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Relativity and Quantum Mechanics: the elevator pitch

Tuesday, August 24th, 2010

Physicist: A woman on the subway, about two stations away from her stop, asked us “what are relativity and quantum mechanics?”
So, this is a two-stop elevator pitch for the two most pivotal sciences since slicedbreadology.

Elevators: Wonderful, mechanical rooms, quietly skirting the ever-thinning line between broom closet and robot.

Relativity: Speed is just distance over time (as in “miles per hour”). Normally when you change you’re own speed, the speeds of everything else changes (for your point of view). For example, if you’re walking slowly down the street everyone else will be moving quickly (and, for the sake of this example, in the same direction), but if you pick up the pace and walk normally, then everyone else will barely be moving at all.

But the speed of light is different. No matter how you move, it will always stay the same. Since that particular speed refuses to change, distance and time have to change instead.  Relativity is the study of how distance and time change with speed, and the consequences that follow from those changes.

Quantum Mechanics: When you look at very, very small objects, like individual particles, you begin to find that they don’t behave right. If particles were like ordinary objects, but smaller like tiny billiard balls, then you’d expect them to act like ordinary (but tiny) objects. Instead, they “ooze” from place to place, move through impassable barriers, exist in several places at the same time, and interfere with each other. It’s impossible even to say exactly where they are.
All of these are impossible (or at least very unlikely) behaviors for solid “particle-ish” objects. But all of these behaviors are explained, and even expected, if all of matter is actually some kind of wave.
So, quantum mechanics is the (more-accurate-than-every-other-science) study of the universe from the perspective that everything, at the lowest levels, is made up of some kind of waves.

Posted in -- By the Physicist, Quantum Theory, Relativity | 2 Comments »

Q: How can electrons “jump” between places without covering the intervening distance?

Thursday, August 12th, 2010

Physicist: Frequently in quantum mechanics you’ll find that particles are restricted to only a certain set of states or locations, and yet somehow they can move from one to the next. It’s like moving between islands without crossing any water.
“Classically” (19th century, pre-relativity, pre-quantum) this is impossible. If you see a particle in one place and then see it again, but in a new place, then of course it must have traversed the distance from one to the other.
But here’s the essential difference between quantum mechanics (correct) and classical physics (wrong): particles aren’t solid objects that have a genuine position, instead they’re waves that are “smeared out”.

A standing wave (like a guitar string, or an electron orbital) usually has “nodes” where the wave is always zero.

The string remains stationary in the middle, but the wave has no problem getting past that point. This image stolen, unrepentantly, from http://people.rit.edu/andpph/text-string-vibrations.html.

But of course there’s a big difference between the wave being zero at a node, and the wave being unable to get to the other side of that node. It’s the difference between the string in the above picture, and what you would have if you nailed that string to a piece of wood at the node and left the bottom half dangling.

For you calculus buffs out there, the difference is hidden away in the derivatives.

If you have a particle (wave) like this and you measure where it is many times you’ll find that it’s on the top about half the time, on the bottom about half the time, and never ever in the middle (ever).

The shape of the standing waves formed inside a microwave oven. The peaks and troughs move up and down, but the lines in between are stationary and zero. This is why microwaves have rotating plates, to keep food from sitting in the nodes (and staying cold) or from staying in the peak areas (and burning).

One of my personal fave examples of “large scale” quantum weirdness is microwave ovens.  A microwave oven creates a standing wave of photons with a “plus-shaped” (+) node, which leaves the center of the chamber especially cold.  The chance of finding a photon anywhere on the plus-shaped-node is zero.  So there are four different cells that it should be impossible for the microwave photons to move between, but they don’t seem to mind at all.  It’s not like particles exist or something.

So thinking of things like electrons as particles leads to incorrect conclusions.  Thinking of them as waves is really  the way to go.

Posted in -- By the Physicist, Physics, Quantum Theory | No Comments »

Quantum mech, choices, and time travel too!

Friday, August 6th, 2010

Physicist: Recently I sent a series of emails back and forth with a reader that seem interesting enough to post. Conversations (near a chalkboard especially) are the best way to learn just about anything.

 

Q:
I wake up at 6am.
I brush my teeth get dressed and go downstairs.
I eat my breakfast at 7am and I’m just about to leave out the door.
when i notice there is an apple and an orange in a fruitbowl on a table next to me
at exactly 7:02am. I CHOOSE to take the apple.

