Sorry, not in this post, but upcoming. I’m scheduled to do another episode of Bloggingheads.tv with David Albert, and we’ve decided to spend the whole hour talking about quantum mechanics. Start with the basics, try to explain this crazy theory and some of its outlandish consequences in ways that anyone can understand, and then dig into some of the mysteries of measurement, superposition, and reality.
So — what do you want to know? What are the really interesting questions about QM that we should be talking about?
One thing I don’t think we science-explainers get as clear as we could is the idea of the Wave Function of the Universe. It sounds scary and/or pretentious — an older colleague of mine at MIT once said “I’m too young to talk about the wave function of the universe.” But it’s a crucial fact of quantum mechanics (arguably the crucial fact) that, unlike in classical mechanics, when you consider two electrons you don’t just have a separate state for each electron. You have a single wave function that describes the two-electron system. And that’s true for any number of particles — when you consider a bigger system, you don’t “add more wavefunctions,” you beef up your single wave function so that it describes more particles. There is only ever one wave function, and you can call it “of the universe” if you like. Deep, man.
Here is another thing: in quantum mechanics, you can “add two states together,” or “take their average.” (Hilbert space is a vector space with an inner product.) In classical mechanics, you can’t. (Phase space is not a vector space at all.) How big a deal is that? Is there some nice way we can explain what that means in terms your grandmother could understand, even if your grandmother is not a physicist or a mathematician?
(See also Dave Bacon’s discussion of teaching quantum mechanics as a particular version of probability theory. There are many different ways of answering the question “What is quantum mechanics?”)
I forgot to add that the above post was a response to Neil B., post 139.
In response to the discussion between Neil B. and Jason Dick regarding decoherence:
Neil, I think the key in efforts to use decoherence to answer the measurement problem is assuming there is no collapse of the wave function. That is, all branches of the wave function still exist, but the observer becomes entangled with one of the branches so from his perspective the others are unobservable.
Let’s focus on the case of a wave (which is spread out in space) turning into a localized particle.
The wave can be viewed as a superposition of particle states at each different possible position. That is, if S1 is the wave function of a particle at “position 1”, and S2 is the wave function of the particle at “position 2”, then the full wave function of the wave is:
S = a*S1 + b*S2 + C*S3 + …
where here a, b, etc. are appropriate coefficients, and * just represents multiplication.
Now, let’s also consider the wave function of the observer. Let’s say before he observes the wave, he’s in state O, and once in observes the wave in some state Sn (for n an integer) he’s in state On.
So, before observation, the combined state of wave and observer is:
(a*S1 + b*S2 + C*S3 + … )*O
and after observation the combined state of wave and observer is:
a*S1*O1 + b*S2*O2 + C*S3*O3 + …
Note that no matter what state the observer is in, he sees the particle as having a particular position (whether it’s S1, S2, etc.) That is, he’s only seeing the particular branch of the wavefunction (meaning the particular term in our sum) that he’s living in. If we could see the wavefunction of the whole system (including the observer), we could see that we still have a non-localized wave, but the observer can’t perform a quantum mechanical mearsurement on this system because he’s part of it. There’s no state where the observer sees everything, only a set of states where he sees the wave function collapsed to various positions.
To make this point another way, the part of the wave function corresponding to the observer can no longer be factored out of the full wave function — so there’s no way to describe the state as an observer who sees the full superposition (that is, who sees the sum over all the terms).
So, from this perspective there is no true wave function collapse. When we observe apparent wave function collapse, we just say “I happen to have landed in the branch of the wave function where the particle appears to have this particular definite position. And although I can’t see it, I know that the full wave function contains other copies of me who observe the particle in other particular positions. But each of us sees it as collapsed to some position, because we all measured its position and thus entangled ourselves in this way.”
This can be seen as fixing the measurement problem, because it makes measurement no different than anything else (while still explaining the apparent collapse of the wave function that we observe). If two electrons interact, they get entangled, but with no collapse of the wave function. Likewise, if a human and an electron interact, they get entangled, but with no collapse of the wave function. Everything is on a level playing field.
