Q: Why is Schrodinger’s cat both dead and alive? Is this not a paradox?

Posted on April 13, 2013 by The Physicist

One of the original questions was: A basic rule of logic is that something cannot contradict itself. It is impossible for P to be true and not true. Doesn’t Schrödinger’s cat violate this law and therefore invalidate logic?

The experiment

Schrödinger proposed this thought-experiment to demonstrate how ridiculous quantum super-position is.  Basically the multiple states of a single atom (decayed and not decayed) causes a cat to be in multiple states (living and dead).

Physicist: The resolution to this comes from a careful look at what is meant by the “state” of something.  Turns out, logic is safe from Lil’ Schrödinger’s claws.

There’s a big difference between “reasonable” and “logical”.  To see the difference, find a calm, reasonable person and talk to them, and then (this is more difficult) find a professional logician and try to talk to them.

Talking to professional Logicians: more effective than water-boarding.

Talking to professional Logicians: among the more frustrating conversations you’ll ever have.

It’s pretty reasonable to say that a single thing must be in one state or another, especially if those states are mutually exclusive.  It’s obvious.  It’s common sense.  In fact, it’s so reasonable/obvious/sensible that disagreeing with it would be a good way of being laughed out of every fancy science salon of the 19th century (or at least the occasional salon with sober members).  Logic, on the other hand, has nothing to do with physical reality (neither does being reasonable for that matter).

Logicians start with a big bucket of postulates and symbols and statements, and then run with them.  None of it needs to be “physically motivated” or even remotely intuitive.

Clearly, this reads

Clearly, this reads “P is possibly true if and only if P is not definitely untrue” and also “P is definitely true if and only if it is not possible for P to not be true”.

The statement that things must be one way or another (specifically, that each state is mutually exclusive of the others), is a whole new logical statement on its own.  The statement even has a name: “counterfactual definiteness“.  Overly-complicated terms like that are just made up so that people will think that physicists are wizards-of-smartness.  A better term for things needing to be in a definite state is “realism”.  While realism is “obviously true”, is isn’t necessarily true (not “logically true”), and point of fact: isn’t true.

There’s a famous no-go theorem in quantum physics called “Bell’s theorem” that says that, given the results of a variety of experiments involving entanglement, “local realism” is impossible.  This means that things always being in single states requires the exchange of some kind of faster-than-light signals.  Or conversely, if no effects can travel faster than light, then things must be allowed to be in multiple states.

It’s pretty natural to jump to the conclusion that things are communicating faster than light.  Losing realism is philosophically, even mathematically, a bitter pill to swallow.  Unfortunately, there are a lot of problems with faster than light stuff (like this one!).

It turns out that the universe doesn’t seem to have any problem dropping realism.  Things are perfectly happy being in multiple states at the same time: particles being in multiple positions or energy states, single events happening at multiple times, or (admittedly reaching a little past our grasp) being in multiple states of living and dead.  The last of course has never been observed in the lab (and probably never will be), but this is a well-studied property otherwise.  We’ve seen multiple-stated-ness in every physical system we’re capable of measuring the effect in.  So far, there doesn’t seem to be any limit to the scale at which quantum weirdness shows up.

In short, it does make sense to say that things must be in a single state or another, but it isn’t necessarily “logical”.  The universe couldn’t care less about what makes sense.

Answer gravy: This bit threatened to derail the flow of the post.

Realism is technically a statement that limits the exact nature of what kind of states are allowed.  For example, only the states |living\rangle and |dead\rangle are allowed.  When the cat is both living and dead it’s technically just in multiple states in certain “measurement bases”.  So the cat could be in the single state \frac{1}{\sqrt{2}}\left(|living\rangle+|dead\rangle\right).

We see this all the time in the polarization of light, for example.  A diagonally polarized photon is in a single state, |\nearrow\rangle.  But, if you insist on looking at it (measuring it) in terms of horizontal and vertical polarizations, then you find that it must be in multiple states, |\nearrow\rangle = \frac{1}{\sqrt{2}}\left(|\rightarrow\rangle + |\uparrow\rangle\right).  This moves the problem from being a purely philosophical/logical problem, to one of defining what is meant in detail by the word “state”.

The answer to whether Schrödinger’s cat is in multiple states becomes a resounding “Yes!  Unless some very specific measurement is set up, in which case: no!”.