I then make a choice whether I should choose to take my car or save gas and take the bus.
I CHOOSE to take the bus.

SUDDENLY. when i walk into the office, a strange event occurs and time starts moving backwards.
It goes back and back and back until finally it’s 6am in the morning and i wake up,
brush my teeth get dressed and go downstairs to eat my breakfast.

Here’s my tough quantum mechanics question for you,
at exactly 7:02am will I still choose to take the apple?

And if this process were repeated over and over and over again for about a million times. will the choice I make ALWAYS BE the apple?

 

A:
I’m working on a (to long) post about Bell’s theorem. The thought experiment you propose, about going back in time, is one of the better ways to understand it.

To actually answer your question: if choosing the apple is based on some quantum mechanical process in your brain (and there’s a good chance that at least some part of it is), then every time that choice is made the result is random. Time travel or not.
Part of the weirdness comes from the fact that every possible thing that can happen does. So (even when you time travel) some versions of you take the apple and some versions don’t.

 

Q:
Ok, remember how I took the bus in the thought experiment?

My question is, does quantum mechanics also apply in a reversal of time?

For instance, lets say that time started to slowly reverse.
Will I always get onto the bus backwards and head home.

OR.

will my car magically appear (even though i didn’t take it)
and will I backwards drive home in that?

So the concluding question is,
do quantum principles apply in a reversal of time as it does when time moves forwards?

 

A:
You’ll often hear “everything that can happen does” so if a particle can take two different paths it will actually take both.
If I understand your question correctly, the answer is yes. It turns out that “everything that could have happened did”. The “branching” goes both forward and backward in time. This is demonstrated by things like the “Franson experiment” that demonstrates the interference of a single photon with an earlier version of itself.
Driving a car, for example, will leave telltale signs that later make it impossible for you to have actually taken a bus. Chair fibers, leaving tire tracks, you’ll remember it, etc.
But if, in every way, you could have done either one, then you did both (no magically appearing cars).
This is actually the backbone of the Feynman path integral technique.

 

Q:
So are you telling me that just next to us, could exist a place where the Nazis won the second world war?
A place where there exists a flying spaghetti monster? (to quote richard dawkins)

Or even a place out there in a dimension somewhere where there exists an all knowing omnipresent, omnipotent, all encompassing being who “watches over us” etc..

 

A:
Sure. BUT, it’s impossible to interact with things that are even a little bit different. For example, a stream of identical photons (lasers) will all interact with each other strongly. You can see evidence of this in effects like speckling. Non-coherent (regular) light is made up of all kinds of different photons, and the best way to figure out how they’ll behave is to assume that they’ll ignore each other. This is sort of a metaphor, and sort of a concrete example.
So while, yes, there are almost certainly universes where the Nazis won, it doesn’t matter. It’ll never have any impact on our universe whatsoever.
A good rule of thumb is: if there is any conceivable way, whatsoever, for anything to tell the difference between universes, then they can’t interact (from the perspective of that thing that can tell the difference).

 

Q:
Couldn’t that then solve the entire God dilemma? I mean if in only one of these infinite dimensions there existed an all encompassing all knowing all powerful entity, wouldn’t this entity then transcend all dimensions? (since he is all encompassing)

 

A:
If you want to consider God, then it’s best not to do it in any kind of physics based context. That being said:
Remember that if two universes are even slightly dissimilar, they won’t interact at all. By “slightly dissimilar” I mean something like a single electron being conspicuously out of place.
So any existing Gods that follow the most basic laws of logic and quantum mechanics will be stuck in their native worlds.
If you’re not worried about Gods that follow physical laws, then, again, physics is literally the worst possible forum.
Also, you have to be careful with this kind of reasoning. You can make up just about anything and claim that it should exist in every version of the universe.
The rule “anything that can happen does” carries a bit more heft that it seems to at first. If something can’t happen, then it doesn’t happen in any version of the universe.
For example, spaghetti can neither fly nor think, so the FSM (pasta be upon him) can’t exist in any universe, no matter how much anyone dresses like a pirate.

Posted in -- By the Physicist, Philosophical, Physics, Quantum Theory | 2 Comments »

Q: Will we ever overcome the Heisenberg uncertainty principle?

Sunday, August 1st, 2010

Physicist: Nopers!

