I guess I have mistaken your misgivings somewhat; I will try to say something new to address them better instead of repeating myself unhelpfully.
mr. vodka said:
In regard to your reply on (1):
I'm familiar with the mathematical formalism and what it states, but that's not really what my question was about, it was in response to your "Examining some realistic cases may suggest to you that this is not so strange. Suppose I want to measure the position of an electron by [...]" where the reason for the change in momentum when measuring the position was due to the specific 'construction' of the position measurement apparatus; it seems one can easily think of another measurement apparatus that will destort the momentum more or less, but as you say yourself the math does not imply such things, so it's questionable that the math and your physical example are talking about the same principle (why do you think it's obvious they are?).
You are correct that it is possible to construct various measuring apparatuses for some observable, some of which will cause more or less disturbance to the incompatible observables. The minimum disturbance is determined by the commutation relations and the corresponding uncertainty relations. But nothing prevents a measurement from causing *more* uncertainty in incompatible variables than the minimum specified by the uncertainty relations.
Consider: I have an electron traveling along a tube of 1 meter in diameter and many meters long. Suppose the electron's initial state is a wave packet with a sharply defined momentum, but spread out over a significant portion of the tube in position space. I now perform the following crude position measurement: I irradiate a 1 meter long section of the tube with electromagnetic plane waves of enormously high frequency. If I detect any scattered photons I have localized the electron to this cylindrical region of 1 meter in diameter and 1 meter in length. This is a very large uncertainty in position and the corresponding uncertainty in momentum demanded by the uncertainty principle is quite small. But in fact the electron has received, from recoils, a potentially huge amount of momentum in an unknowable direction, so after the measurement the electron has a very large uncertainty in momentum.
mr. vodka said:
A side remark: any measurement of position will make it collapse into a position eigenstate (and thus all information about momentum will go away), yet in your physical reasoning of the experiment, there seems to be a gradual increase of the removal of momentum-information, based on how well you measure the position.
This raises a point you should think about when considering these questions: the measurements we most like to think about when doing the math are highly idealized (as what mathematical physics is not?). A position eigenstate represents a particle localized to a volume of size literally zero, which is manifestly unrealistic. No physical process can really be said to measure the position observable we use so often in quantum mechanics. Instead consider as a slightly more realistic example the observable
\lfloor \hat{x} \rfloor
That is, the floor of the position operator (let's confine ourselves to one spatial dimension, and also fix some unit of length so that the floor of the position makes sense). It's effect, for instance, on the position eigenstate localized at x=2.5 is
\lfloor \hat{x} \rfloor | x=2.5 \rangle \;\; = \;\; 2 | x = 2.5 \rangle
and its effect on some wave function psi(x) is
\lfloor \hat{x} \rfloor \psi(x) \;\; = \;\; \lfloor x \rfloor \psi(x)
Any wave function in position space entirely contained between x=n and x=n+1 for some integer n is an eigenfunction of this operator (note that there are infinitely many such eigenfunctions for any n; the spectrum of the operator is infinitely degenerate). A measurement of this operator has the following effect on the wave function
\psi(x) \to \left\{ \begin{array}{lr}<br />
\psi(x) & : x \in [n,n+1)\\<br />
0 & : otherwise<br />
\end{array} \right.
on obtaining eigenvalue n.
This is a crude representation of an approximate position measurement, which specifies the approximate position in discrete increments of size 1. This observable is not too different from x, but obviously it has limited precision. It is more or less the observable that is measured when you follow the plate-insertion measurement procedure I mentioned a couple posts above.
Now, if I measure this observable I know that my wave function is contained within some position interval of size one, and accordingly there is some minimum dispersion in momentum space of the wave function after the measurement. This dispersion increases if I make the plates closer together. Of course, if I do this my measurement corresponds to a slightly different operator, perhaps floor(Ax)/A, which represents a measurement when the plates are separated by a distance 1/A. By increasing A I make an increasingly precise position measurement, and correspondingly the minimum momentum uncertainty after a measurement increases. In the limit where A goes to infinity and the plates have zero spacing, floor(Ax)/A goes to x and I am conducting infinitely precise position measurements, which correspond to measuring the operator x, and which induce infinite uncertainty in momentum.
In consideration of the above you might consider it an abuse of language how much I have talked about "a fairly precise measurement of position x" or the like. Really, perhaps, I ought to say, "A measurement of the observable floor(Ax)/A where A is fairly large but finite."
mr. vodka said:
Does this not raise doubt on the issue whether your physical conception strikes the heart of what the math tells us about a measurement?
I'm not sure what you mean exactly by this, and maybe this is your main point. Can you expand on it and say whether or not what I wrote above touches on it at all?
mr. vodka said:
In regard to your reply on (2):
" I disagree with your statement that QM "only says something about the result of a measurement." [...] and we can watch the time evolution of this overall system to see what is physically happening during the measurement."