Posted in -- By the Physicist, Logic, Physics, Quantum Theory | 18 Comments

Q: What kind of telescope would be needed to see a person on a planet in a different solar system?

Posted on April 8, 2013 by The Physicist

Physicist: When talking about telescopes there are two quantities to take into account; the “light gathering power” and the “resolving power” of the telescope.  “Light gathering power” is just how much light can be collected by the telescope.  “Resolving power” is a measure of the smallest angle that the telescope can reliably detect.

Telephoto lenses

Telephoto lenses need to be large because the amount of light that bounces off of a distant object and that then goes through the lens is fairly small.  Begin wide means they can gather more light.  They need to be long for other, more subtle, reasons.

Because light is a wave it has a way of spreading out (technically: diffracting).  The smaller the telescope the more the waveness becomes a problem.  If the angle between two distant points is θ, the light in question has a wavelength of λ, and the size of your telescope is D across, then the smallest resolvable angle is approximately \theta = \frac{\lambda}{D}.

What’s a little weird is that this D isn’t just limited to the size of the mirror or lens of a single telescope.  By cleverly networking telescopes together you can make them act like a single large telescope.

The VLA, or "Very Large Array", was named in honor of Professor Wilhelm Von Verylarge

The VLA, or “Very Large Array”, was named in honor of Professor Deirdre Von Verylarge.  By combining information from all of these radio-telescopes together they behave like one very large telescope that is effectively 36km across (the dishes are mobile and can be separated by at most 36 km).

Coincidentally, something that’s a large distance L away, and that’s a size S across, takes up an angle of approximately \theta=\frac{S}{L}.  So, if you want to be able to see something, you need \frac{S}{L}\ge\frac{\lambda}{D}.

Visible light has a wavelength of about half a micrometer (one two-millionth of a meter), people are about a meter across (assuming a spherical person), and the Earth is about 13,000,000 meters across.  So, using ground-telescopes that are perfectly constructed and networked, we could spot something person-sized from about 1/400th of a light year away, or about double the distance to Pluto.  For comparison, the distance to the nearest star is about 4 light years.  So, using ground based telescopes we can’t come even remotely close to seeing a person standing around on a planet in another solar system.

The nearest known, reasonable, candidates for being an Earth-like planet (as of April 2013) are about 20 light years away (HD 20794 d, Gliese 581 c, and Gliese 667C c).  Spotting dudes and ladies on one of these worlds requires, at minimum, a telescope array that’s at least 100 million km across.  That’s an array more than half the size of Earth’s orbit.  The good news is that an array like that (under absolutely ideal circumstances) isn’t that difficult to create.  Setting aside that the telescopes would each need to be essentially perfect for their size (Hubble-quality), all we’d need to do is set them up in solar orbits about the size of Earth’s orbit.  This is a lot easier than sending them to another planet, and about as hard as sending them to crash on the Moon.

So, assuming that we could set all that up, the problem stops being one of resolving power, and becomes one of light-gathering power.

On a sunny day we’re hit by about 1021 (1,000,000,000,000,000,000,000) photons (give or take) every second.  Assuming that a fair fraction of those escape into space, then that number, which seems large, is all that distant aliens have to work with.  Over 20 light years that scant 1021 photons means you would need a telescope array with an area of more than 500 million square kilometers to catch just one photon per second.  That’s the size of the surface area of Earth.  In the mean time there’s a lot of other light flying around, and single photons are pretty hard to detect so… the signal-to-noise ratio would be small.

Creating an array capable of seeing big stationary things like rivers and mountains on other worlds wouldn’t be too difficult, because you can just use tremendously long exposures to overcome the whole light-gathering issue.  This is a pretty standard trick in astronomy; the Hubble Deep Field took a more than a week of total exposure time.  There would be some issues with the fact that those distant planets are moving and whatnot, but there are clever ways around that too.

People, and probably aliens too, move around a lot.  So if you want to get a picture of one, you need the exposure to take less than, say, a second.  Unless you catch E.T. literally napping.  I would wildly guestimate that you’d need at least a few thousand photons per second to overcome the signal noise enough to say for certain that you’re looking at something real.