The Heisenberg uncertainty principle, while normally presented in physics circles, is actually a mathematical absolute.  So overcoming the uncertainty principle is exactly as hard as overcoming that whole “1+1=2″ thing.

The uncertainty principle (the “position/momentum uncertainty principle”) is generally presented like this: you have some particle floating along and you’d like to know its position and its momentum (velocity, whatever) so you bounce a photon off of it (“Bounce a photon off of it” is just another way of saying “try to see it”).  A general rule of thumb for light (waves in general really) is high frequency waves propagate in straight lines, and low frequency waves spread out.  That’s why sunlight (high frequency) seems to go in a perfectly straight line, but radio waves can spread out around corners.  For example, you can still pick up a radio station even when you can’t see it directly.

So, if you want to see where something is with precision you’ll need to use a high frequency photon.  After all, how can you trust the results from a wandering, low frequency photon?  But, if you use a precise, high-frequency, and thus, high-energy photon, you’ll end up smacking the hell out of the particle you’re trying to measure.  So you’ll know where it is pretty exactly, but it’ll go flying off with some unknown amount of momentum.  Any method you can come up with to measure the momentum will require you to use low-frequency, low-energy, gentle photons.  But then you won’t be able to figure out the particle’s position very well.

Low frequency photons (like radio waves) don't tell you much about where a particle is, but they doesn't knock it around much either (so you can measure its momentum better). High frequency photons (like sunlight) are terrible at measuring momentum, but can tell you position well.

So far this seems more like an engineering problem than a problem with the universe.  Maybe we could arrange things so that the high frequency photon hit softer or something?  There was a lot of back and forth for a long time (still is in some circles) about overcoming the uncertainty principle, but in the end it can never be violated.

Rather than being something that’s merely very challenging like, “you can’t break the sound barrier”, “what goes up must come down”, and “you can’t be the world’s best kick-boxer and be the world’s most handsome physicist”, the uncertainty principle is a mathematical absolute.  So, unless the basic assumptions of physics are completely wrong (and they’ve held up to some serious scrutiny), the uncertainty principle is in the company of things like “you can’t go faster than light”, “energy and mass are conserved”, and “modern mathematicians don’t have beards” (has anyone else noticed this?).  What follows is answer gravy.

Answer gravy: This gravy has some lumps.  If you know what a “Fourier transform” is, and are at least a little comfortable with them, then this could be interesting to you.

The square of a quantum wave function is the probability of finding it in a particular state.  For example, the “position wave function” can tell you the probability of finding a particle at any position. To get the probability from the wave function, all you have to do is square the wave function.

If you’ve got the quantum wave function f(x) for the position of a particle, then you can find the the momentum wave function, g(p), by taking the Fourier transform of f.  That is, g=\hat{f}.  Now, you can define the uncertainty as the standard deviation of the probability function, which is a really good way to go about it.

A probability function (blue), with its uncertainty or standard deviation (red). Like you'd expect, the particle is most likely to be near zero, but it's not certain to be near zero.

The uncertainty principle now just boils down to the statement that the product of the uncertainties of the square of a function, f, and the square of its Fourier transform, \hat{f}, is always greater than some constant.  In what follows you’ll find some useful stuff such as Plancherel’s theorem and Cauchy-Schwartz.

\begin{array}{ll}\sigma_x\sigma_p=\sigma_{|f|^2}\sigma_{|\hat{f}|^2}\\=\sqrt{Var(|f|^2)}\sqrt{Var(|\hat{f}|^2)}\\=\left(\int x^2|f|^2\,dx\right)^{\frac{1}{2}}\left(\int\xi^2|\hat{f}|^2\,d\xi\right)^{\frac{1}{2}}\\=\left(\int |xf|^2\,dx\right)^{\frac{1}{2}}\left(\int|\xi\hat{f}|^2\,d\xi\right)^{\frac{1}{2}}\\=\frac{1}{2\pi}\left(\int |xf|^2\,dx\right)^{\frac{1}{2}}\,\left(\int|\widehat{f^\prime}|^2\,d\xi\right)^{\frac{1}{2}}&(\widehat{f^\prime}=2\pi i\xi\hat{f})\\=\frac{1}{2\pi}\left(\int |xf|^2\,dx\right)^{\frac{1}{2}}\,\left(\int|f^\prime|^2\,d\xi\right)^{\frac{1}{2}}&(\textrm{Plancherel})\\ \ge\frac{1}{2\pi}\int|xf f^\prime|\,dx&(\textrm{Cauchy-Schwartz})\\=\frac{1}{2 \pi} \int |x| \, |f| \, |f^\prime| \, dx \\\ge \frac{1}{2 \pi} \left| \int x |f| f^\prime \, dx \right| \\= \frac{1}{2 \pi} \left| \int (x) (\frac{1}{2}|f|^2)^\prime \, dx \right|&(\frac{1}{2}|f|^2)^\prime=|f| f^\prime\\= \frac{1}{4 \pi} \left| \int |f|^2 \, dx \right|&(\textrm{integration by parts})\\=\frac{1}{4 \pi}&(\textrm{the total probability is always 1})\end{array}

