Thursday, July 30, 2009

Schrodinger's Cat

Schrodinger's Cat is a famous thought experiment. It goes like this:
A cat is kept in an impenetrable box, and no one can look inside. There is also a single radioactive atom. After exactly one half-life, there would be a 50% chance that the atom has decayed, and 50% chance that it hasn't decayed. But since we can't observe the atom, we must describe it in a quantum mechanical mixed state, 50% decayed and 50% not decayed. Within the box, there is a detector which determines whether it has decayed. If it has decayed, then the cat is killed. If not, then the cat is left alive.

However, since we have not opened the box, we must describe the cat in a quantum mechanical mixed state, 50% dead, and 50% alive. So the cat is dead and alive at the same time. When we open the box, the quantum state collapses, and we either find a completely dead cat or a completely alive cat.
Before I get into explaining it in more detail, I'd like to ask, what's your favorite joke involving Schrodinger's cat? Mine is this Saturday Morning Breakfast Cereal comic.

Moving on!

Some people think that the takeaway message of the Schrodinger's Cat thought experiment is that it's completely absurd. "A cat which is dead and alive? Absurd!" But there's more to it than that. It's that the explanation seems unnecessarily absurd. Couldn't we just say that there's a 50% chance that the cat's dead and 50% chance that it's alive? Why do we need to add in this whole idea of being dead and alive simultaneously if no one ever sees the cat that way anyways? It's like we're saying that something really weird is going on, but only when we're not looking at it. It's as preposterous as painting my moustache green, only to hide it with a huge fan whenever anyone looks my way.

The thing is, the weirdness of quantum mechanics may seem unnecessary for the cat, but it is entirely necessary for smaller systems, such as single particles or photons. There's even a proof, known as Bell's Theorem, that at least some weirdness is absolutely required.

Consider single photon. If we build a detector to observe its polarity, we will always see it polarized vertically or horizontally.

Since we only see it one way or another, it would be quite absurd to say that it's simultaneously horizontal and vertical. We only see it one way or another, so why would we even consider that it could be both at once? But in fact, there is a quantum state which is a mix of horizontal plus vertical. There's also another quantum state which is horizontal minus vertical. And we can see both of these states if we just rotate our detector 45 degrees. Now we have the two diagonal polarizations.

If you try to measure either diagonal polarization without rotating the detector, then we will have a 50% chance of seeing horizontal and 50% chance of seeing vertical. But before we observed them, they were both vertical and horizontal simultaneously, so to speak.

But how is the cat anything like those photons? No matter how you look at the cat, it will always be either completely dead or completely alive. Rotating your head won't help. In principle, you can build a detector which can see the cat in a simultaneously dead and alive state. But in practice, such a detector is utterly impossible. You'd probably have to look at every single particle in the cat all at once. And you'd have to isolate the cat, making sure that no stray observers destroy the mixed state by looking at it.

Cat illustrations taken from Griffith's Introduction to Quantum Mechanics

But I could be wrong. As I said, the detector is utterly impossible to build. So if I can't build it, how would I prove that it's possible at least in principle? Perhaps there really is something fundamentally different happening to the cat which is not happening to the photons. It depends on which interpretation of quantum mechanics we take.

The most common interpretation is the Copenhagen interpretation. The Copenhagen interpretation says that at some point, the state of the system "collapses" into a definite state. So when we open the box and look at the cat, the cat "collapses" into a completely alive or a completely dead state. Or perhaps the collapsing happens earlier? Perhaps the cat observes its own survival/death, collapsing itself into a state that is either completely alive or completely dead. Or perhaps the collapse occurs even earlier than that? There is a detector inside the box which determines whether the radioactive atom decayed or not. Perhaps this detector collapsed the atom by observing it.

Might we go even further, and say that the atom collapses itself? Perhaps. But an atom is a relatively small set of protons, neutrons, and electrons. Perhaps we will eventually build a detector which can measure the atom in a simultaneously decayed and not decayed state. So we probably don't want to go as far as the atom.

We can go in the opposite direction too. Perhaps the quantum collapse happens even later than when you open the box. Perhaps you, too, are in a mixed state, simultaneously happy that the cat lived, and sad that the cat died. "Ridiculous," you say. "When I look at the cat, I am 100% sure that it is either dead or alive." That doesn't prove anything at all. You could be in a mixed state, simultaneously being absolutely sure the cat is alive, and absolutely sure that the cat is dead. Maybe you exist in this mixed state until you tell a friend about it, allowing your friend to "collapse" you.