So, to answer a somewhat more detailed question; to get a picture of an alien that’s person-sized, standing on a world 20 light years away, so that it takes up one pixel in the image, using an exposure time of about one second, would require an array of telescopes with exposed mirrors and lenses with an area totaling more than several thousand times the Earth’s surface area and spread out over a region about the size of Earth’s orbit.

This isn’t technically impossible, but it would be “expensive”, and would require substantially more materials than are likely to be reasonably found in our solar system.  It probably isn’t worth it to get a blurry, tiny picture of some alien picking it’s nose 20 light years away and 20 years ago.

Of course, if you wanted to see farther, you’d need a much larger telescope array.

Posted in -- By the Physicist, Astronomy, Physics | 8 Comments

Q: Is Murphy’s law real?

Posted on April 1, 2013 by The Physicist

Physicist: The mathematical statement of Murphy’s Law, as used in scientific communities, is tremendously complex.  But the common form, “everything that can go wrong will”, is fairly accurate and more than sufficient for most applications.

The short answer is: yes, Murphy’s Law is real.  There are a lot of basic logical reasons for this.  For example; nothing lasts forever, so eventually every part of every machine will eventually break down.  Or, because being in traffic involves spending more time getting from place to place, you can expect to spend disproportionately more time in traffic than not.  But, as you’ve no doubt noticed, using logic and random chance alone it’s impossible to explain away the preponderance of horrible happenstances that show up seemingly without pause, everywhere, at all times.

“Coincidences” like that are a strong indication that a physical law is in play.  We can clearly see that Murphy’s law is both real, and unfairly applied to people colloquially known as “clumsy”.  Robert Oppenheimer, in addition to some entirely forgettable work he did in physics, pioneered research into Murphy’s law but studying his own unfortunate condition.  Bob’s affliction was first brought to the attention of his colleges when it was noticed that when he was in the lab, everyone’s muffins and buttered scones were 42% more likely to land upside-down.  Even sandwiches with particularly binding peanut-butter were more likely to open on the way down.

(Left) Murphy, the discoverer of "Murphy's Law" shown here after a flossing incident. (Right)

(Left) Wilhelm Murphy, the discoverer of Murphy’s Law, shown here suffering from complications after a gum-chewing incident. (Right) Oppenheimer, extended and modernized our knowledge of Murphy’s Law.  This the only known picture of Oppenheimer with his eyes open.

Oppenheimer, after he was politely asked not to work in the laboratories, made several starling discoveries such as the fact that Murphy’s Law is “recursive”, “pessimistically optimal”, and “robustly unfair”.  The recursive nature of the Law is one of the more obvious.  It says, in effect, that Murphy’s Law can’t be out-smarted.  For example, washing you car is almost certain to make it rain. However if your intention is to make rain, then washing your car will probably just make someone slip and fall.

Oppenheimer was involved in one of the better examples of Murphy Recursion.  During a celebration of his accomplishments and “clumsiness”, some of his fellow scientists constructed a lever attached to a prop chandelier, such that when Oppenheimer walked in and inevitably pulled the lever the chandelier would drop.  However, they forgot to take into account the recursive nature of Murphy’s Law, and the lever didn’t work.  Of course, had they tried to take Murphy Recursion into account, something else would have gone wrong.

Many people known to be “unlucky” or “followed by a black cloud of misfortune” are the sad victims of the fact that Murphy’s Law is demonstrably unfair.  This is the essential reason behind why computer problems only exist until you try to show someone else.  The range of “what can go wrong” varies wildly from person to person.  For Roy Sullivan (for example) being struck by lightning is something that can go wrong (and did go wrong seven times).  Despite specifically trying to avoid storms and clouds, he could barely leave his house without some lightning bolt setting him on fire.

So, Murphy’s Law is certainly very real, and can even be measured qualitatively.  However, it can’t be anticipated or taken into account.  We can only wait for terrible, unfortunate things to happen, and hope that they won’t be too bad.

Posted in -- By the Physicist, April Fools | 6 Comments

Q: Why doesn’t life and evolution violate the second law of thermodynamics? Don’t living things reverse entropy?

Posted on March 24, 2013 by The Physicist

Physicist: In very short: nope.

