Originally posted by twhiteheadIn addition to Feynman's lectures, you can try his books. He wrote some nice books for laymen.
Sorry for not following up on this thread. I genuinely want to understand more about electron spin and photon polarity. However, the responses convinced me I need to do some reading first before asking questions or I won't understand the responses. (and my questions won't be very good).
I am in the middle of watching a Yale course on evolution, so when I ...[text shortened]... then 9% on what we don't yet know. What I liked about Feinman is he pushed the 1% up to 90%.
Ok., here's a question. Suppose we have an electron in a state where it's spin is the superposition 1/sqrt(2)( |1/2> + |-1/2> ) we observe it's spin and suppose it comes up heads, the wavefunction now is |1/2>. The entropy is given by:
S = - Sum( p_i*log2(p_i))
where p_i is the probability of a measurement giving spin i, the sum is over all spin states, and I've used the Shannon form of Entropy to highlight the informational aspect and avoid fiddly constants. Initially the entropy is:
S = - 1/2*log2(1/2) - 1/2*log2(1/2) = log2(2) = 1 bit
After the observation the entropy is:
S = - 1*log2(1) - 0*log2(0) = 0 bits
So the entropy of the system seems to have dropped. With some waving of hands it's possible to attribute the entropy loss of the particle with an entropy gain for the observer, but without a decent theory of observers there seems to be an entropy anomaly with quantum measurement. Comments?
Originally posted by DeepThought"Observation" is essentialy a many-body process, so you can't really look at entropy in this way. You could calculate entropy in a closed system (although I never used it in my work).
Ok., here's a question. Suppose we have an electron in a state where it's spin is the superposition 1/sqrt(2)( |1/2> + |-1/2> ) we observe it's spin and suppose it comes up heads, the wavefunction now is |1/2>. The entropy is given by:
S = - Sum( p_i*log2(p_i))
where p_i is the probability of a measurement giving spin i, the sum is over all spin ...[text shortened]... eory of observers there seems to be an entropy anomaly with quantum measurement. Comments?
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Originally posted by KazetNagorraYour point about the observation being "many body" is fair, this is why I'm wondering about a theory of the observer to produce an idealized one to get a decent handle on this question. I don't think there's anything wrong with my entropy calculation, although I'll grant you that entropy is usually applied to somewhat larger systems. The formulism should be resilient to application to simple systems. The electron is isolated except when being observed, so its entropy is well defined between measurements.
"Observation" is essentialy a many-body process, so you can't really look at entropy in this way. You could calculate entropy in a closed system (although I never used it in my work).
This is a basic problem though, in something like Stern-Gerlach it's not quite clear what's happened to the electron after it's been measured. Being able to repeat the spin measurement is critical to being able to define an entropy, the probabilities are probabilities of outcomes of future experiments on the otherwise undisturbed electron.
Undergraduate courses in quantum theory normally justify expressions like [x, p] = ih/, in terms of chaining idealized momentum and position measurements. But if you measure the position of a photon your apparatus will have absorbed it and it's not available for the subsequent momentum observation. So the kinds of problems that exist for my entropy argument exist for fairly uncontroversial foundational arguments as well.
We could rehash this in terms of qubits. A qubit is prepared in quantum register with wavefunction N( |0> + |1> ), where N is the normalization. We then measure the qubit and store the result in a classical register. Now since, in the information theoretic approach, entropy is information, the loss of the bit of entropy from the qubit is balanced by the bit of data in the classical register. But here I'm treating the classical information as being present when it's certain, when the quantum information is maximized when it's uncertain; which doesn't really make sense.
The proof that the statistical definition of entropy is a non-decreasing function of time depends on the normal evolution of occupation numbers. It doesn't take account of a quantum measurement as the thing that alters the state. There are two attitudes to this, from an old thread, as well as what you've just said, your opinion is that a quantum measurement is just an interaction and the normal entropy rule still applies; with the possibility of a new theory that makes more sense than (traditional) quantum theory. My opinion is that EPR like experiments involve something fundamental to physics which doesn't happen at classical scales. Keeping track of what happens to the informational entropy may shed light on this.
Originally posted by DeepThoughtWell, unless you can say something about the entropy of the environment after observation, I don't think there's much you can conclude from your gedanken experiment.
Your point about the observation being "many body" is fair, this is why I'm wondering about a theory of the observer to produce an idealized one to get a decent handle on this question. I don't think there's anything wrong with my entropy calculation, although I'll grant you that entropy is usually applied to somewhat larger systems. The formulism shoul ...[text shortened]... ping track of what happens to the informational entropy may shed light on this.
Undergraduate courses in quantum theory normally justify expressions like [x, p] = ih/, in terms of chaining idealized momentum and position measurements. But if you measure the position of a photon your apparatus will have absorbed it and it's not available for the subsequent momentum observation. So the kinds of problems that exist for my entropy argument exist for fairly uncontroversial foundational arguments as well.
Often p = -ih d/dx is taken as an axiom of quantum theory, from which you can derive the commutation relations and Heisenberg's uncertainty principle (alternatively, you can take the commutation relation as an axiom, and derive the momentum operator). There are situations where you can measure position and momentum right after each other. In those cases, you will find that the latter measurement will have a high uncertainty, in accordance with Heisenberg's uncertainty principle.
Originally posted by KazetNagorraWell ok, but in order to do a momentum observation you have to transfer some momentum out of the particle in order to be able to detect it, with some rule to convert the transferred momentum to a total momentum. In the idealized case the new particle is a plain wave with infinite uncertainty about the location. Each subsequent momentum observation should give the same result. But this must happen with zero momentum transfer, otherwise the measured momentum would change. So how is this momentum measured, while being left undisturbed?
Well, unless you can say something about the entropy of the environment after observation, I don't think there's much you can conclude from your gedanken experiment.
[b]Undergraduate courses in quantum theory normally justify expressions like [x, p] = ih/, in terms of chaining idealized momentum and position measurements. But if you measure the posi ...[text shortened]... rement will have a high uncertainty, in accordance with Heisenberg's uncertainty principle.
Part of the problem for me is that I did theoretical physics and simply don't know enough about the experimental set-up to have an intuition of how one takes find the limit of the momentum transfer as the observations become more ideal.
What I was getting at with that paragraph was that this kind of "how do you do that" problem exists with the foundational aspects of quantum theory anyway, so I've got an excuse for labeling my observer as non-entropy increasing.
Yeah, the introductory course I did basically started with "here's a trick with calculus", and justified the formalism from that.
Originally posted by DeepThoughtThe "idealized" plane wave cannot exist in nature because it is not a normalizable state. It is possible to have a wavefunction with a momentum distribution sharply peaked around some value though.
Well ok, but in order to do a momentum observation you have to transfer some momentum out of the particle in order to be able to detect it, with some rule to convert the transferred momentum to a total momentum. In the idealized case the new particle is a plain wave with infinite uncertainty about the location. Each subsequent momentum observation shou ...[text shortened]... cally started with "here's a trick with calculus", and justified the formalism from that.
Unfortunately I am a theorist too (and not a very good one) so I don't know the fine details of what experimentals can and cannot measure.