Hi I’m about half way through writing a realluy long post on collapse and decoherence in response to your question. I’ll post it tomorrow when I get the chance.
Upon reading this again I think I have have hard time formulating sentences with the word superposition. Thanks for taking the time to answer though 
I think the amazing number of falsehoods that are confidently stated in this thread just proves that you cannot accurately learn quantum mechanics without doing math. Sorry. You just can’t. In a college physics class, these misconceptions would almost all be quickly dispelled because everything was based in solid logical argument rather than fishy semantic interpretations and youtube videos.
The “almost all” caveat is intended to cover the total inexplicability of quantum interference, which is necessarily axiomatic. Beyond that, though, it’s all just a case of following the equations and seeing where they go - quantum erasure, tunneling, whatever. Feynman says at the very beginning of the third volume of his Lectures on Physics that two slit interference is “impossible, absolutely impossible to explain in any classical [read: intuitive] way, and has in it the heart of quantum mechanics. In reality, it contains the only mystery. We cannot make the mystery go away by “explaining” how it works. We will just tell you how it works. In telling you how it works we will have told you about the basic peculiarities of all quantum mechanics.”
So I’d say give up trying to learn quantum mechanics from peoples website and from pop science books. You’ll only get more confused, and probably for no good, because the explanations will probably be inaccurate.
That doesn’t mean you shouldn’t try by any means, though. Quantum mechanics is amazing, truly beautiful, and learning it was one of the most inspiring and incredible experiences of my academic life. But learn it the right way, with a textbook designed to teach it correctly. Feynman’s book is still probably the best out there.
Well, I’m afraid that I must disagree. Feynman’s Lectures may be a very good adjunct to a text book, but it most definitely is not the best to start out with.
I would guess that most profs would recommend David Griffith’s Introduction to Quantum Mechanics or Shankar’s Principles of QM. But in addition you’ll need to know math at least through linear algebra, and Hamiltonian and Largrangian classical mechanics.
Also just because you understand the math does not mean that you’ll understand QM; Instead you’ll clearly understand why you’ll never understand it. Maybe you should reread Feyman’s quote, "I think I can safely say that nobody understands Quantum Mechanics.
One of the postulates (The postulates of a physical theory as a set of statements from which the rest of the theory can be derived) of quantum mechanics is that the state of a quantum mechanical system is completely described by a state vector (aka wavefunction). This vector lives in state space, which is basically the set (or more properly vector space) of all the possible state vectors of the system.
At this point don’t worry too much about ontology: state spaces are completely abstract mathematical structures, there’s no point trying imagine what a state vector actually is, it’s just a mathematical way of describing a physical system. Infact each quantum state is described by an infinite number of different state vectors in a state space.
If you were to look at the postulates of quantum mechanics you could group them in to the following three catergories (not that I’m claiming this is the only way or even most logical way to catergorize them):
[ul]
[li]Those which describe the state vector and how it behaves (mathematically)[/li][li]Those which describe what happens to the state vector when a measurement is made[/li][li]Those which describe how the state vector relates to results of measurements on a quantum mechanical system[/li][/ul]
Whilst the postulates do describe what happens when a measurement is made, it’s important to realise that they don’t actually describe what a measurement is. In fact currently there still is no 100% satisfactory defintion of a measurment in quantum mechanics. This problem is known as the quantum mechanical measurement problem.
One of the postulates of QM states that, when not being measured, the mathematical behaviour of a state vector is described by a wave equation such as the Schrodinger equation. This equationdescribes the time evolution of the state vector and in particular time evolution it descriobes is of type known as unitary. “Unitary” here means that as the state vector evolves as per the Schrodinger equation it’s ‘length’ does not change.