The second law of thermodynamics is sometimes (too succinctly) stated as “disorder increases over time”.  That statements seems to hold true, what with all of the mountains wearing down, machines breaking, and the inevitable, crushing march of time.  But living things seem to be an exception.  Plants can turn dirt (disordered) into more plants (order), and on a larger scale life has evolved from individual cells (fairly ordered) to big complicated critters (very ordered).

However, there are a couple things missing from the statement “disorder increases over time”, such as a solid definition of “disorder” (it’s entropy) and the often-dropped stipulation that the second law of thermodynamics only applies to closed systems.

Creatures, both in the context of growing and reproducing, and in the context of evolution are definitely not closed systems.  Doing all of that certainly involves an increase in order, but at the expense of a much greater increase in disorder elsewhere.  Specifically, we eat food which, with all of its carbohydrates and proteins, is fairly ordered, and produce lots of heat, sweat, and… whatnot.  Food, and air, and whatnot are what make living things “open systems”.

Whatnot.

Whatnot.

If a creature could take, say, a kilogram of non-living, highly disordered material and turn it into a kilogram of highly ordered creature, then that would certainly be a big violation of the second law of thermodynamics.  However, people (for example) consume along the lines of about 30 to 50 tons of food during the course of a lifetime.  Some of that goes into building a fine and foxy body, but most of it goes into powering that body and fighting degradation (blood and skin and really everything wears out and needs to be replaced).  So, about 0.15% (give or take) of that food matter is used to build a body, and 99.85% is used for power and to fight the entropy drop involved in body construction and temporarily holding back the horrifying ravages of time.

When compared to the entropy involved with turning food into the many, many bodies that make up a species, evolution is barely an afterthought.  In fact, the entropy (as used/defined in thermodynamics) of most animals (by weight) is all about the same.  A person and a mountain lion have about the same entropy as each other, simply because we weight about the same.

The big exception is photosynthesizing plants.  They really can turn a kilogram of inert, high-disorder dirt, air, and water into a kilogram of low-disorder plant matter.  But, again, they’re working with a bigger system than just the “plant/dirt/air/water system”.

There's a huge drop in entropy between the incoming sunlight

There’s a huge increase in entropy between the incoming sunlight and the outgoing heat that’s radiated away from the Earth.

Sunlight is a bunch of high-energy photons coming from one direction, which involves relatively little entropy.  A little later that energy is re-radiated from the Earth as heat, which is the same amount energy spread over substantially more photons and involves a lot more entropy (relatively).  This huge increase in entropy, between the incoming sunlight and the outgoing heat, is the “entropy sink” that makes all life on Earth possible (with just a handful of exceptions).  In particular, green plants take a tiny amount of the sunlight that hits the Earth and turns some of the energy into sugars and other useful plant-ey material.  It all eventually turns into heat and radiates away, but instead of doing it all at once it does it through a few links in the food chain.

You can think of this huge sunlight-to-re-radiated-heat increase in entropy like water going over a waterfall, and life as being like a hydro-electric dam.  It all ends up at the bottom of the falls, but sometimes it can do some interesting stuff (life and other useful mechanical work) on the way.

Posted in -- By the Physicist, Biology, Entropy/Information, Evolution | 5 Comments

Q: Does quantum mechanics really say that there’s some probability that objects will suddenly start moving or that things can suddenly “shift” to the other side of the universe?

Posted on March 17, 2013 by The Physicist

Physicist: In a word; nope.

The Heisenberg Uncertainty Principle is a statement about how “certain” some combinations of quantities can be.  The most commonly referenced is the “position and velocity” version of the Uncertainty Principle, that says that the more exact the position of a thing (any thing) the less certain its velocity, and vice versa.  It’s basically because of the Uncertainty Principle that you’ll hear about how quantum mechanics predicts that “particles have some small chance of jumping across the universe (position uncertainty)”, or “there’s some possibility that all the atoms in a book will suddenly start moving and it’ll jump off the shelf (velocity uncertainty)”.

And in fact, if you apply Schrödinger’s equation directly (which essentially describes how quantum wave functions change with time), it does seem as through there should be no problems with things suddenly jumping around.  If you apply it directly you find that if you have a particle confined to a particular region, then any amount of time later there’s some chance (no big) that it can be anywhere else, which is pretty exciting.  Unfortunately, Schrödinger’s equation is an approximation in very much the same way that Newton’s equations of motion are approximations of the (correct) relativistic equations of motion.