So, there’s the Heisenberg uncertainty principle: \sigma_{|f|^2} \sigma_{|\hat{f}|^2} \ge \frac{1}{4 \pi}.  A physicist would recognize this as \Delta x \Delta p \ge \frac{\hbar}{2}.  The difference comes about because the Fourier transform that takes you from the position wave function to the momentum wave function involves an h, and \hbar = \frac{h}{2\pi}.  (For the physicists out there who were wondering what happened to their precious h’s)

Posted in -- By the Physicist, Equations, Math, Physics, Probability, Quantum Theory | 2 Comments »

Q: If gravity is the reaction matter has on space, in that it warps space, why do physicist’s look for a gravity particle? Wouldn’t gravity be just a bi-product of what matter does to space?

Saturday, July 31st, 2010

Physicist: Isn’t that weird?

The name “quantum mechanics” comes from the fact that, at its most base, quantum mechanics requires all particles and energies to come in discrete (one might say “quantized”) packets.  At some point a bunch of physicists starting asking awkward questions like; the matter is quantized, the energy is exchanged in quantized packets, so why do we assume the force is smooth and continuous?

Compounding this awkward line of questioning was the fact that photons were already known to carry electromagnetic force.  Literally, photons are little oscillating bits of electric and magnetic fields, which is exactly what the electric and magnetic forces are.  So the next obvious question was “do the other forces have ‘force carriers‘?”

You’re damn right they do.  Photons for electromagnetism, W and Z bosons for the nuclear weak force, and gluons for the nuclear strong force.  There’s every force but gravity!  Each of carriers were predicted by the (then new) study of “quantum field theory”, and have since been observed.  The theory itself is gorgeous and works beautifully.  In fact, it barely makes sense to think of anything in the universe (including space) as not being quantized.

So, some physicists are looking for evidence of the existence of gravitons (the gravity particle), because it would really tie things together nicely.  There are a couple drawbacks however…  In order for something to be detected it has to do something.  Gravity is a really, really weak force, and a graviton is the smallest amount of that force that can exist.  Most physicists have already given up any hope of detecting the graviton directly, and instead are looking at extremely indirect methods.  The drawback there is that the graviton (if it exists, and if our theories hold up) is a very strange particle, and is described using amazingly nasty math (even more nasty than the normally nasty math of quantum field theory).  So it’s difficult to even figure out what those indirect methods should be.

To actually answer the question: some physicists are looking for the graviton because it “fits”.

Posted in -- By the Physicist, Physics, Quantum Theory | No Comments »

Q: Will there always be things that will not or cannot be known?

Wednesday, June 30th, 2010

Mathematician: Unfortunately, limits to knowledge seem to be built into the nature of the universe, and even into logic itself.

Relativity: Einstein’s theory of special relativity implies that no information can travel faster than the speed of light. That means that information from sufficiently recent, sufficiently far away events will not have had the time to propagate to us yet, making detailed knowledge of such events impossible. In physics speak, we say that these events are outside of our “past light cone“, “space-like separated” from us, or just “elsewhere”. As long as new events of this type keep happening, there will always be things about which we do not and cannot know.

Quantum Mechanics: The Heisenberg uncertainty principle states that the uncertainty \Delta x we have in a particle’s position and the uncertainty \Delta p we have in the particle’s momentum cannot both be very small at the same time. In particular, the product of these uncertainties is greater than a constant (\Delta x \Delta p > \frac{\hbar}{2}). This implies a fundamental limit to the knowledge that is possible because we can know x accurately or p accurately, but not both.