Or maybe you never collapse at all, and nothing ever collapses. Perhaps the entire universe is in a mixed state of having simultaneously N living cats and N+1 living cats. This interpretation is known as the Many Worlds interpretation, because it implies that the world is simultaneously in many different states. It's like we're all together in this huge box which will never be opened. (FYI, this is the interpretation I advocate, so I might be biased in my presentation.)

So that's the basic idea of Schrodinger's Cat. I hope this raised all sorts of new questions... you can always ask.

Eduard said...

There are a lot of questions which are difficult to answer. For the Schrödinger Cat Question I get a similar feeling as with "Is there a life after death" or "Is there a god". Can we not state that the questions must be classified in good ones which can be clearly answered (eventually in a many pages long mathematical prove) and bad ones where several centuries of human thinking gives no well accepted result? There are so much very pleasent questions which can be answered, why loose energy (attention) with the "bad" ones. I want that Fermat's Last Theorem was a good question.

Secret Squïrrel said...

If you set up a panel with two slits in it and shoot bunches of cats through the slits, a CAT scanner on other side will detect what state the cats are in.

However, if you shoot a single cat at the panel, the CAT scanner is unable to determine whether the cat is alive or dead.

miller said...

Eduard,
I agree that the question "Is the cat dead and alive?" is a bad question, in the sense that it is unanswerable. However, I think it is a good question to ask why it is a bad question. After all, we could ask the same question about an atom (is it decayed and not-decayed?) and potentially get sensible results.

So why is it a good question for atoms, but a bad question for cats? And where exactly does it change from a good question to a bad question? Scientists have studied this issue and have already gotten some sensible answers out of it.

Secret Squirrel,
I'd say instead that the CAT scanner can determine whether the cat is dead or alive, but it changes the result by observing it!

Ólafur Jens Sigurðsson said...

But what does the cat have to say about this? Does he now know if he is dead or alive (if we follow the Copenhagen interpertation that is, not the many worlds one)? :-)

The problem with these kind of speculations is the word "knowing", why does someone or something (a measuring device or a human) have to interact with something to collapse the states, why can't the particle/cat know for themselves in which state they are and leave the rest of us in uncertainty?

Do I (or someone else for the matter) need to hear if a tree fell in a forest in Siberia for it to fall at all.

miller said...

Hmm, I'd say that the cat is in a superposition of two states: 1) The cat is alive and knows it and 2) The cat is dead, so it knows nothing. Honestly, I do not think there is anything special about "knowing", since you could just say that the cat's knowledge is in a mixed state too. But that's just the Many Worlds interpretation talking.

Even under the Copenhagen interpretation, "knowing" still isn't necessarily anything special. If that tree falls in Siberia, perhaps it causes a chain reaction of air movements which causes one more air molecule to hit your hand in the next day. Even though you're unaware of it, you have "observed" the tree falling, and thus collapsed its quantum state.

Secret Squïrrel said...

A tree falling in an unpopulated forest makes the same sound as one hand clapping.

So I've heard anyway...

Ólafur Jens Sigurðsson said...

The observation is essential to the Copenhagen interpertation, since before any observation is performed the state of the object can be a superposition or .. just about anything that it is allowed to be. I am not familiar enough with the many world interpertation to comment on that, but to me it sounds like a crazy idea that doesn't really explain anything, just one more idea that is untestable to our reality. I would even go so far as to say that it contradicts with our experience of reality but I can't back that up with anything solid, just a feeling that I can't put into words :-)

miller said...

Yes, under the Copenhagen interpretation, "observation" is special; "knowing" is not.

As for the Many Worlds interpretation, you cannot say that it contradicts our experience while simultaneously saying it is untestable. It can only be one or the other.

I would say that the Many Worlds interpretation is untestable for the foreseeable future. In the mean time, we can only speak of preferences. I prefer the Copenhagen interpretation because it is simple and practical. I prefer the Many Worlds interpretation because it predicts that no matter how large a system we consider, the quantum nature of the system never magically disappears; it simply becomes invisible through known mechanisms such as decoherence.