The wave function of a particle, which describes its position, tends to spread out like this

Soon after a particle’s position has been measured to be near zero, the wave function of a particle (which describes the probability of it being found at that position) tends to spread out like this, getting wider and wider as time goes on.  According to Schrödinger’s equation the tails on both sides approach, but never quite reach, zero.

Schrödinger’s equation was a massive break through and provided a lot of insight into a lot of problems.  But despite that, it doesn’t work perfectly.  In general, if you have a theory and it doesn’t line up perfectly with special relativity, then you only have part of a theory.  The fact that Schrödinger’s equation is “non-relativistic”, as evidenced by the fact that it predicts that sometimes particles will blink from place to place faster than light, made a lot of physicists extremely nervous.  It took a couple more years (1926-1928) until Dirac fixed the problem with the Dirac equation, which is more or less the same, but adheres to relativity.

The Schrödinger Equation (top) and the Dirac equation (bottom).

The Schrodinger Equation (top) and the Dirac equation (bottom).  The Dirac equation takes into account relativity.  Heck, it’s even got a “c” for light speed in there.

Newton’s equations of motion are very accurate, but only up until they disagree with relativity.  For example, they imply that there’s nothing special about light speed, and you can totally go faster.  Similarly, Schrödinger’s equation is remarkably accurate in most day-to-day, electron-shell type calculations, but makes big mistakes when relativity needs to be taken into account.

Long story short, even when considering the Uncertainty Principle, nothing can ever end up someplace else that would normally require faster than light travel.

As for books suddenly jumping off of shelves; the universe according to the laws of quantum mechanics is a seriously weird place.  But ultimately, laws are laws.  In this case, the conservation of momentum and energy.

If you take the predictions of quantum mechanics at face value (and why not?), everything that can happen does (in a very specific, many-worlds, sense).  But that “can” is pretty iron-clad.  Something that’s possible, even if it’s very unlikely, will happen in one some versions of the world*, but a book (or any other object) suddenly moving involves some extra energy suddenly being added to the universe, which is no good.

So, winning the lottery 75 times in a row, while making blind free throws for a couple weeks: sure.  Books jumping off of shelves: ridiculous.

* “World” is definitely not the right word for this, because it evokes images of other dimensions à la Sliders and leads to general confusion.  Neil Stephenson uses “narrative” which seems like as good a word as any, and hits a little closer to the mark.

Posted in -- By the Physicist, Physics, Quantum Theory, Relativity | 9 Comments

Q: Using modern technology, are we any closer to turning lead into gold than alchemists were hundreds of years ago?

Posted on March 11, 2013 by The Physicist

The original question was: With the current technology, it is possible now to transmute lead into gold, or whatever element into another? What transmutations should have tried the ancient Alchemist instead of the famous lead-gold one, in order to find an easy and useful success?

Physicist: Lead to gold: no.  But you can change some elements into others.  The yield is famously tiny, and the process is prohibitively expensive.  Before the late 19th century, no body had ever observed one element turning into another, and until the 20th century there was no equipment on Earth that had the faintest prayer of successfully changing one element into another (on purpose).

Back in the day, when chemists (alchemists) were getting good at purifying samples and making fancy chemicals, they got pretty cocky about turning stuff into other stuff.  But while you can use basic chemical reactions to turn hydrogen and oxygen into water, or flour and water into bread, there’s no combination of chemicals and reactions that even start to change one element into another.  Alchemists back in the day, being unaware of these sorts of things, got very excited about lead-to-gold stuff, philosopher’s stones, and life from nothing.  Many of them were legit scientists of the day, so we legit scientists of today have inherited a lot of their symbols and short-hand (though not their methods, by and large).

Newton loved himself some alchemy.

Newton loved himself some alchemy.  Sure he did calculus and science, but he also did pioneering work into finding the holy grail and even a variety of crazy pursuits.

Fancy chemicals and molecules are different because they use different combinations of elements, but elements and isotopes are different from each other because they have different numbers of protons and neutrons in their nuclei.

There are basically three ways to change the number of protons and neutrons in the nucleus of an atom.  Fusion, radioactive decay, and neutron bombardment.

Fusion is tricky.

Fusion is tricky.

