No, it means it’s in the single state you get when you add the two together and normalize the resulting state.
Look, quantum mechanical states are normalized vectors, and they obey the mathematics of vectors.
If I want to construct the superposition of two states (1,0) and (0,1), I can add them together, giving (1,1). Then I stick a constant in front so the length of the resulting vector is 1 (this step is normalization). The answer is sqrt(1/2) * (1,1).
Clearly sqrt(1/2) * (1,1) is not the same vector as either (1,0) or (0,1).
I am also amazed that you are saying when Feynman says, “The particle take all possible paths between A and B” that if I say, “So, the particle is on all those routes at once” he would say, “No, you misunderstood me, you need to do the math to understand.” :rolleyes:
What was Feynman saying when he redefined QM? When Schroedinger and company were on about probabilities and Feynman came in and said that is not quite right, the particle takes all possible paths, what are you telling me Feynman was really saying?
He did not say they were not quite right. On the contrary, he said you could do the math a different way and get identical predictions. The approach itself has mathematical constructs internal to the calculations that have to do with all possible paths, but no where along the way do those paths become real things.
Quantum computers don’t work by being in two states at once, which is what you seem to be saying and what people often say. They do use superposition, but I hope my examples are helping to clarify that superposition does not really mean it’s in two states at once. (1,1) is a different state than (1,0) or than (0,1) . . . (1,1) is not two states at once.
Feynman provided the math. No one would have cared if he had just said “The particle takes all possible paths” (if indeed he ever phrased it that way) and not elaborated on what he meant.
I cannot find a quote from Feynman on much of anything much less this. I did provide Brian Greene attributing that to Feynman. So, one removed from the source and Greene may have been playing fast and loose but that would surprise me.
Further, I think it would be simplicity itself for scientists like Greene and whoever to say, “It as if the particles were taking all paths at once but to be clear we do not literally mean they are.”
A few more words yet far clearer. Why do I not see it written that way when this is discussed?
I think that the problem here is that the attempt to map a precise, mathematical statement onto everyday, English words that have their roots in a classical understanding of the universe is going to lead to a lot of vagueness and confusion.
I can think of several different interpretations of “the particle is simultaneously in state |0> and state |1>.” As Pasta said, it might be in |0> + |1> or |0> - |1> (normalization neglected). But these are two orthogonal states (assuming |0> and |1> are orthonormal). Quantum mechanically, they are as different as they can possibly be! But we’ve used the same, fuzzy English to describe them. And, if we’re trying to describe the real world rather than an idealized thought experiment, the particle is probably in a mixed state that is best described by a density matrix – i.e., it’s not described by a pure quantum state at all. Would this state also translate to the same English phrase, “the particle is simultaneously in state |0> and state |1>”?
Frankly, what you want to regard as “real” is a matter of taste. The path integral formulation of quantum mechanics gives exactly the same predictions as the wave mechanics and matrix mechanics formulations. So who really cares whether the wavefunction is a “real” thing or particles “really” take multiple paths? Even in classical physics, these questions arise. Are magnetic fields real? If magnetic fields are real, what about the magnetic vector potential? After all, the vector potential depends on the (arbitrary) choice of gauge, so it must be a mathematical fiction, right? But (back to quantum mechanics) the Aharonov-Bohm effect predicts phase shifts even in a region of zero field (but nonzero vector potential), so maybe the potential is real and the field is a fiction.
I think that our brains just didn’t evolve to deal with this level of abstraction, so they’re constantly engaging in reification to cope with the alienness of it all. And so we end up believing that the sloppy, English translation of the physics is what describes “reality.”
Practicality, I would guess, in achieving the particular goal of exciting the general public in science. While Greene does an admirable job in his public relations quest, I often cringe at the white lies he tells. To be sure: if he didn’t, his audience would probably become disinterested, and perhaps that is worse than them being interested in a slightly squishy science. (This sort of thing is ubiquitous in popular science writing, of course. It’s not a Greene-specific issue.)
i think feynman actually does use the phrase “taking all paths” in reference to reflection of photons in his book about QED. iirc it was the chapter where he was explaining how the photon model of light was able to explain things that the ray model assumed - straight light movement, snell’s law, equal reflection angles, etc.
however, he does stress repeatedly and emphatically that the paths are probabilistically determined.
So, if I understand you correctly, you are saying math can describe what is happening while the English (or any) language is wholly incapable of conveying what is going on.
I wrote upthread a very simple statement of how scientists could explicitly say they do not mean that a particle takes all paths in a literal sense. It was a simple, clear and concise sentence yet the answer to that amounts to, “They are trying make it more exciting than it is.” “They” being other physicists trying to sell books.
And I still do not have an answer, in English and not math, how the double slit experiment gets the results it does.
I get that IF we want to find a particle traversing A → B we use probability to determine where it might be at a given moment.
However, we have the observer effect. If we do not look then the particle appears to go through both slits.