Well I think this might be semantics: you seem to be calling the wave function physical, as in really there in some sense. This might very well be, but I don't see this as an appendix of the mathematical formalism: all the math seems to be telling me is the probability on some outcome when performing a measurement (even if another measurement apparatus is part of the investigated system, as in your subsequent example).
All right; I was just trying to make the point, which maybe was unnecessary, that a measurement is not some extra-physical process that proceeds by different rules than normal physical processes.
mr. vodka said:
If I choose to see the wave-function (when not measured) to be a physical representation or a statement of ignorance is interpretation.
To an extent, yes. But you might note that interpretations in which the wave function is an expression of ignorance of so-called "local hidden variables" are in contradiction with the QM formalism, and experimentally ruled out. ("Non-local hidden variables" are allowed and realized in the Bohmian interpretation.)
mr. vodka said:
The rest of your post argues against the piece of physical reasoning I typed out in my last post, and of course I welcome you to do that (not sarcastic in any way), but I wonder if you realized that the intent of my post was to indeed argue that the use of such physical reasoning is easily inconsistent?
Right; I tried to acknowledge this in the very last part of my post and to provide a strategy for reasoning physically while being less likely to stumble into error. But when you are not confident what the equations imply, the only way to be sure is to write out the equations.
As an aside, I'm not sure if maybe you are using "physical reasoning" to mean something like "classical reasoning." If so, I haven't been interpreting it this way.
mr. vodka said:
Well, I can already hear you say "but no, correct physical reasoning is allowed, but the continuity you used is simply not correct physical reasoning" and I again would agree with this, or at least that nobody has any reason to believe that it would be correct, because it is indeed not a part of the math, but my point: you did use it in your previous post: in your previously quoted example where you measure the position of a particle, you're working with the physical idea of a particle moving continuously, having a well-defined position and momentum, aspects you are now telling me I cannot use. So did I misunderstand your earlier use of it or what is the deal?
Rereading my initial examples, particularly the charge-plate-insertion example, I realize I have a confession to make. To determine the origin of the change in the electron's energy in the charge plate insertion position measurement technique, I first reasoned internally about wave functions and then translated my thoughts into classical language. I think there is a tendency in QM to do this often, maybe because relatively few effects are purely quantum mechanical; many have analogs in classical mechanics, and discussing them in classical language is easier and keeps us from obscuring this connection. But we are talking about the incompatibility of observables, which is purely quantum mechanical, so use of classical language is perhaps inexcusable. Here, then is a quantum mechanical discussion of plate insertion.
Say we have an electron in some energy eigenstate of a potential well. We are going to insert a number of plates that the electron cannot penetrate, dividing the well into a number of small regions. A plate can be modeled as a very narrow region in which the potential spikes to a very high value, making it impenetrable to the electron. Let's imagine inserting the plates somewhat gradually, so that the potential spikes grow continuously from zero to their large (but finite) size. Imagine the shape of a potential spike as some continuous function like a Gaussian.
What happens in the region of a plate as we insert it? Initially the potential is flat at zero. Suppose we are considering a small enough region so that the wave function in this region is approximately constant. A bump in the potential appears and starts growing. Now, a classical particle located on the left slope of the bump would feel a force to the left (F = -dV/dx). Quantum mechanically, the wave function in the region of the left slope develops a momentum component to the left. Inversely, the wave function in the region of the right slope develops a momentum component to the right. Said another way, the probability current, which was zero in the stationary state, starts to point away from the growing bump in the potential. As a result, the magnitude of the wave function begins to decrease at the spike. As the spike grows larger these effects only get stronger, and the wave function falls to zero in the region of the spike. However, we clearly saw that as the spike was growing, it change the momentum components of the wave function.
Another way to reason about this process is: Suppose we insert the plates fairly quickly, so that overall wave function does not change much during the insertion process, except in the region of each plate. At each plate, the wave function must fall to zero, since the electron is being excluded from these regions. But since the overall wave function was largely unaffected, near the border of a plate the wave function still has its original nonzero value. Accordingly the wave function varies quite rapidly in space as you move away from the edge of a plate. But rapid spatial variation of the wave function corresponds to large kinetic energy. So the wave function has acquired significant high-momentum and high-energy components as a result of the plates driving the wave function to zero in certain regions.
Probably the way to insert the plates that creates the minimum disturbance to the particle's momentum and energy is to insert the plates very very slowly ("adiabatically"). If we do this, the "adiabatic theorem" tells us that if we start out in, say, the ground state of the square well, we will end up, after the very slow modification of the potential by plate insertion, in the ground state of the new potential. So the new wave function will have the particle in the ground state of each of the new little infinite square wells formed by the plates. Clearly this state has higher energy and greater momentum dispersion, and we can see where the energy and momentum come from by considering the processes described two paragraphs above, which occurred slowly over the process of insertion.