There is however another kind of time evolution of the state vector described by the postulates of quantum mechanics and that is the instantaneous “collapse” of a quantum state due to a quantum mechanical measurement (a quanutm mechanical measurement here is merely defined circularly as one which collapses the wavefunction). Collapse is described mathematically by a projection of the state vector (on to the eigenvector of an operator of a quantum mechanical observable to be precise) and projection is never unitary, except in the degenerate case. The degenerate case being the projection of a state vector on to itself, which would describe what happen if you made a measurement and made the same measurment immediately after. So collapse itself is never unitary.
To elaborate a little on what collapse is (matehmatically) each quantum mechanical observable (e.g. postion, momentum, etc) is represented by something known as a Hermitian operator (don’t worry too much about what this is, basically it’s more maths), each such operator has a set of eigenvectors (the eigen- prefix is just from their association with the operator another way to describe them would be as the charateristic vectors of the operator in question) and each eigenvector is associated to different outcome of the measurement. When the state vector/wavefunction collapses it will instantly ‘flip’ to one of these eigenvectors at random. The likelihood of obatian a certain measurement is linked to the relationship between the state vector just prior to collapse and the eigenvectors associated with the measurement.
Don’t worry if you did not understand all of that the key point of this is: the time evolution of a state vector as decsribed by a wave equation like the Schrodinger equation is always unitary, the time evolution described by collapse is never unitary. (of course feel free to ask about anything).
So we can conclude a (linear) wave equation can never describe the collapse of the wavefunction
There are caveats aplenty that haven’t been mentioned, but I think it’s best to explain it this way first. In my next post I’ll try to explain how you combine quantum mechanical systems together and how decoherence can result.
I think it’s also worth pointing out that I would never describe myself as an expert in quantum mechanics. I believe there’s a couple of people on the board working in the general field and they might want to interject and I’d certainly want to hear from them if they do.
Question about randomness:
Is it pretty much proven that the randomness can’t be accounted for by any other factors?
For example:
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If the entire universe was connected in some unknown way such that particles aren’t as “isolated” as we think they are during experiments. For example: if the particle going through the slit was influenced by the state of the entire universe, due to the flux in the universe we wouldn’t really be able to calculate what it’s going to do. I understand you can’t exceed the speed of light for information xfer, so maybe it’s a fabric of forces operating at the speed of light we simply do not know about. Maybe it’s such a weak force that only influences particles/energy during this wave collapse. Is possible?
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I was thinking about a 3D projection onto a 2D, plane, would a 2D occupant be able to accurately determine all rules? Seems like there is information lost, which could appear in 2D as random/undeterminable. Could be something similar going on?
I started out with both Griffiths and Feynman in my undergraduate class, and I found Feynman much, much more helpful. For me, the matrix formulation was just a lot easier to grasp the first time I was going through the material, and I found I had a sort of vague intuition about what should happen just from looking at a piece of Dirac notation. I never got anything like that in Griffiths. Maybe it’s just me - I’ve always thought that the matrix formulation works a lot better
Yes, exactly. I was careful to never use the word “understand” in my post; you can learn QM but you won’t understand it. That’s what I was trying to say in my post. I apologize if I was unclear.
Yes, it has. The proof is called “Bell’s Inequality.” It says that if you want a “hidden variable” theory of quantum mechanics, you’re going to have to make it “nonlocal.” This means that particles will have to know about each other without being anywhere near each other and act based on that knowledge. If I wave an electron in my bathroom, it might affect a proton in your cranium.
To most physicists, this is too much to deal with. Nonlocality might as well be magic, and we just choose to reject it. But there are some who subscribe to the Bohm interpretation and just say “Yeah, well, so what if its nonlocal? At least it isn’t random!”
If you are a Bohmian, that comes with some heavy changes - there’s no more real uncertainty principle, particles act with the knowledge of the entire universe, and so on. To my knowledge - which is definitely limited in this area - no one has yet come up with an experiment that can test the Bohmian interpretation versus the more orthodox “Copenhagen” interpretation that its all just random.
Very nice. I look forward to part 2.
