rahul M · Post by **Rahul M** » 02 Jul 2011 02:11

why would frequency be changed if energy of emitted photon is same as that of incident photon ?

btw, you can check feynmann's QED for a treatment of light-matter interaction in terms of photons.

sanjeev ji, it's not about lab conditions, light slows down while passing through atmosphere as well (in the sense it is used in those experiments).

GuruPrabhu · Post by **GuruPrabhu** » 02 Jul 2011 03:51

Rahul M wrote:why would frequency be changed if energy of emitted photon is same as that of incident photon ?

There is no mechanism to ensure that. Energy levels are discrete and finite. So, how can a material be able to absorb and re-emit photons of all arbitrary energies.

Plus, you can check absorption and emission spectra of materials. They are also not necessarily the same.

You can also check the typical lifetime of an absorption and re-emission process. Plus, the laws governing re-emission produce an exponential decay curve. This implies that your mechanism will have:

1. Long delays (10s of nanoseconds per absorption and re-emission), while the transmission delays over a 1 cm are the order of 10s of picoseconds.

2. The delay time for any photon will be a probabilistic quantity, thus making the refractive index a probabilistic quantity, which it is not.

===============

You can do a thought experiment. Take a jar of liquid hydrogen. The energy levels of hydrogen are well known. Now, if you shine a laser light of a wavelength which does not correspond to any energy level difference of hydrogen, what will happen?

Will the photons slow down or not?

===============

btw, you can check feynmann's QED for a treatment of light-matter interaction in terms of photons

QED does not deal with light-"matter" in the sense of "bulk material". It deals with light-"particle" interactions, i.e., with charged particles like electrons.

To be more precise, QED has a current that describes a vertex of a photon and a fermion and an anti-fermion. The interaction is current-dot-current.

GuruPrabhu · Post by **GuruPrabhu** » 02 Jul 2011 07:36

GuruPrabhu wrote: The interaction is current-dot-current.

Before some Guru jumps on me for my careless lingo, the "dot" above refers to "an inner product of two complex bi-spinors". The bi-spinors in this case contain "gamma-mu" to form a "vector" for the purpose of parity transformation.

SriKumar · Post by **SriKumar** » 02 Jul 2011 20:00

GuruPrabhu wrote:Well, the reasoning is quite simple really. Physics believes that the universe arose as described by big-bang cosmlogy. This event happened about 14 billion years ago, and the universe then evolved according to the laws of physics. No one will claim that *all* laws are known - the claim is that there were *some* set of laws that governed the evolution of the universe. No one also claims that they know *how* these laws came into being.

Not to belabour the point further but I did want to get clarified- by 'laws' of physics, what is meant? Does it 'merely' and exclusively refer to a set of observations of various systems in the universe, and a verbal and/or mathematical statement of those phenomena?

On a semi-related note, I was trying to see why are mathematical manipulations important i.e. what is their sancitity. If you look at the basis for a simple but widely used and powerful mathematical operation- differentiation of a function, it is: (f(x + delta x) -f(x))/delta x as delta x -> 0. The whole thing seems like a very reasonable thing to do (and eminently more explainable) when it is represented as a geometric curve, and it is suggested that the solution to this equation gives us the slope of the curve f(x) that that point, which is also the rate of change of f(x) with x. But from a purely mathematical point of view, it boggles my mind that someone would have decided to run this calculation for a range of functions just to see what it gave them, without a geometric description of the problem.

. Why would it even be interesting? Like, go back to 500 A.D. in India. 'Aho Sahasrabuddhi! Aham ganitham pathAmi'. What do you think we'd get if we took this equation f(x) and did an f(x+ sukshma change) - f(x) and divided it by the sukshma change. (sukshma = small). Kim vadati, Sahasrabuddhi?. Why would anyone think of even doing such a manipulation and more importantly, find it interesting. sin(x) gives cos(x). X^n gives (n) x^(n-1) etc. Big deal.

.

