What is a wave function and its collapse?

[Kathryn0505]

I think that the wave function is the description of a particle's position at a point in time. But I'm not precisely sure what an eigenfunction is and how it is different. I know that certain eigenfunctions return certain values, and that even if you do not have a certain eigenfunction you can consider it (the wave function?) to be the superposition of certain eigenfunctions. When you measure, however, the particle "collapses" to one of the composition functions. But what does this actually mean for the particle?

Any help is appreciated!

 

[glengarry]

A "wavefunction" is just the mathematical description of a harmonically oscillating standing wave. Each possible harmonic solution is an eigenfunction of this guy: $\nabla^2\phi=0$

Also, there is no such thing as "particles" in quantum mechanics. This is a "classical" concept that is not warranted by the mathematical formalism.

All we are doing is applying [linear] Hermitian operators (euphemistically called "observables") to a non-linear field of values, which are given by the square of the wavefunction in question.

And when this application is done an infinite number of times, then the quantum mechanical expectation value--given by the inner product of vectors in state space--is said to have been realized.

(Or something like that ;)

 

[Kathryn0505]

Wow, I think I understood almost none of that haha

If they're not particles, what are they?

 

[glengarry]

You are not supposed to "understand" quantum mechanics. You are just supposed to sit down, shut up, and calculate.

If you were to ask Schrödinger himself (the developer of the quantum mechanical wavefunction), he would say that all of reality consists of actual, dynamically oscillating standing waves. Now, if you want to know "how" such entities can be seen to be empirically "localized," then you might want to check out this thread, and ask me questions there.

 

[jamesmo]

Though glengarry was blunt, he is correct. "Sit down, shut up and calculate", more gently put means that the quantum mechnical world is very different than the macroscopic world. The wave-particle duality is better explained as a probability field. More like the poker player placing bets on what the next card might be. Every card is uniquely determined when it gets flipped over (particle, eigenvalues), but until it is flipped over, we can describe what the next card might be as a bunch of probabilities (waves, eigenfunctions or bra-kets operations).

I stretched that cards analogy a bit, but it is a good conceptual way of looking at it.

Hope that helps

 

[Kathryn0505]

Okay, I read all about Schrödinger's cat from like 50 different sources. I guess I'm confused about whether or not the cat is actually both alive or dead, or if it's just better to think of it that way because we don't actually know yet so we must think about it in probabilities.

 

[Fra]

I think that the wave function is the description of a particle's position at a point in time.

Yes and the word description here is important.

When you measure, however, the particle "collapses" to one of the composition functions.

Correction, the particles _description_ collapses.

But what does this actually mean for the particle?

In a measurement theory, this is not a well posed question. A theory of measurement tries to predict results of measurements and actions in between measurements.

All that is importat is what your description of the particle says, as that is what determines your actions towards it. The same goes for all other parts of the setup (including detectors, etc)

/Fredrik

 

[Fra]

I guess I'm confused about whether or not the cat is actually both alive or dead, or if it's just better to think of it that way because we don't actually know yet so we must think about it in probabilities.

According to your description of the cat (wavefunction) it is both, or both possibilities are real. So the expected action in between the measuremens is as if it is both. Or more properly, your action is so that you acount for ALL possibilities at once, not randomly one of them.

IF you make a parallell to risk-analysis and risk-weighted actions you'll find a decent easy to understand analogy.

/Fredrik

 

[Fredrik]

I think that the wave function is the description of a particle's position at a point in time.

Not just the position, but all the properties of the system. Actually, that's a bit too strong. The assumption that "a wavefunction is the mathematical representation of all the properties of the system" can be taken as the definition of a kind of many-worlds interpretation. (It would take too long to explain why, so I won't). If we prefer not to commit to a particular interpretation, we should be saying that the wavefunction is "the mathematical representation of the statistical properties of an ensemble of identically prepared systems". (The assumption that that's all the wavefunction is, can be taken as the definition of the ensemble interpretation).

But I'm not precisely sure what an eigenfunction is and how it is different.

It's just a special case of the above. It's the wavefunction that "represents the system" (see above for a more accurate statement) right after a measurement, when it's known with certainty that a subsequent measurement of the same observable will have the same result.

When you measure, however, the particle "collapses" to one of the composition functions. But what does this actually mean for the particle?

It's important to understand that QM doesn't answer questions like that. QM tells us how to calculate probabilities of possibilities. It doesn't tell us what "actually" happens. It's the "interpretations" of QM that try to tell us what actually happens. I said "try" because I don't think any of them have been developed to the point where it can be said to give us a complete picture.