What’s more, the vast majority of physicists agree that quantum mechanics demonstrates the universe is random at a fundamental level. This means that some events, like the time at which an atom will decay, can be predicted only probabilistically. We can say how likely an atom is to decay in a given time interval, but we will never be able to say precisely when the decay will occur, placing another limitation on what knowledge is possible. (Physicist’s note: After the decay you still can’t say when exactly it happened because according to quantum mechanics the exact time doesn’t actually exist!)

Mathematics: Gödel’s  first incompleteness theorem states (essentially) that any mathematical system  that is able to express elementary arithmetic (and doesn’t contain any contradictions) must contain true arithmetical statements that cannot be proven within that system. Essentially this implies that there will always be true mathematical statements that we cannot prove.


Add to all of these theoretical considerations the enormous (and possibly infinite) number of things that could be known about our physical universe, and the (most definitely) infinite number of true mathematical statements that could be known, and it is clear that there will always be knowledge that is beyond our reach.

Posted in -- By the Mathematician, Math, Philosophical, Quantum Theory, Relativity | 2 Comments »

Q: How it is that Bell’s Theorem proves that there are no “hidden variables” in quantum mechanics? How do we know that God really does play dice with the universe?

Tuesday, June 22nd, 2010

Physicist: Bell’s theorem, and its philosophical fallout, is one of the most profound discoveries since relativity.

Bell’s theorem states (among other things) that the universe is fundamentally unpredictable, and that quantum mechanical things (for example: everything) are not actually in one state.  If a box could contain either a blue marble or a red marble, then when you open it you’ll see either on or the other.  In “reality” it was one color or the other before you open the box.  In QM, it can be both before you open the box (it’s actually still both afterwords, but moving on…).

Einstein (and most other physicists of the time) believed that if you knew everything about a system of particles (no matter how big) that you could theoretically predict what that system will be doing in the future, perfectly.  Homeboy thought that the only reason that the movement of air molecules seems to be random, is that we can’t perfectly measure that exact position and velocity of every single one.  So he thought that every particle truly is in some particular state, but that we merely don’t know for sure what that state is (the marble in the box has only one color, but we don’t know what it is).

The idea that randomness and unpredictability are caused by unknown (or unknowable) things is called “hidden variable theory” (The ‘Stein believed in this).  For example; 2, 2, 3, 6, 0, 6, 7, 9, 7, 7, 4, 9, 9, … is not random, but seems random.  It would be really hard to predict the next term (7) if you don’t know the hidden variable.  (BTW, the “hidden variable” is: this is the decimal expansion of \sqrt{5})

Bell’s theorem essentially boils down to a proof that the result of an experiment doesn’t exist until the measurement is made (so it can’t be predicted).  Hidden variable theory presupposes that the particles involved are in definite states, which means that the result of a measurement already exists before the measurement is made.  For example: before you open a gift what you’ll see is already set in stone.  The gift is a set thing before you open the box.  This is not the case for most quantum mechanical systems.

Here’s one of the experiments that demonstrates Bell’s theorem, and two ways to look at it.

An entangled pair of photons is created and fired in opposite directions. En route the polarizers are randomly oriented, then the detectors measure whether or not the photons pass through. This is done hundreds of thousands of times to measure the relationship between 1) the difference in angles between the polarizers and 2) the probability of measuring the same result.

The experiment: Step 1: Create a pair of entangled photons and fire them in opposite directions.  Entangled particles always yield the same result when they are subjected to the same measurement, and are likely to yield the same result for similar measurements.

Step 2: Randomly orient the polarizers, after the entangled pair is created, but before either is detected (this is hard to time, and is really fast).  This is done so that the photons “don’t know what to expect” and “can’t compare notes”.  Information about polarizer A would have to travel faster than the speed of light to get to photon B before photon B hits it’s own polarizer.  So, without faster than light effects (which don’t exist for many, really good reasons) the photons are each acting independently.  The orientation is random so that the photons can’t “plan ahead”.

Step 3: Measure the polarization.  If the detector “clicks” then the photon made it through the polarizer, and therefore has the same polarization.  If the detector doesn’t click, then the photon had the opposite polarization and was stopped.

The probability of the measurements being the same (for an entangled pair) is P = \cos^2{(\theta)}, where \theta is the difference in angles between the polarizers.  It is tricky to see why, but this probability is impossible if you assume that the result of a measurement exists before the measurement is made.  Here’s why.

The possible polarizations for polarizer A (red) and polarizer B (blue).