There are some issues with practical fusion.  To date we’ve managed to fuse deuterium (hydrogen) into helium, which is the easiest fusion there is, even then just barely, and only to useful effect in the middle of very big bombs.  To use fusion to make gold (which is the way gold is created in nature) you need a super nova, which would probably be expensive.  Also impossible.

A lot of atoms have unstable nuclei that will occasionally “pop” and turn into another element or isotope.  So, technically, being patient is one way to turn a sample of some material into another.

Start with some Uranium 235, then wait a few billion years, and you've got lead.

Start with some Uranium 235, then wait for several hundred million years, and you’ve got mostly lead.

Unfortunately, the material in question has to be radioactive beforehand.  Radioactive atoms “decay” until the number of protons and neutrons are in balance (not equal, just balanced in a particular way), and for heavy elements that balance is almost always reached with lead or thallium.

The last way to change one isotope to another, and the only technique we can really use and control, is “neutron bombardment“.  Neutron bombardment isn’t the best option, so much as it’s the only option.  The idea is that since neutrons are electrically neutral (hence the name) they can enter and join the nucleus of an atom without being repelled by the positively charged nucleus (this repulsion is why this technique doesn’t work with protons, and why fusion in general is so difficult).

Bombardment is how plutonium is manufactured from uranium.  Bombarding a sample with neutrons sometimes makes the atoms in question decay into higher elements, and almost always makes them more radioactive (so this is a “bombard then wait” sort of thing).  In some cases it makes them so spectacularly radioactive that they immediately fly apart, and if they also produce a spray of neutrons, then you’ve got yourself the makings of a bomb or a power plant.

Here’s a map of all of the known isotopes and their preferred means of decay (many isotopes have several ways they can decay).  The full chart, in detail, can be found here.  It’s a very big picture.

All of the isotopes, with the number of neutrons increasing as you go up, and the number of protons increasing to the right.

All of the isotopes, with the number of neutrons increasing as you go to the right, and the number of protons (which is the “atomic number” or “element number”) increasing as you go up.  The black squares are the stable isotopes, and this region is called the “valley of stability”.

The different colors indicate different decay paths.  For example, pink is β+ decay, which turns a proton into a neutron, and an extra anti-electron, which in this case is the “radiation” we detect flying out.  So, on the chart the pink isotopes decay down and to the right, by losing one proton and gaining one neutron.

How to "play the game". At best, we can cause a tiny fraction of a sample of an isotope to move one to the right (gain one neutron).

How to “play the game”. At best, we can cause a tiny fraction of a sample of an isotope to move one to the right (gain one neutron).

By looking at the chart you can figure out what elements can reasonably be made from others using neutron bombardment.  For example, you might look at this little part of the chart (picture above) and think that you should be able to create gold by bombarding platinum 196 (which is directly below gold 197, the only stable gold isotope).  This would add a neutron, which changes some of the sample into platinum 197, which would then execute a β- decay, moving up and to the left, and turn into gold.  As it happens, this is exactly how you create gold from platinum.  That β- decay has a half-life of about 20 hours, so once you irradiate your platinum you only have to wait a few days before extracting the trace amounts of gold from your sample.
There’s also an isotope of mercury, mercury 196, that can be turned into gold (it’s above and two to the left from gold 197).Lead, on the other hand, is in a bad position to form gold.  Using neutron bombardment you can move to the right on the chart, but if you follow the decay path from every heavy isotope of lead, they all lead back to either lead or bismuth.

All we can do in add neutrons (white arrows), but that just takes puts us on "decay paths" that lead back to lead or bismuth.

All we can do is add neutrons (white arrows), but that just takes puts us on “decay paths” that lead back to lead or bismuth.

So, using the one and only technique available to us, we definitely cannot turn lead into gold.  Not even a little bit.  Platinum and one fairly rare isotope of mercury, sure.  But not lead.

Also, both the platinum and mercury processes are substantially more expensive and dangerous than digging gold out of the ground.  Among other things you need to get your hands on a neutron source, which is generally an extremely radioactive (illegal and expensive) metal, or a multi-billion dollar accelerator used to blow apart heavy isotopes into buckets of neutrons.  There are easier ways to lose money.

Posted in -- By the Physicist, Particle Physics, Physics | 6 Comments