The downside of this is that the other branches still exist, so really this amounts to postulating a virtually infinite number of unobservable copies of myself (one for each possible outcome of the experiment.) Unless these copies have real existence, then we have to admit that the wavefunction collapsed after all, and the measurement problem returns. In this way, it’s kind of like the Many Worlds Interpretation.
Whether this is a philosophically acceptable solution to the problem is debateable. Some people would say that postulating all that unobservable but still existing stuff is a worse problem than what we started with.
Hmmm, wave function is a superposition of “particle states” at each position? But look at the expanding wave from an ultraviolet photon emission. It is, typically an expanding spherical shell. I don’t even know what “photon positions” it would or could be considered constructed out of. How wide are those photon positions, about a wavelength wide? Anyway, when this shell encounters a screen, the photon jiggles a single atom, causing say a flash of visible light from a few green photons at that spot. I don’t see how entanglement with an “observer” occurs right there, since the interaction just happens and then we see the other photons later.
In any case, you still employ the back-door taken for granted, IMHO (innocently of course but I’d like you to scrutinize that) that the observation as a “given” has the power to do something special. For example, you say:
Now, let’s also consider the wave function of the observer. Let’s say before he observes the wave, he’s in state O, and once in observes the wave in some state Sn (for n an integer) he’s in state On.
But if there’s really “no collapse” to explain to begin with, even that doesn’t get off the ground and the waves just all interpenetrate and shift each other around like the waves on a ripple tank – it’s still just classical physics, you still can’t get it off the ground.
In response to Neil B.‘s post, above:
We can write the wave function as a superposition of basis states for any basis of our Hilbert space.
For simplicity, let’s talk about the state of a single particle. Even when you shoot particles through a double slit one at a time, you still see interference fringes when you measure the overall pattern that accumulates on your detector screen. However, when you use a detector to determine which slit each particle passes through, the interference fringes go away. This is the standard example to illustrate that individual particles propagate as waves unless you collapse the wave function by measurement.
Free particles live in an infinite-dimensional space, and the usual basis for this space is plane waves. It’s infinite-dimensional because there are infinitely many plane waves — one for every possible momentum vector.
The plane waves have a fixed value of momentum and total uncertainty in position. But we could also choose as our basis states which have a fixed value of position and total uncertainty in momentum. These are essentially just the Fourier transform of our original states. They look like a plane wave in momentum space, but in position space they look like a Dirac delta function (a distribution which is non-zero only at a single position).
So, to answer one of your questions, the particle position states I’m talking about really have no width at all, and the plane wave is equal to a superposition of infinitely many of them. Of course in the above example I was pretending that we had detectors which could measure the exact position of the particle — in real life we’ll at best get a small range of positions with some level of confidence, but I don’t think that distinction really matters for the point I’m trying to make.
At that point you are correct that there is no entanglement with the observer. The particle is merely entangled with the apparatus of observation. Then, when the light from the apparatus hits the observer’s eyes, he becomes entangled with it. In my above description I simplified things a bit. I could take P = particle, A = apparatus, O = observer, and then describe the sequence of entanglements as:
(a*P1 + b*P2)*A*O –> (a*P1*A1 + b*P2*A2)*O –> a*P1*A1*O1 + b*P2*A2*O2
(where here for brevity I’ve pretended the particle is in a superposition of only two states instead of infinitely many)
However, this distinction about when what gets entangled with what doesn’t affect my overall point. The point is the observer is always entangled with the system being observed by the time he observes it.
I think what you’re misunderstanding is that from the observer’s perspective something special does happen when he makes the measurement — he sees the appearance of wave function collapse. But in this view of things, the wave function collapse isn’t real, it’s an artifact of the fact that the observer has become entangled in the wave function himself.
Let me restate this point because I think it’s really the crux of what I’m saying
– In the traditional Copenhagen interpretation of quantum mechanics, measurements changes the wave function in a different way than any other sort of interaction. In technical terms, the way the wave function normally evolves is called “unitary”, and the way it changes when there’s a measurement is “non-unitary”. This gives rise to the measurement problem: “Why should measurement be different than everything else?”