 

[zonde]

I think that the wave function is the description of a particle's position at a point in time.

It would be more prudent to say that wave function is the description of an ensemble of particles. And in that case it would be more meaningful to speak about distance in time (from point of emission to the point under consideration) for each particle from ensemble. If all parts of ensemble have the same distance in time to the same position i.e. ensemble evolves coherently then it's time independent evolution.

But what does this actually mean for the particle?

As for individual particle measurement reduces ensemble and so it changes particle's "position" in ensemble.

 

[jamesmo]

Okay, I read all about Schrödinger's cat from like 50 different sources. I guess I'm confused about whether or not the cat is actually both alive or dead, or if it's just better to think of it that way because we don't actually know yet so we must think about it in probabilities.

The cat is neither dead nor alive. It may be one or the other. Dead or alive is your eigenvalue. The eigenfunction is going to be the probability that she is alive or dead at any given moment.

The difference between a real cat and the QM-cat is that if the cat really dies, there will be a stink. The real cat will have a state, independent of the observation. With our QM-cat, the act of observing the QM-cat's state, set's that state. Until you make the observation the QM-cat can be either alive or dead, but you have no way of knowing without observing the QM-cat.

So the question is: was the cat dead already? or did you kill the cat by looking at it?

 

[Kathryn0505]

So we don't know if the cat is either dead or alive until we actually check to see. It's just that we have to consider all possibilities before we check, so it is considered both until we know for sure?

 

[jamesmo]

So we don't know if the cat is either dead or alive until we actually check to see. It's just that we have to consider all possibilities before we check, so it is considered both until we know for sure?

More to the point, the cat is neither dead nor alive until we check. "Checking" extracts the eigenvalue. The cat cannot be in either state until we check. It isn't merely that we don't know if the cat is alive or dead, it is that we cannot know through any means the state without looking.

 

[Kathryn0505]

Okay, I have a completely different question now. In the double-slit experiment, the particles can be considered waves as they produce an interference pattern. But when you go to measure which path they actually took, they act like particles when they hit the screen. This illustrates the principle of complementarity, that they are not waves and particles at the same time.

But I don't understand how knowing which slit the particle passes through destroys the interference pattern.

 

[jamesmo]

Okay, I have a completely different question now. In the double-slit experiment, the particles can be considered waves as they produce an interference pattern. But when you go to measure which path they actually took, they act like particles when they hit the screen. This illustrates the principle of complementarity, that they are not waves and particles at the same time.

But I don't understand how knowing which slit the particle passes through destroys the interference pattern.

You have answered your own question, "knowing which slit" is the key to understanding this problem. Knowing is the same thing as measuring. When you take the measurement, you have selected the eigenvalue. With that, there is no "probability" running around to create interference.

 

[Kathryn0505]

Wait, what? Is it that measuring actually affects the particle and destroys the pattern? Or is the pattern something I don't understand? I thought the pattern was the light and dark bands. How does it relate to probability?

 

[Fredrik]

More to the point, the cat is neither dead nor alive until we check. "Checking" extracts the eigenvalue. The cat cannot be in either state until we check. It isn't merely that we don't know if the cat is alive or dead, it is that we cannot know through any means the state without looking.

This isn't quite right. What you're saying is what the Schrödinger equation predicts, but the Schrödinger equation applies only to systems that are kept isolated from their environments, and there's no way to keep an object as large as a cat isolated from its environment for very long. So in the real world, the cat is going to be either dead or alive before we look. "Just looking" can never be a "measurement" in the QM sense. A measurement is always an interaction that creates correlations between eigenstates of the system and macroscopically distinguishable states of a system which for all practical purposes can be considered classical (because it interacts strongly with its environment).

The point of the Schrödinger's cat thought experiment is to show that the linearity of the Schrödinger equation implies that if microscopic systems can be in superpositions, the same thing must be true for macroscopic systems. But that stuff I said about interactions with the environment wasn't known until half a century later.

 

[Fredrik]

But I don't understand how knowing which slit the particle passes through destroys the interference pattern.

It's not about "knowing". The interaction between the particle and the device that determines its position changes the state of the particle, to a state that's localized at one of the slits.

I think the best way to think about the double slit experiment is the way that's explained in this fantastic book. There's no math in there. The book is based on a lecture series for people who aren't physicists. The videos of the lectures have been posted online as well. I prefer to read the book, but there's no harm in watching the videos as well: http://vega.org.uk/video/subseries/8.