Algebraic approach: Restricting the possible angles of the polarizers to 0° and 45° for A, and 22.5° and 67.5° for B, run the experiment. Here’s what’s about to happen:

1) If you could predict the outcome of each version of the experiment, then you could find a definite value of L (see below).

2) For strictly (unarguable) mathematical reasons L = ±2.

3) Experimentally we find that the average value of L is 2√2.

4) But this is a contradiction, so we cannot actually make useful predictions.

Now it’s happening:

If polarizer A is at 0° and the detector clicks then you’d say “A0 = 1″, and if the detector doesn’t click then “A0 = -1″.  Similarly, you can define B67.5, A45, and B22.5.  Just for the hell of it, take a look at: L = A0B22.5 + A45B22.5 + A45B67.5 - A0B67.5 = (A0 + A45)B22.5 + (A45 - A0)B67.5

L = (A0 + A45)B22.5 + (A45 - A0)B67.5 = ±2, since either (A0 + A45) = ±2 and (A45 - A0) = 0, or (A0 + A45) = 0 and (A45 - A0) = ±2.  So L = A0B22.5 + A45B22.5 + A45B67.5 - A0B67.5 = ±2 ≤ 2.

So if you could fill out each of these values (A0, A45, B22.5, B67.5), then L = ±2 ≤ 2.

However, you can’t make all of these measurements simultaneously, so you can’t actually get A0B22.5 + A45B22.5 + A45B67.5 - A0B67.5 for each run of the experiment.  The best you can do is find one of these four terms each time you run the experiment.  For example, if the polarizer A was randomly set to 45° and the detector clicked, and polarizer B was randomly set to 22.5° and the detector didn’t click, then you just found out that A45B22.5 = (1)(-1) = -1 for that run.

You can however find the expectation value by running the experiment over and over and keeping track of the results and polarizer orientation.

E[A0B22.5] + E[A45B22.5] + E[A45B67.5] – E[A0B67.5] = E[A0B22.5 + A45B22.5 + A45B67.5 - A0B67.5] ≤E[|A0B22.5 + A45B22.5 + A45B67.5 - A0B67.5|] = E[2] = 2.

So E[A0B22.5] + E[A45B22.5] + E[A45B67.5] – E[A0B67.5] ≤ 2.  This is one version of “Bell’s Inequality”, and it holds if each term (A0, A45, B22.5, B67.5) has a value.

Using the fact that the chance of getting the same result is P = \cos^2{(\theta)}, and that each term is 1 when the results are the same ((1)(1) or (-1)(-1)), and -1 when the results are different ((1)(-1) or (-1)(1)), you can calculate each term.  For example:

E[A_0B_{22.5}]=P(same)-P(different)=\cos^2{(22.5)}-(1-\cos^2{(22.5)})=\frac{1}{\sqrt{2}}

You’ll find that:

E[A_0B_{22.5}]+E[A_{45}B_{22.5}]+E[A_{45}B_{67.5}]-E[A_0B_{67.5}]=\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\frac{-1}{\sqrt{2}}=2\sqrt{2}

Holy crap!  2\sqrt{2}>2!  But that’s a violation of Bell’s inequality!  But the existence of each measurement (whether or not you actually do that measurement) is all you need for Bell’s inequality!  So if the inequality is false, then the result of those measurements don’t exist if the measurement isn’t made!

God plays dice with the universe.

Maybe, if you're clever and have ready access to a time machine, you could go back and do all the measurements you didn't make the first time. Then all the results would have to exist! They'd just have to!

Me and my time machine vs. quantum mechanics: If the results exist, but you just didn’t happen to do all the measurements, why not get a time machine?  Then you could do one measurement, go back, do a different measurement, go back, do a different measurement, …  Then every possible result would be known.

However, once again that correlation probability (P = \cos^2{(\theta)}) screws things up.

So, for example, if the photon goes through at 50°, and then you go back in time, change the polarizer to 51°, and repeat the experiment, then there’s a 99.97% (cos2(1°) = 0.9997) that the photon will go through again.