– In this alternate interpretation, the evolution of the full wave function is always unitary. The wave function never actually collapses. However, because the observer inevitably entangles himself with the observed system in the process of making a measurement, from his perspective there appears to be wave function collapse. The full wave function is now a sum over all possible observers and all possible experimental outcomes, and it is only from that perspective that the lack of collapse is still apparent.
I should be clear that I personally find this sort of thing philosophically unsatisfying, but I do believe it’s a logically coherent interpretation in which no wave function collapse occurs.
No pun intended in my use of the word coherent. 🙂
To condense even more:
Before measurement, the wave function is a sum of many terms. After measurement, the wave function appears to the observer to only consist of one term. There are (at least) two possible explanations for this:
(1) The other terms of the wave function went away.
(2) We now have a superposition of observers, each observing a different one of the terms of our original wave function. The total number of terms is unchanged.
With (1), we have the traditional measurement problem.
With (2), we don’t have the traditional measurement problem, but we are left with a full wave function which contains a copy of ourselves and our experiment for each possible experimental outcome, all of which are unobservable to us. Whether accepting the existence of all this unobservable stuff is an acceptable price to pay to resolve the measurement problem is debatable.
Is anyone out there not a fan of decoherence and would be willing to take a professional crack at discussing disagreements and problems? I appreciate the effort TimG took to explain it, and at this point I should digest that and read and fiddle more before putting up more generalized conceptual complaints. However I still have a sense of misgiving, and I wonder how good this is even aside from weird multiple universe issues. Also, there must be some critics of what I’d like to call “Art Deco” out there; what are they saying? I remember, was it Penrose not being real impressed, and mentioning perhaps the Renninger issues of null results having consequences of wave function redistribution, etc.
tx all for your time, forbearance, and patience!
I don’t have much problems accepting point (2) in TimG’s post above. The superposition you end up with is a unitary transformation of the initial state, so you could just as well interpret as representing the intitial state.
If you consider the entire mulitiverse, then time evolution become trivial. In the MWI, it is more natural to consider the multiverse static. The wavefunction then satisfies the equation:
H|psi> = 0
So, the time evolution that we experience is simply an illusion. The multiverse doesn’t change at all. All that happens is that in the same multiverse that I exist in, my “time evolved copies” also exist. All the possible states that I can possible be in exist and they each contain some subjective notion of time and personal history.
Typo:
H|psi> = 0
This may not be very well expressed, but if E=mc2 and I strike a match, that turns matter into energy. At what point does energy start to become matter again? Is it when we measure/observe it and the wave becomes a particle? Fundamentally it travels as a wave, yet any effort to measure it requires an interference that results in a particle. Is this what plants do when they photosynthesize light, absorbing it and turning it into mass?
The idea of C2 doesn’t seem to make much sense as compounding the speed of light, so is it a function of volume, that amount of energy expressed within the x times the y coordinates is squeezed into the volume of the mass? When energy is released, it is as a wave in all directions.
That way, the wave collapses, but the energy isn’t lost and will eventually be released as a wave again. Often by the process of measuring/observing how much energy is in the mass, such as striking the match.
I feel I should add one more comment on decoherence. In particular I want to clarify that decoherence has a meaning beyond attempts to resolve the measurement problem.
Basically, decoherence is what happens when a system in a superposition interacts with some other system. (It could be any two systems, not necessarily an atom and an observer like in my above example.) You get a situation like I described above where you can’t “factor out” the original system, so the only way to see the superposition is to perform measurements over the composite system. If we’re talking about an atom interacting with a macroscopic system (such as the surrounding environment), then we can’t possibly measure the quantum state of the combined system, and the superposition is effectively lost.
Decoherence is an observable effect, and it happens regardless of our interpretation of quantum mechanics. Someone trying to build a quantum computer, for instance, has to worry about decoherence effects. (Quantum computers take advantage of the fact that the bits can be in superpositions of 0 and 1, so destroying these superpositions is a problem for them).
