Showing posts with label quantum mechanics. Show all posts
Showing posts with label quantum mechanics. Show all posts

Sunday, January 26, 2014

What is Spin? A Concrete Explanation.

To say that a particle has "spin 1/2" is to say that it must be rotated through 720 degrees before it can return to its original configuration.  This is not something normally witnessed in the world of classical mechanics, and so this aspect of quantum mechanics is often piled up with unhelpful metaphors and mysticism.

I wrote a post previously trying to point out that quantum mechanical spin is just a degree of freedom.  Spin tells you the components of a particle in a combination of two wave states with the same energy.  You can make pseudospins and isospins with any two such states, no matter what they are.  When you rotate the system, the components get mixed up -- just like angular momentum states.  You have to rotate the system by 720 degrees before the components get mixed up enough to be un-mixed up (i.e. back to there they were).  That's all it is.

What gives spin states this weird property is that the space of rotation is three dimensional, but the spin "vector" is only two-dimensional.  Rotations of typical vectors with three components (even if one of those components is zero) work just the way you'd think they should.  But, it's not completely surprising that 2D objects in 3D space don't rotate like 3D objects in 3D space.

To illustrate where spin comes from, and how it contrasts to orbital angular momentum, consider the case of rotation in 2 dimensions.  The best way to talk about rotations is to start at the unit circle.

Thursday, January 23, 2014

Everything Cool is Impossible


Physics has known for a long time how to build a time machine.  The possibility in a real spacetime geometry was first noted by Van Stockum, but this possibility was only really first analyzed by Frank Tipler in the 70's.  All you need is a massive rotating cylinder.  And also it has to be infinitely long.

This illustrates how frame dragging
can lead to time travel 
Since then, at least a dozen other possibilities have been proposed for time travel to the past, and physicists have proven that these spacetime geometries result in what are called "Closed Timelike Curves" (CTCs), which are trajectories a massive object could follow to go back in to its own past.  We know that they would work within the theory of General Relativity.  But, they're all impossible.  They either require the universe to be rotating (it isn't), they require infinitely large systems (we can't make them), they require negative-mass matter (no such matter exists), or they require you perform your time travel within the interior event horizon of a Kerr black hole (which is fine, but then you can't leave).

This situation is worse than merely having a concept of physics that excludes time travel, or that merely says that time travel is impossible.  For if time travel was excluded by theory, then we could always say the theory was incomplete.  What we have instead is a system that fully allows time travel possibilities without prejudice, as long as we're able to break some other law of physics to get there.  It's not just the stubborn "no" of a parental figure; it's like having your parents describe step-by-step exactly what you can do to eat chocolate cake for breakfast, and one of those steps is "eat infinite broccoli".

Physics also knows how to effect FTL travel.  The speed of light puts a prohibitive barrier on
our ability to explore the stars, but a number of work-arounds have been proposed.  Technically, relativity only prohibits local FTL movement, but says nothing of global FTL travel.  So if you can distort space and time in just the right way, you can move however fast you want.  One of the more frequently explored proposals is wormhole travel.    Wormholes produce a kind of "short cut" in spacetime, and it is actually a Federal Law that when you want to discuss how wormholes work you must draw two dots on a sheet of paper, "A" and "B", draw the straight line connecting them, then fold your paper so "A" and "B" touch and jab a pencil through it.  While going along the line you draw may take billions of years, going through the wormhole may take minutes.
My lawyers also recommend I show you this diagram

Sadly, you can't make a wormhole.  And even if you made a wormhole, the throat collapses when you try to travel inside of it, so you can't even use the wormhole for travel anyway.

Another proposal is the Alcubierre warpdrive.  This contracts spacetime in the front and expands it in the back, producing what some call a "wave" of spacetime contraction that "tips over" the light cones inside the warp bubble.  Locally, you're moving slower than light, but globally you may be moving, in theory anyway, as fast as you want.

But you can't make the Alcubierre warp drive either.  If you took the mass of the universe and made it negative, the Alcubierre warp drive requires ten times that number in negative-mass matter to move a standard-sized spaceship.   To clarify, we haven't even found one single particle of negative-mass matter.

Science knows how to make a Bag of Holding, and can even make a Bag of Holding that slows down time (see chapter 3 here).  You can store a lifetime supply of hot pies and ice cream in the same box, and whenever you take them out the pie is still oven-fresh and the ice cream still ice cold, and so even twenty years later you can serve yourself delicious pie a la mode.  But, like so many awesome things, it requires either negative mass or impossible mater distributions and can't be made.

I just made a post about how the Bag of Holding (aka, Van den Broeck Bubble) can be exploited to, potentially, travel to parallel worlds (if any even exist).  This one is a lot more speculative, requiring ideas way beyond established science, but is at least partially based in what we already know about general relativity and curved-space geometry.  It isn't really scientific, but if we wanted to know if there were other universes, this has potential to actually find them.  But it also requires not only negative mass, but infinitely much of it.  So we won't ever be able to try.