I'm also going to repeat my earlier recommendation: You should stop thinking about wavefunctions as something "real" that "exist out there", and start thinking of them as a part of the mathematics you can use to calculate probabilities of possibilities. I also have to repeat what I said about QM and its interpretations. QM can't and doesn't tell us what "actually happens". (This is particularly clear in the approach taken by Feynman in his book). That's what the "interpretations" are for, but the interpretations are not as well understood as the theory, so you shouldn't expect anyone to be able to tell you what "actually happens".

 

[Kathryn0505]

Okay, new question (sorry!). A superposition state is the sum of all the possibilities of all the states it could be in. Until we actually measure and know what state it is in. But that's not to say that it's not in a certain state until we measure? We just aren't sure yet. Just because we don't know yet doesn't mean something's not a certain way. (That was probably poorly worded).

Briefly, as far as entanglement goes, measuring one particle gives information about the other. But does it change what the second one is doing? Or just give us information about it?

 

[glengarry]

You are simply trying to read more into QM than is warranted by the formalism. All that QM is saying is that a certain result will be attained a certain percentage of times, given a particular, "classical" experimental setup. Because of this, the only real "it" that we can talk about is the equipment that we are using in order to make measurements.

So, we are simply not allowed (from within the context of "formal" QM) to speak of any kind of independently subsisting "its." And neither are we allowed to speak about any such "its" as being the "cause" of an experimental result, because it is only the result itself that is the entire concern of QM.

Whenever quantum theorists speak in term of "particles," they are really just using a kind of heuristic (metaphorical) device that gives "naive" minds something to latch onto while the mathematical formalism is being learned.

 

[Fredrik]

A superposition state is the sum of all the possibilities of all the states it could be in. Until we actually measure and know what state it is in. But that's not to say that it's not in a certain state until we measure? We just aren't sure yet. Just because we don't know yet doesn't mean something's not a certain way. (That was probably poorly worded).

The wording is good enough. It's clear what you mean, and what you're saying here is not the way it is. The terms are not the possible states the system might be in right now. The first clue to that is that you can expand the state in several different bases. For example, if we write the eigenvalue equations for two observables A and B as

Au_a=au_a
Bv_a=bv_b

a state \psi can be expressed either as

\psi=\sum_a\langle u_a,\psi\rangle u_a

or

\psi=\sum_b\langle v_b,\psi\rangle v_b

Each u_a is the state that the system can be in immediately after a measurement of the observable A that had the result a, or to put it differently, the mathematical representation of the statistical properties of an ensemble of systems of the same type that were all prepared by a measurement of A that had the result a. The statistical properties of such an ensemble are as simple as they can get, as far as measurements of A are concerned. If we measure A again, we will get the same result with certainty.

The above doesn't conclusively prove that the system isn't always in a specific but unknown eigenstate, but Bell inequality violations do exactly that. The assumption that any measurement is simply "finding out what state the system is in" has consequences called Bell inequalities, which QM predicts do not hold. Experiments have proved that assumption wrong by showing that Bell inequalities don't actually hold in the real world.

Most QM books include derivations of some sort of Bell inequality. I like the derivation in Isham's book (215, 216) of a particular Bell inequality for spin-1/2 particles, called the CHSH inequality. That's a really good book by the way. It's probably exactly what you need right now, so I think you should consider buying it. (I guess you'll have to if you also want to see his proof that QM predicts a violation of the inequality. I think it's on the next page and can't be previewed at Google Books).

Briefly, as far as entanglement goes, measuring one particle gives information about the other. But does it change what the second one is doing? Or just give us information about it?

I think you need to think of an entangled "pair" as a single entity.

 

[zonde]

Okay, new question (sorry!). A superposition state is the sum of all the possibilities of all the states it could be in. Until we actually measure and know what state it is in. But that's not to say that it's not in a certain state until we measure? We just aren't sure yet. Just because we don't know yet doesn't mean something's not a certain way. (That was probably poorly worded).

Let's say we have source producing 100 particles on average in 1 second and we have one measurement that should produce 30 clicks on average and another measurement that should produce 70 clicks.
If you assume that each particle from those 100 is in certain state that produces click for first measurement with probability 0.3 depending from some classically random conditions of measurement and for second measurement probability of click is 0.7 then this will work in some cases but won't work in a lot of other cases.
On the other hand if you assume that from those 100 particles there are 30 that will produce click with certainty and 70 that won't produce click with certainty and vice versa for second measurement it won't work either.
That's the problem with certainty of state.

Briefly, as far as entanglement goes, measuring one particle gives information about the other. But does it change what the second one is doing? Or just give us information about it?

It seems that you can get quite far assuming that it just gives us information about it. At least as long as you do not assume that each individual particle from ensemble possesses all the properties of ensemble as a whole.