In the Feynman lectures I posted the other week, Feynman made an excellent point about why we ought to keep all these untestable theories in mind. Though the different theories make identical predictions, they tend to make very different suggestions as to future theoretical developments. For example, there are a few QM interpretations (so far unnamed) which have been largely rejected because they don't transition very well to relativistic QM.

Ólafur Jens Sigurðsson said...

Ahhh .. yes, you are right about the contradicting and untestability part of course, my bad there. I should rather say it is counter intuitive and untestable (but the former applies to all QM theories so nothing new there except that the MWT seems a bit more counter intuitive then others, it's a bit difficult to swollow that there exist many different versions of one self and that each time we do something we branch ourselves into many parts, just doesn't fit with our experience of every day life, we can't "feel" the branching taking place).

That is a very good point in mr. Feynmans lectures in deed, keeping an open mind and thinking about other possible solutions is one of the things that defines a scientist, thats why I allways get a bit irritated when scientists talk about QM effects such as entanglement as some kind of known scientific fact (there are experiments that can't be explained with our knowledge today unless we use ideas like that, but that doesn't necceserely mean they are true because the basis can be wrong). I do this myself sometimes because I know people don't want to get all philosophical about the basis of QM when they are talking about some experiment performed in a lab that involves two laser beams that exhibit some kind of strange behaviour that we call entanglement.

But these are interesting discussions I think and glad that there are some people out there that take the time to think about them.

miller said...

I must admit that Feynman opened my eyes too when he said that. He gave a pretty compelling reason to see equivalent theories as more than just mathematical tricks.

I'm not sure entanglement is such a great example, because as far as I know (correct me if I'm wrong) all of the QM interpretations have entanglement in them.

I know popular science would have you believe that entanglement is all mysterious and rare, but it's really not. For example, in 2-D space, you might have a particle which is 50% localized near (0,0), and 50% localized near (1,1). As soon as you measure its x-coordinate, you know that its y-coordinate is nearly the same. So you could say that its x-coordinate is entangled with its y-coordinate.

Entanglement between multiple particles is the same idea, except it's in a higher dimensional space. For 2 particles, you would have a 6-dimensional space (x1,y1,z1,x2,y2,z2), and they can all be "entangled" together. Of course, that's way too hard to think about, so we tend to make approximations in which there is no entanglement. But you really need entanglement, especially when we get to the topic of identical bosons and fermions, since entanglement is what gives them their nature.

Mr. Nichols said...

Another thought experiment: A man is secluded on a tropical island. He has constructed an air tight, sound proof hut. The hut has a two doors. One for entry/exit. One is for a small window. The man inside the hut theorizes that the environment outside the hut exists in two states, windy or not windy. When the man opens the door, he collapses the system and observes either windy or not windy. He could place a windy detector (pinwheel) outside the hut. Then, he could open the window and observe the detector also collapsing the wind.

Hopefully, this sounds absurd. My point is, because we fully understand weather systems, we know wind is not predicated on the man's observation. If we fully understood quantum mechanics, we would understand that quantum do not depend on our observance either. Except for you multi-worlders who cannot accept the definite reality of anything.

miller said...

I think for an example of the absurdity of quantum physics to be informative, we must be able to pinpoint precisely what is absurd about it, and what led to this absurdity.

The idea that the state of the weather should depend on the single man in the hut relies on a very peculiar Copenhagen interpretation. It assumes that wavefunction collapse is real, but can only be caused by this one man. Furthermore, it neglects the possibility of observing without knowing (which I mentioned in an earlier comment).

Anonymous said...

Thank you guys for your comments on this/these subjects. I have been very fascinated by the idea of MWT and am thankful for the different points-of-view on the subject matter

xavier Magnate said...

All dumb and inept explanations! All way off and years behind. Years of morons being lead to study theoretical chew toys for clowns!!!!! Back to sub freezing levels and gravity fools. It's not what you observe the state its in etc. WHAT IS THE CONSTANT THAT KEEPS ALLOWING PHOTONS TO BE DISRUPTED IN THE FIRST PLACE. IN A STEADY STATE OF CONSTANT CHANGE BUT AMBLE TO BIND, ADHERE, REPEAT OR DUPLICATE. BACK TO GRAVITY MORONS!!!!!

miller said...

Xavier,
Thanks for the feedback. I hope you stick around, as we tend to be short of capital letters around here.