As for the measurement problem, as I said above it can be stated as:
“The evolution of the wavefunction is always unitary — except when someone makes a measurement, in which case it’s non-unitary. What makes measurement different than everything else?”
But really, the evolution is only unitary for closed systems. That is, we expect the quantum state of a system that isn’t interacting with anything else to evolve unitarily. But when we have a situation like above, where the system we’re studying is really a subsystem within some larger closed system, then the evolution of the subsystem need not be unitary.
Like I said, I don’t think this really resolves the measurement problem, since even though for all practical purposes the superposition is destroyed, at least in principle it still exists within the state of the full system (i.e., the system being studied plus the macroscopic environment it’s entangled with). One way around this is to note that the observer himself inevitably becomes intangled in the wave function, leading to a sort of “many worlds” picture like I discussed above.
An alternative is to deny the reality of the wave function — to say that it is merely a mathematical device for predicting the results of experiments. If you initially have a quantum system in a coherent superposition of state A or B, then after decoherence if you look at the state of the system alone (ignoring the quantum state of its environment, which is presumably unmeasurable) you can find a certain probability of A and a certain probability of B, but not the superposition. As far as I can see this doesn’t really explain why the system persists in state A after you’ve measured it once — but with this mindset we essentially say “Quantum mechanics allows us to calculate probabilities of different measurement outcomes based on our current knowledge of the system. Once we know the result of the measurement, we no longer need quantum mechanics to know what we’d measure.” (Well, until some time passes, at any rate.)
I should reiterate that I’m sort of playing devil’s advocate here — I don’t personally find either of these explanations particularly satisfying. If wave functions don’t exist, then the question of what actually exists that explains the predictions of quantum mechanics reamins open. I can’t stomach the extreme positivism in saying “The goal of science is just to predict measurement outcomes.” Someone once made the point that if we discovered a magic oracle that could correctly predict the result of any experiment, no one would consider that the end of science. We’d want to know how the oracle worked, and the reasons why those answers were the right ones. In my opinion the goal of science is to create a conceptual framework that accurately describes the real world and is predictive.
As I’ve said, I also don’ t care for the sort of “many worlds” interpretation you get from assuming that the wave function has real existence and never collapses. Postulating an infinite number of alternate versions of ourselves which are unobservable even in principle seems to badly violate Occam’s razor, and is in my opinion too high a price to pay to resolve the measurement problem.
Personally, I don’t think a satisfactory resolution of the measurement problem yet exists — unless it turns out that some as-yet undiscovered physics beyond quantum mechanics actually causes wave function collapse. There are some theories to that effect, but so far none has been successfully tested (although one can always hope).
TMG “The evolution of the wavefunction is always unitary — except when someone makes a measurement, in which case it’s non-unitary. What makes measurement different than everything else?”.
A “quantum” of anything will always, repeat ALWAYS!, know where you are/exist, long before you can locate a quantum? by default of scale, a quantum has always measured “you” prior to your ability to locate a “quantum”. Think about a quantum needle that exists in a macro haystack, the needle will detect you macro movement, whilst you will not “register” the needle anywhere!
By default, the evolution of any observation is stacked one-way, from the quantum outwards, in gravitational terms it’s like walking on the surface of the Earth, I am the “quantum” and I know with certainty that the macro Earth is below, but WRT all else that is going on upon the Earth’s surface, does it know with certainty where I am shuffling my feet ?
Paul, I’m not sure whether you’re being sarcastic.
If you’re not, what kind of “knowledge” are you referring to? How does a field with only a few components code for the knowledge of everything?
The way I see it, it goes the opposite way. The macro system “discovers” that it’s linked to the observed quantum when the quantum forces it to separate into distinct possibilities, and the macro “chooses” which to become.
(I’m not referring to a consciousness or “cat” argument. I’m referring to “dumb luck”. That is, Markov Processes or Martingales or whatever.)
The decision propagates from the macro to the quantum, starting with how much blurriness the macro can tolerate and “deducing” that of smaller and smaller portions.
If that doesn’t make sense, what am I missing?
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