bbbeard said:
But the amplitude is not a probability. It lacks the essential characteristics of a probability. It is not real and non-negative. It is does not sum to 1.
But it does sum to 1, that's why it is a probability. And all the
net probabilities he is talking about are non-negative. So he
is talking about probabilities, it is just that the
contributions to the net probabilities can be negative. That is what he sees as the crucial innovation of quantum mechanics, and that really is a lot of it.
The wave function in quantum mechanics is directly linked to a mechanical flux as well. If you write down the current
j = -(i hbar/2m)*(ψ
* grad ψ - (grad ψ
*) ψ)
then it follows
∫d
3x j(x,t) = <p>
t/m
where <p>
t is the expectation value of the momentum operator at time t.
But that isn't what sets quantum mechanics apart from classical physics, that much is quite similar to classical wave mechanics. What sets quantum mechanics apart is the "quantum" part, and that is also where the probability comes in-- you have to be able to talk about probabilities if you need to talk about what quanta do. You just don't need that with classical waves, so with waves it's all just amplitude interference, never negative probabilities. So the key difference is that quantum mechanics is invoking negative probabilities. Why don't we find it strange that opening a second slit can reduce the light intensity at a point, but we do find it strange that opening a second slit can make it less likely for a
particle to show up at some point? It's because we think that particles should obey probabilities rather than amplitudes. Aaronson is saying we can still think that way-- we just overlooked negative probability contributions.
Except the real-number analog doesn't work. In the standard notation, even if you implement real-valued Sx and Sz, you wind up having to introduce complex numbers as soon as you try to formulate Sy.
That's why it is just an analogy. It extends to the complex numbers in a fairly straightforward way that he didn't want to get into in that simple description. All you have to do is say that if you have amplitudes a and b, and probability |a+b|
2, you can think of that as:
|a+b|
2 = |a|
2 + |b|
2 + 2 Re(ab
*).
The first two contributions are the way independent probabilities add classically, but it is the third term that can be negative. Aaronson interprets that third term as a negative independent probability contribution, because it can be larger in magnitude than either of the first two terms (a feature that classical probabilities never have). The chances of a and b can be less than the chances of either a or b by themselves, so that's what he means by negative probability contributions.
The "preservation of the 2-norm" in classical wave theory is equivalent to the conservation of energy, isn't it? Maybe I'm misunderstanding something here.
It is not quite equivalent, because it's a conservation of probability, not a conservation of energy. Conservation of classical probabilities doesn't involve interference, though conservation of wave energy does. There is a very important marriage of these two effects that is very much the distinguishing character of quantum dynamics.
Well, first, "flux" is not a conserved quantity.
Yes it is, you are referring to "flux density", which is not conserved. Those terms get used rather loosely.
Second, there are "discrete outcomes" in classical wave theory. Consider the eigenmodes of a plucked string, or the oscillations of gas in an organ pipe.
That is not what I mean about a discrete outcome. I'm not talking about the quantization of the frequencies of a guitar, which is not an example of a quantized action or a quantized energy, it is just an example of a quantized frequency, that's all. What I mean by a discrete outcome is a discrete set of possible outcomes for a
quantum of some kind-- be it a particle or a quantum of action (i.e., whether first or second quantization).
Third, a system doesn't have to have a discrete eigenspectrum to be quantum mechanical.
Again, the discreteness I refer to is not the eigenvalue, it is the particle or the action being tracked. In short, it is the "quantum" in quantum mechanics.
Of course there are "negative contributions to probability" in classical probability theory:
P(A ∪ B) = P(A) + P(B) - P(A ∩ B))
But note, that third term can never be larger than either of the first two. Thus, it is not really its own independent effect, it is merely the overlap of the first two effects. Aaronson is implying a kind of independence between the probabilities of a particle going through either of two slits, for example, so it is something much more than just subtracting the probability of going through both (if that were possible). For example, you could draw a green line and red line on the floor, which overlap slightly, and ask what is the probability of a classical particle crossing either one. That would be a result like yours above. But that isn't interference, it's just subtracting off the double-counting. Aaronson is saying that a quantum particle can have a probability of crossing the green line that actually cancels out part of the probability of crossing the red line, in a way that has nothing to do with double counting and indeed can give a zero result for the whole business. That's fundamentally quantum mechanical, even though it certainly borrows from the classical concept (as quantum mechanics generally does).
Again, it's not a probability until you marry the amplitude to itself. I.e. the amplitude can be rotated in state space but it's not a probability until you square the amplitude. The wave function of an electron is not a probability distribution; only the squared amplitude is.
Both Aaronson and I agree with that, it's not a problem.
And this is one place where I think Aaronson is missing the boat on "quantum weirdness". The wave function is much more than a "probability wave".
Ah, but that's his whole point. He is saying it is not much more than that, as long as you allow for a type of negative probability contribution. The probability of going through one slit and hitting a box on the wall, and going through the other slit and hitting a box on the wall, are indeed probabilities even in quantum mechanics-- they just don't have to add, they can subtract (what really happens is some of the probability contributions are complex, like ab
* and a
*b, but they add to something real so their combination can be thought of as a single probability term or go whole hog and talk about complex probabilities).
Yes, quantum mechanics is interpreted in terms of probability. But if you apply an operator to a ket, i.e. if you make a measurement on a quantum state, you can get all sorts of strange outcomes.
But he covers the completely general case when he talks about density matrices and unitary operations. The only bit he is sweeping under the rug is he tends to explain the situation only for real coordinates, rather than the full complex-coordinate description. But I see that as just a streamlining trick, what he is saying generalizes like orthogonal --> unitary for the evolution and symmetric --> Hermitian for the operators.
But if the ket is not an eigenket, you get a linear combination of all the eigenkets of the operator. This is not a true statement for the probabilities.
Yet what you are saying here is just what he was describing with that "quantum coin" analogy. It can all be said at the probability level, if one sees a machinery going on behind those probabilities. In other words, he's not really saying you get quantum mechanics from negative probabilities, he is saying you get quantum mechanics from a fairly natural behind-the-scenes mechanism just by not requiring it to deal only in positive contributing probabilities (with subtractions due only to double counting).
I suppose I am jaded enough, or maybe just came to the game late enough, that I don't find quantum mechanics all that weird. But if "weird" is the word you want to use, then I'd say a lot of weirdness has not so much to do with the probabilistic aspect and more to do with how wave functions represent particles. If you write down Schrodinger's equation for a single electron, you get six degrees of freedom (three position and three momentum) (not counting spin). But if you write down the Schrodinger equation for N electrons, you find that there are 6N degrees of freedom that are antisymmetrized. All of a sudden you have to deal with a Slater determinant. I find that a lot "weirder" than the "probability wave"...
I agree that multiple identical particles brings in a whole new level of quantum weirdness (as does entanglement of all kinds), but if we stick to single-particle descriptions, it is still a bit weird that wave mechanics is applicable. But I would say this type of "weirdness" is very much a holdover from classical thinking. I view quantum mechanics as the unification of classical particle and wave mechanics, so unification is usually not considered weird, but what's weird is that no one was looking for unification of those things (any more than they were looking for unification of Maxwell and Newton when relativity came along, which is why relativity is often also considered weird). I thik Aaronson is arguing QM is
not that weird, what makes it weird is simply our classical prejudices like positive probabilities.