Pictured: A guy wearing a green screen.
Not Pictured: An invisibility cloak 
Science has pretty recently discovered (less than ten years ago) how to make a literal cloak of invisibility.  It involves bending light in just the right way.  We know what that just-the-right-way way is, and we even know how to make materials that bend light in just that way.  Sadly, it only works for a single frequency (i.e. color) of light at a time.  There's no way to be completely invisible, because there don't exist materials with  the right optical properties naturally.  So you can be green-invisible, but you'll still be perfectly visible in red and blue.  I guess you'll just look slightly more purple?

I recently calculated (as part of my research) how to make a slightly different kind of cloak, namely a shadow cloak.  Also something you'd read about in fantasy books, the shadow cloak works on the same spacetime distortion principle as for a black hole, but now modified to work with optical materials (so not requiring it be made of actual black holes).  A perfect realization  would allow light to enter, but trap it there.  If you were wearing it, you would appear to be not just covered in a black garment, but actually swathed in shadows.  (Look at a black object, then look at an unlit hole; there's a big visual difference)  You'd also probably heat up a lot (since all the energy is trapped), which would make this kind of material perfect for solar panels, increasing their efficiency probably to near 100%.  But you can't make the shadow cloak, because it requires material parameters that are both infinite and negatively infinite.  Like with the invisibility cloak, you can only realize this (if at all) for a single color of light at a time.  Which vastly diminishes its coolness.

You can probably see where my knowledge tends to specialize, but physics knows a lot more cool things in the quantum domain, such as teleportation devices and solutions to the P=NP problem.  All of which, we know how it would work, and only minor technicalities render it impossible.  Things like wavefunction collapse, quantum decoherence, and the no-cloning theorem.

Any time there's something cool in physics, there's something else that renders it impossible.

Again, this isn't the situation of wanting to do something incredible and merely lacking a theoretical model to describe it.  Our formulations of physics account for it exactly.

It's just that all the cool stuff is impossible.

More and more, it just seems like the Universe comes equipped with fail-safes against our ever doing the cool things of science fiction.

Thursday, August 1, 2013

The Bottomless Starbucks Gift Card and Quantum Immortality

I have recently acquired an item of rare wonder and power.  An artifact of legend, forged in a mythical age.  I am now the owner of the Bottomless Starbucks Gift Card.

From Piled Higher and Deeper
How this enchanted relic came in to my possession is common enough.  Believe it or not, it was given to my mother (a middle school teacher) as an end-of-semester present.  She, seeing no need to for it, did bequeath it unto me.  And I, a grad student in physics, have found very much need for some extra coffee money.

I've gone through a number of these re-gifted Starbucks cards from my mom, almost all of which were for $5.  They got me about two uses, then I'd switch to the next.  I seriously carried four or five of them around, gradually burning through them.  But the Bottomless Card... that's the last one I came to.

I have no idea how much money is on it, or was on it.  I go up to the counter, order whatever I want, show them the card, they swipe it, and there's always still enough money left for next time.

There is an interpretation of Quantum Mechanics that is called Everett's Many-Worlds Hypothesis.  This is often misunderstood and abused by science fiction authors, and philosophers as implying something stronger than it actually does -- the actual existence of parallel universes with alternative versions of ourselves (like in His Dark Materials).  This isn't quite what it means; it's more like every quantum measurement, rather than resulting in a collapse of the wave function, actually results in the further entanglement of the observer with one of the terms in the superposition.  The parts of the universal wave function describing us continue to exist but now in a superposition, one with every possibility of the measurement.  It's kind of the same thing, but not really.
From Asbtruse Goose

Everett's is a popular interpretation and appears frequently cited in "popular science" articles and books.  It is not the strict implication of quantum mechanics, nor is it anything more than a philosophical framework built around quantum mechanics, but it's there and cited a lot.  Most of the appeal is the fantasticality of it; alternative universes, Narnia, cool!  There's also some physicists who prefer it for philosophical reasons; for instance, Everett's hypothesis would recover a deterministic universe, which was believed to be broken by quantum measurement.  Actually, Frank Tipler -- physicist, transhumanist, many-worldsist, and all around weird dude --  proposed an experiment to test Everett's based on the convergence of quantum interference patterns, to see if "probability" were in fact "leaking" to another universe.  I have no idea why no one has done this experiment yet, but he's put it out there.

Tipler's experiment is slightly more sane than another proposal: Quantum Immortality.