One result from probability says that P(x=z)\ge P(x=y)+P(y=z)-1.  Do this twice and you get P(w=z)\ge P(w=x)+P(x=z)-1\ge P(w=x)+P(x=y)+P(y=z)-2.  So if you measure in the 0° direction to find A0, then go back and change the angle by 1° and repeat this until you’re measuring at 90°, then:

P(A_0=A_{90})\ge P(A_0=A_1)+P(A_1=A_2)+\cdots+P(A_{89}=A_{90})-89 =90\cos^2{(1^o)}-89=0.9726

So, if you go back and forth in time to measure whether or not the photon goes through at 1° increments, then there’s a 97% chance that by the time you get to 90° you’ll be getting the same result you did at 0°.  However, in reality P(A_0=A_{90})=\cos^2{(90^o)}=0.

But this is a contradiction.  So the results of each measurement (A0, A1, A2, …, A90) can’t all exist.

If I had to guess, every time you go back in time the experiment is completely reset, and the experiment becomes completely random again.  The reason (such as it is) is below this unsettling picture.

Wait. Wait... Why?

But why?!: It turns out that the reason that the results of a quantum event can’t be predicted, is that every possible result of that event plays out.  So if you ask “will I see the photon go through the polarizer?” the answer is “yes, some versions of you will see the photon go through” and an equally valid answer is “no, some versions of you will not”.

If different versions of you will see every possible result, then the result can’t be predicted, and doesn’t really exist one way or the other until after the measurement is done.  At that time the different versions of you will disagree on the result.  But don’t worry too much.  You’ll never meet you’re parallel-universe twins.

Posted in -- By the Physicist, Philosophical, Physics, Probability, Quantum Theory | 2 Comments »

Q: Do virtual particles violate the laws that energy can be created or destroyed? Have virtual particles ever been observed? In any other instance can energy ever be destroyed or created?

Monday, June 7th, 2010

Physicist: Almost. There’s a version of the uncertainty principle that says that the amount of energy and the amount of time involved in an event can’t both be certain.  You can think of this version of the uncertainty principle as the universe making clerical errors.
Generally a virtual particle will pop into existence, do whatever it does, and then pop out before the universe catches it.
For example: the gluon (pronounced “glue on”) is the virtual particle that holds the nucleus together. But the time that it can exist is so short that it can’t even get from one side of the nucleus to the other. This is a big part of why big atoms fall apart (uranium, plutonium,…).
Unfortunately, only “real” particles can be measured. Virtual particles have to be inferred. We can observe gluons by introducing enough energy that they don’t have to rely on clerical errors to exist (I’m talking about particle accelerators here).  But virtual particles can only be detected in terms of the effects they have on other particles (like holding an atom together).
Aside from the uncertainty principle, everything obeys conservation of energy. And even with the uncertainty principle the extra energy gets ironed out faster than you can blink.

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Q: Spectroscopy?

Sunday, May 23rd, 2010

The complete question was: What is spectrum? What spectrum does the absolute black body have? Why do different bodies have different spectra? What is spectroscopy and how is it used in science? Why do different elements in star spectrum have different frequencies or what?

Physicist: If you have a sample of light, from a star, or some kind of lamp, or whatever, then its spectrum describes how much of each frequency of light shows up in that sample.  For example, a laser has a very “sharp” spectrum (all concentrated at one frequency), while sunlight has a very “broad” spectrum (many frequencies).  “Spectroscopy” is the science of gleaning information about something by looking at the spectrum of light it emits, or even absorbs.

The spectrum of a perfectly black body is, not surprisingly, called the “black body spectrum”.  The the intensity (I) of light at a given frequency (\nu) in the black body spectrum is given by I=\left(\frac{2h\nu^{3}}{c^2}\right)\frac{1}{e^{\frac{h\nu}{kT}}-1}, where h, c, and k are Planck’s constant, the speed of light, and Boltzmann’s constant respectively.

What’s amazing about this formula is that the only variable is temperature (T).  So the spectrum of a perfectly black object is determined entirely by its temperature.  The black body spectrum is also a very good approximation for the spectrum emitted by pretty much any thermal source.  Such as light bulbs, hot irons, fires, people, stars, etc.  In this case light is emitted by atoms slamming into each other and losing energy as light “splashes” off (smacking atoms jiggles their electrons, and jiggling charges is what makes light).

The peak of the black body spectrum moves to higher frequencies as temperature increases. Albireo, a binary star system, is a dramatic example of two different temperatures being indicated by two different colors.

By looking at the spectrum of a light source you can (often) tell what the source of that light is made of, what its temperature is, and even what the light has passed through before it gets to you.  The electrons in atoms can only exist in certain, discrete energy levels.  As such, the light that they can emit or absorb corresponds exactly to the amount of energy that can be gained or lost by jumping between energy levels.  The set of light frequencies that a particular element emits is called that element’s “atomic spectra“.