Why do the physicists seem to like randomness vs non-locality. If you assume the universe is bizarre, then, from a non-physicist point of view, non-locality seems like the lesser of 2 evils (maybe?).
The primary problem is that Bohmian mechanics requires you to jump through a lot of hoops to get the right answers. These hoops are often way more mathematically complex than the ones you need to just do the standard Copenhagen quantum mechanics. It’s not clear what you’ve gained, and you’ve added a lot of complexity to the theory by demanding deterministic causality. And after all that, it’s not clear how the Bohmian interpretation is testably different, even if it makes you feel more comfortable that the world is deterministic.
Another problem is the speed of light. If particles can communicate instantly over arbitrary distances, then they could send information at faster than the speed of light. If that’s true, then they can actually communicate backwards in time according to some reference frames, which violates causality - perhaps the most basic axiom in all science!
Bohmian mechanics gets past this disaster by denying relativity’s fundamental precept that “everything is relative” and claiming that some priveleged reference frame does actually exist that everything else is measured off of. The reasons that this solves the problem is too abstract to understand, but very smart people say it, so I believe them. It still makes me uneasy: why does the universe have one privileged frame to view everything from?
Finally, Bohmian mechanics just shifts us to a new question: How do the particles talk to each other instantly over any distance? What’s the mechanism that allows that? It seems that most Bohmians would say “never mind how, they just do.” For a theory that claims to return us to our happy intuitive sense of deterministic causation, this isn’t very satisfying.
I’m sure there are more issues. I’m totally positive that there are mathematical objections which I’m not qualified to answer. But that’s my sense, as someone who has only dabbled a bit in this discussion, a debate known as the “foundations” issue in quantum mechanics.
I’ve noticed there were a few errors in my last post due to constant re-editing, I hope it’s still readable. Be aware though I’m not the greatest proof reader.
Another of the postulates of quantum mechanics tells you how to combine quantum mechanical systems (for example a system describing an individual particles) in to a larger quantum mechanical systems (for example a system containing several particles). To combine two quantum systems in to a larger quantum system you must take the two state spaces describing each system and combine them in to a ‘larger’ state space by taking their tensor product (again just maths). I’m using ‘larger’ loosely here because what’s is increasing in size is the number of dimensions and that’s assuming they both have a finite number of dimensions to start with when generally speaking state spaces of quantum mechanical systems have an infinite number of dimensions! For two finite dimensional vector spaces of dimensions n and m the dimension of their tensor product space is n*m.
For a macroscopic object such as a set of measuring equipment I calculate the region of 10[sup]27[/sup] particles. That’s only a rough calculation, so you can dispute my figures, but the point is a macroscopic system has a huge number of particles which as I will explain below is the reason that decoherence occurs.
An interesting property of combing two state spaces of two quantum mechanical systems is that the state space of the combined system will contain state vectors which will not allow you to think of the system in terms of its two subsystems. The states correspond to such state vectors are known as non-separable or entangled states. To illustrate this a little better let’s say we have a system of two particles which where the combined system in an entangled state. We cannot assign a state vector to each individual particle, instead we must describe the particles in terms of the two particle-system. Though on the other hand if they’re not entangled then we can assign each individual particle a state vector and treat them as if they were two separate quantum mechanical systems.
Mathematically entanglement is just a property of the tensor product and physically the consequence of entanglement is that the behaviour of entangled particles is not independent.
Now it’s important to note that the state vector of the combined system is still governed by the postulates of quantum mechanics and the postulates of quantum mechanics apply equally to a system of one particle as it does to a system of a gazillion particles. In particular let us be aware that unless a quantum mechanical measurement (i.e. a measurement that collapses the state vector) is made then the system’s time evolution will be described by a wave equation and it will be unitary.
Next on to decoherence.
Slight delay, I had a handwaving explanation of decoherence, but I’ve seen a couple of flaws in it. So I’m trying to think of a better way of explaining it