Quantum Immortality is - roughly - proposed to work in the following way.  You have a quantum gun; whether it fires a bullet when you pull the trigger is tied to some quantum mechanical superposition, so there is always a chance it won't fire.  In the Everett interpretation, each time you do this, your wave function splits in to two "worlds": one where the gun fires, the other where it doesn't.  The experiment calls for you to point the gun at your head and pull the trigger.  In Everett's interpretation, each time you do this, your wave function splits in to "dead" and "alive" parts; therefore, even if you do this 10,000 times, there still exists some version of you in some "universe" that is still alive.  Therefore, if you pull the trigger 10,000 times and live, you can conclude that you live in the "world" where you're still alive.

Here's an illustration from Super Mario World, where Mario keeps splitting and one Mario copy always survives:


The Quantum Immortality experiment doesn't require that you point the gun at your head.  It basically just states that if you keep making a quantum observation and keep getting the same result, then it makes more sense to assume you live in a universe that is a segment of a multiverse than that you just keep getting lucky.  You could even do this experiment with...

... a Starbucks Gift Card.

I have no idea how much money is on the card.  Each time I swipe it, I make an observation of whether or not there is sufficient money for my purchase.  There always is.  Always.  It's been weeks, and I still have enough money.  I've even started ordering fancy-fru-fru drinks and it keeps working.  It always works.

So now you can see how it works.  So long as I don't directly observe the exact amount of money on the card, there is no exact amount of money on the card!  Between "No Money" and "Yes Money", I also happen to live in the universe where the Card always splits to the "Yes Money" side of things.  Always.

And that is how I came upon the key to eternal coffee, and the strange mysteries that went to forging its powers.

[P.S. I'm not going to bother explaining every thing wrong with the Quantum Immortality proposal, nor my wonky application of it to an inherently non-quantum event.  Suffice it to say, almost none of it is scientifically rigorous, and Everett's interpretation is pretty dumb, even if it makes for fun science fiction.]

Sunday, May 26, 2013

Whether Something Can Come From Nothing, and Quantum Mechanics

It is very popular  in certain circles that place a high value on the classical scholastic arguments for the existence of God to ask "why is there something rather than nothing?"  Ex nihil, nihil fit, is the Latin phrase, that from nothing, nothing comes.  If there is something, then why?  How did it get here?

It is then popular in certain circles that place a high value on scientific understanding --- people who perhaps don't understand math well enough to study it for real, but who nonetheless appreciate human efforts to understand the natural world in terms of rational processes and read as much of it as they can understand --- to make the rebuttal claim that, according to the physical understanding of quantum mechanics, something can come from nothing.


You can see an example of this conversation in the below video:


The idea is that in quantum field theory, study has shown that even in the state representing a vacuum, i.e. a system with zero particles, there is still the constant process of random particle-antiparticle pair creation and annihilation going on all the time.  You start with zero particles, and for brief instances you have two particles.  Or, in higher order interactions, four, or one hundred and twenty four.  Therefore, something -- particle-antiparticle pairs -- can come from nothing -- the quantum vacuum.

This idea is right, and it's wrong.  I think both people are talking past each other, and in this post, I would like to try to clarify.

I'm not a field theorist.  I've had some grad classes in it, but it's not anything in which I'm an expert (in fact, there probably isn't anything in which I'm an expert, but it's a helpful caveat).  Still, what I'm about to say is very basic to field theory (if anything in field theory can be called "basic"), and I'm more or less directly citing the text Field Quantization by Greiner and Reinhardt (available on Amazon for only $\$20$!).  What follows is a very, very brief outline of how quantum field theory leads to the understanding of the quantum vacuum, but also how the results therein do not mean what many people think it means.  I have some wikipedia links throughout, so that hopefully people who do not understand math can at least follow along with what I'm trying to say -- the math isn't important, but the physics is.

The Uncertainty Principle and Energy Non-Conservation, part 2

Quantum mechanics is typically interpreted to mean that the conservation of energy can be violated as long as the time scales involved are short.  An old professor of mine used to summarize it as "there is such a thing as a free lunch, if you can eat it fast enough."

Here's how the argument goes.  From quantum mechanics, we get the uncertainty relation
$$\Delta E \Delta t \geq \hbar,$$
where $\Delta E$ is the uncertainty -- or statistical spread -- of the energy, and $\Delta t$ is the uncertainty of the time.

Following this, physicists reinterpret the uncertainty $\Delta E$.  Rather than representing a quantification of our lack of knowledge about the energy of a system, this is interpreted as being, somehow, the amount of "free" energy that a system can borrow in violation of the First Law of Thermodynamics.  So if we have mean energy $E$ and uncertainty $\Delta E$, it means we "actually have" energy $E$, and then Nature gracefully lends us $\Delta E$ to overcome some energy barrier, which we quickly repay in time $\Delta t$.

However, that puts us at 
$$\Delta E \geq \hbar/\Delta t$$
which puts no limitation how much energy we can borrow.  Or, rather, it puts a lower bound; we must borrow at least $\hbar/\Delta t$ worth of energy.  Or, we could borrow even more!   If this is true, then we have infinite energy forever!

The oil companies will go bankrupt!