Left: the spectra of Argon, Helium, Hydrogen, and Mercury. Right: by passing the light through a difraction grating or prism you can tell what kind of gas is in it.

Different atoms have different spectra because the higher the atomic number, the higher the number of protons in the nucleus, and the greater the pull on the electrons.  The electrons in turn stack up and have bizarre magnetic interactions.  The interaction between electrons in an atom are very non-linear, and really complicated.  So adding one new electron will change the spectrum completely.  In fact, beyond hydrogen, the atomic spectra can’t be accurately calculated without a good computer.

Elements also have an “absorption spectra”, that corresponds exactly with emission spectra.  For example, big chunks of the infrared light frequencies are in the absorption spectrum of CO2.  Hence the famous green house effect.

The spectrum of sunlight, as veiwed from space (the veiw is clear from there). Rather than make one long rainbow, this was looped (like text on a page) to save room. The gaps in the spectrum tell us what gases are present in the outer layers of the sun.

Because each element (and molecule) has it’s own spectrum, we can look at a light source and see immediately which chemicals are present.  And by measuring (very carefully) how intense each line is we can tell how much of each chemical is present.

Even slicker, the atomic spectrum of each element is the same everywhere in the universe.  So if we look at a star and its hydrogen lines (which tend to be the clearest and most dominant) are all shifted to lower frequencies, then we know that that star must be moving away from us.  This is caused by the Doppler effect, and is called “redshift” because the lines look redder.

Spectroscopy is in use here on Earth to quickly determine what substances, and how much, can be found in a sample.  Generally by shining light (of a well known spectrum) through it.  For example, you can quickly check ozone levels, humidity, and even the size of particulate pollution, from space by watching sunlight filter through the atmosphere.  There are better methods (chemical based measurements) so spectrographic techniques are not en vogue, but they can be used in a pinch.

Also, radar guns and infra red thermometers are basically spectrometers with a single, specific function.

But the science of spectroscopy is mostly at home in astronomy circles, since they have literally nothing else to work with.  Zoologist can smell what they study, electrical engineers can be shocked.  But astronomers have to stare at the sky really hard, and measure spectra.  Even the discovery of planets around other stars comes down to measuring the red-shifting and blue-shifting of the parent star as its planets make it wobble.  We know what interstellar gas and dust clouds are made of by looking at star light filtering through them, and measuring the absorption spectra.

Really, everything we know about stuff outside of the solar system (everything beyond “look, stars!”) is based on spectroscopy.

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Q: If energy is quantized, what is the least amount of energy possible? And how did they measure it?

Wednesday, May 12th, 2010

Physicist: The name “quantum mechanics” is an old name, but also about the best name.  The name came about because it was noticed that the light created by passing electrical current through pure gases results in discrete, separated colors.  First the flow of electricity smacks the electrons into higher energy levels, then the light is created when the be-smacked electrons drop back down into lower energy levels.

The discrete colors indicate that the electron energy levels inside the atoms are also discrete.  One might even say “quantized”.  It may seem a little weird to measure energy using colors, but careful measurement of light frequencies is the best method we’ve got.  We’re pretty good at it.

A tube full of helium (left) and the same light passed through a prism (right).

So that’s where the name comes from.  In nice controlled quantum mechanical systems, like individual atoms or resonant chambers (e.g., microwave ovens), you’ll find that the energy levels are always quantized.  However, the energy levels (and the method of measuring them) depend on what system you’re studying.  Some systems have higher or lower “ground states” (the lowest energy level) than others.

Different quantum systems have different, quantized, energy levels. In this case: Lithium, Sodium, Potassium, Rubidium, Cesium, Mercury, and Neon. These pictures were created using the same method as the helium tube above.

In fact, it’s easy to create a system with an arbitrarily low ground state.  For example, the ground state of a particle contained in a box can be made arbitrarily small by making the box larger and larger.

However, the universe is kind of a dick.  When the ground state is low enough the chance of seeing something in that state becomes lower and lower.  Firstly, because for something to be observed it must do something, which takes energy, and secondly because of the uncertainty principle.  There are some sneaky tricks around this, but they necessarily involve longer and longer measurement times and are, in the end, useless.

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