Amber G. · Post by **Amber G.** » 02 Jul 2011 20:14

RahulM, GP ... From news.harvard.edu (later update on the slow light story..)

Prof Hau' description in her words...

"Two years ago we slowed it down to 38 miles an hour; now we've been able to park it then bring it back up to full speed."
<snip>
...Hau explains that light entering the atomic entanglement transfers its energy to the atoms. Light energy raises the atoms to higher energy levels in ways that depend on the frequency and intensity of the light. The laser illuminating the cloud at right angles to the incoming beam acts like a parking brake, stopping the beam inside the cloud when it is shut off. When it is turned on again, the brake is released, the atoms transfer their energy back to the light, and it leaves the end of the cloud at full speed and intensity.

Hau's team stopped light for one-thousandth of a second. Atomically speaking, "this is an amazingly long time," Hau notes. "But we think it can be stopped for much longer."..

From: Researchers now able to stop, restart light:
http://www.news.harvard.edu/gazette/200 ... light.html

SriKumar · Post by **SriKumar** » 02 Jul 2011 20:18

AmberG, your questions remind me of that book Physics for Entertainment by Yakov Perelman. I bought the first volume but did not buy the second one. If I can find it now, I'd pay 100 times the amount that I bought it for, back then.
Speaking only for myself, I think a subject like physics is best introduced (and taught) through examples that people encounter first hand (and in daily life). Of course, this is not possible for a lot (or even most) of physics but there is enough going on around us that is very interesting and instructive. Your question about trucks and highway noise in the evening is something I experienced first hand. I used to hear trucks in the night from a highway that was almost 1.5 miles away from home, and I used to wonder how that could be.

Amber G. · Post by **Amber G.** » 02 Jul 2011 20:54

^^^ You may also enjoy his "Physics Can Be Fun and Mathematics Can Be Fun"
And scores of other books by Perelman like: Arithmetic for entertainment ...Mechanics for entertainment, Geometry for Entertainment , Astronomy for entertainment etc...
(Along with serious text books)..

Also like many I grew up on Landau and Lifshitz series (uniformly excellent) of text books..
One the most excellent book I bought (about Rs 1 in paper back in India) was Landau's (yes great Landau) popular book "ABC of (or what is - or something like that) Relativity" ...the book was about 100 pages (or less) but certainly the most favorite popular book of mine, I have seen on this subject.

Bade · Post by **Bade** » 02 Jul 2011 20:57

Amber G. wrote: From: Researchers now able to stop, restart light:
http://www.news.harvard.edu/gazette/200 ... light.html

I seem to have difficulty buying into this "light is stopped" business as a description of what is happening. It is not like a single photon is being tagged at the input end of the black box and the "same" photon is coming out at the other end after stopping for some time even if nanoseconds late.

How is this any different from well understood absorption and re-emission of energy from de-excitation from higher energy levels. All that is being achieved from casual reading of the experiment is that due to the cooling of the absorbing atoms the re-emission is delayed significantly and appears as slowing down light.

A free photon cannot be slowed, as then it cannot have zero mass anymore and would behave like any other matter particle.

Here the information in the photon beam is being transferred to the material and the same information is being extracted with a delay, behaving like a memory device and hence its potential application for other uses in computing.

kasthuri · Post by **kasthuri** » 02 Jul 2011 21:04

SriKumar wrote: On a semi-related note, I was trying to see why are mathematical manipulations important i.e. what is their sancitity.... Why would it even be interesting?

Simply put, mathematical manipulations are interesting just because of our psychological urge in abstractions or generalizing things. Without question, there is a beauty in it. Derivative of sin(x) = cos(x) would have been a problem in high school books, but to represent dot product as an inner product and thus defining uncertainty in terms of certain non-commutative operators in function spaces is in fact an *interesting* pursuit. The smaller derivations/manupulations are a by-product for verifying generalizations.

Added later: Manipulations could be of interest in defining counter examples and contradictions too....and they (contra/counter) are integral part of any generalization.