It is not a half a particle going through each slit. Not a smeared out thing that is kinda here and kinda there.
If we do anything to determine which slit it goes through the whole thing falls apart.
So, how does the particle interfere with itself? How does it have to go through both slits to achieve interference without actually going through both slits or dividing itself?
And I am still missing why the example of quantum computers managing to work is not evidence of superposition such that the particle(s) working through the system do not actually work through multiple paths simultaneously. Kinda thought that was the whole point and why code breakers drool at the idea of having one.
The thing is, the particles really are “smeared out” in a sense when you aren’t measuring their position. You may be familiar with Heisenberg’s Uncertainty Principle which captures the idea that the particle doesn’t have a definite position (nor a definite momentum).
And while I wouldn’t exactly say there’s half a particle going through each slit, you could say that the state of the particle has 50% overlap with the state where it definitively goes through the left slit, and has 50% overlap with the state where it definitively goes through the right slit.
It’s perhaps less confusing just to say that the particle is exhibiting wave-like behavior, and waves after all create interference patterns when they pass through the double slit.
The reason code breakers would love to have a quantum computer is that some codes are based on certain problems that are hard to do quickly, such as factoring large numbers, but which a quantum computer could do significantly more quickly. (By “more quickly”, I really mean that the number of bits required to do the factorization doesn’t grow as quickly with increases in the size of the numbers to be factored.)
So yeah, quantum computers can solve some problems more quickly than classical computers. But the reason they’re quicker isn’t that their quantum bits (qubits) are in multiple states at the same time. At any given time the full set of qubits is in one state… but there are many more states afforded to a quantum computer than a classical computer with the same number of bits because in addition to having a bit be 0 or 1 you can have superpositions like 0+1 and even entangled states of multiple bits like 00 + 11. So this lets you run different and sometimes more efficient algorithms than you could on a classical computer.
This is where things get tricky. Let’s compare the statement “the particle goes through both slits” with the statement “the particle is a smeared out thing that is kinda here and kinda there.”
The first statement is a perfectly okay qualitative description of the two-slit system in a path integral description. It doesn’t completely describe the system, but it’s fine as far as it goes. Similarly, the second statement is a vague but also okay qualitative description of the system in a wave mechanics description. In fact, you’ve probably heard descriptions of an atom’s electron cloud that sound just like this: the electrons don’t really orbit the nucleus in little elliptical, classical orbits; rather, they occupy a “smeared out” probability distribution in space. In the two-slit problem, the particle’s wave function is described by a traveling wave packet that evolves according to Schrodinger’s equation. It diffracts from the slits in much the same way that a classical light pulse would, and exhibits a deterministic probability distribution downstream that can be found by computing the evolution using Schrodinger’s equation. The particle’s wavefunction is, indeed, “smeared out,” and so we might choose to say that the particle itself is “a smeared out thing that is kinda here and kinda there” if we wanted to convey the basic idea without a lot of precision.
My point is that the two different mathematical formulations have different English translations, both of which are roughly accurate descriptions of the system. And the two English translations do sound pretty different from one another. This does not mean that one is right and one is wrong – they’re both just imperfect attempts to describe two equally valid ways of looking at quantum mechanics using brief, memorable phrases in everyday language.
But I’m not sure what it would even mean for the particle to take all paths in a literal sense. You can’t observe the particle going along all the paths simultaneously, but you can observe the interference at the end. So did the particle “literally” go along all of the paths? I don’t know – it’s a philosophical question of ontology (which I’m utterly unprepared to discuss in any depth). By the same token, is the wavefunction in the Schrodinger description of QM “real,” or just a useful mathematical trick?
As I said before, and as Pasta alluded to in his post about classical Lagrangian mechanics, this is not a problem that is specific to quantum mechanics. All kinds of notions in the mathematical formulation of classical physics might or might not “literally” exist depending upon one’s tastes. We’ve all learned Newton’s laws, in which there’s something called a force, right? Forces seem pretty real to us. But, if you choose to describe the classical world using Lagrangian mechanics, the notion of force doesn’t need to come up at all. It’s all about finding the trajectory with the least action. Does that mean that one body does not literally exert a force on another body? No, they’re just different ways of looking at the same thing.
I certainly wouldn’t go so far as to say “wholly incapable.” A great deal of understanding can be imparted by non-mathematical explanations. But brief, pithy statements in everyday language do seem to miss a lot of nuance. For example, consider how much confusion there is about “for every action there is an equal and opposite reaction.” It’s not wrong, but it sure is subject to misinterpretation and overgeneralization unless carefully explained using precise definitions of the concepts involved. More rigorous development of scientific and mathematical concepts allows for more precise explanations. Of course, one can be just as wrong while using scientific and mathematical jargon as one can by using plain English.
Okay, post too long, must go to bed. Thanks for an interesting discussion, Whack-a-Mole.