How does the Schrödinger equation lead to the QM theory of measurement?

[LeonhardEuler]

Hi everyone, I've recently been bugged by a question I can't seem to find a reasonable answer to. It's about an apparent contradiction between how the wave equation is supposed to evolve in time according to the Schrödinger equation and the measurement formalism in QM.

Suppose I have any physical system, for the sake of concreteness let's say it's a single slit setup with photons being fired at a single slit one at a time and hitting a screen that records where they hit. The usual way I hear this described is that the photons each have some wavefunction that describes the probability distribution of being at a particular position, and the screen measures the position from the probability distribution of the wavefunction. The same exact experiment will lead to different results for different photons because its wavefunction only gives probabilities.

On the other hand, the wave function is supposed to evolve in time according to the Schrödinger equation. Given an initial condition and valid boundary conditions, there is a unique solution for the wavefunction as a function of time. Here is the apparent contradiction. At the end of the slit experiment the wavefunction is different in different trials with the same exact initial conditions and boundary conditions. I understand that the screen interacts with the photon, but in principle, we could include the screen in the system we describe and put into the S. Eq. If there is a human watching, we could include the atoms in his body as well. I see no way around the fact that, on the one hand, the S.Eq gives a unique solution for the wavefunction at the end, while the experiments give many possibilities.

Does anyone see a resolution to this? I'm sure there must be a simple explanation, but I just can't find it no matter how hard I try.

 

[Count Iblis]

If you describe the system using a wavefunction that includes the observer, then such a wavefunction is of the form:

|psi> = |O_1>|phi_1> + |O_2>|psi_2> + ...

where the |O_i> are normalized and orthogonal states describing the observer making a particular measurement i, and the |psi_i> are un-normalized states that describe the rest of the universe.

The probability that the observer will find itself having made observation i is then given by

P_i = <psi_i|psi_i>,

i.e. the squared norm of the sector of the wavefunction in which observer |O_i> finds itself in. You can conveniently formulate this as the expectation value of the projection operator |O_i><O_i| :

<psi|O_i><O_i|psi> = <psi_i|psi_i>

 

[LeonhardEuler]

I see what you are saying that the wavefunctions give the probability of the observer making a certain observation, but the problem still is: After the observation has been made the wavefunction of the universe is different for different outcomes. Yet the S.Eq gives only one possibility for the wavefunction at the end of the experiment.

The way I usually hear this described, the system is in a superposition of eigenstates until a measurement is made, at which point it chooses one randomly according to the probability dictated by the wavefunction. I guess the question really boils down to this: Can the measurement process itself be described by the S.Eq? It seems not, since the S.Eq gives a definite answer for the wavefunction in the future, while measurements produce an unpredictable wavefunction as a result.

But it seems like many processes which are usually described by the S.Eq are types of "measurement" as far as the system is concerned, like the interaction of a photon with an atom.

As I write this, I'm beginning to see a way out of the problem, possibly. Maybe the wavefunction is sort of a relative thing for different observers. For the observer inside the system, the wavefunction suddenly changes when he observes the screen detect the photon. For an observer who sees the experiment being set up, and then leaves, the position is still in a superposition of eigenstates, because he only knows the probability of the photon being in certain locations. Then, when he walks into the room, the wavefunciton suddenly changes for him, and he needs to update his initial conditions int the S.Eq to predict the future. It sounds really weird, but at least there seems to be no logical contradiction in this.

 

[yossell]

It sounds as though you're describing a version of the measurement problem. The problem here is that, on versions of QM that include collapse, there are two different dynamical principles: the Schrödinger Equation for when measurement isn't taking place; the collapse postulate for when measurement is taking place - and they give different results. This leads to the worry that measuring devices can't be understood as objects that obey the Schrödinger equation. This, I think, is the kind of thing that has lead people, such as Bohr, to say that there's some kind of cut between quantum objects and classical objects.

Dropping collapse would seem to solve this problem: everything obeys the Schrödinger equation. One tries to explain the results of the two slit experiment by arguing that, when one factors in the appearance of the measuring device into the Schrödinger equation, the amplitudes evolve differently from how they would evolve without the measuring device: in particular, the probability curve derived from the amplitudes is peaked near the two holes, rather than wave-like. So it seems as though collapse isn't necessary to explain two-slit results after all and we can live with one dynamics.

But the trouble with dropping collapse is that it now becomes hard to link amplitudes with probabilities and, especially if we think that objects ONLY have sharp values when the wave-function is in an eigenstate of the observable, how to explain the appearance of sharp values at all. At this point, 'interpretational' issues appear, with Everettians and Many Worlds Theorists and others trying to explain how, on their interpretation, they can get explain such things after all.

Overall, I think your worry is a good one, but it leads into difficult issues in the Foundations of Physics, and thus I don't think, at the moment, has a nice, sharp and uncontentious answer.

(all this is just my opinion, of course, and I could easily be very wrong indeed...)

 

[Demystifier]

LeonhardEuler, a possible answer to your question is provided by the Bohmian (pilot wave) interpretation. See e.g. the lectures at
http://www.tcm.phy.cam.ac.uk/~mdt26/pilot_waves.html
In short, the wave function never collapses and is completely described by the Schrödinger equation. However, what we observe are not wave functions, but pointlike particles that move deterministically, guided by the wave function. The uncertainty is a consequence of our ignorance of the actual initial positions of the particles.

 

[Fra]

Maybe the wavefunction is sort of a relative thing for different observers.

I personally think this is the best way to see it.

The wavefunction is supposedly a representation of the observers (inside-information) about it's own environment (ie. the universe). Thus, there is no such thing as THE wavefunction of the universe, because there exists no physical birds-view, there are only "frog views".

But this is still a matter of preference, what you find most plausible. There are people who are more realist minded than me that has no problems whatsoever with realist type bird views. In these views, there is no physical observer attached to the birds view, it's a kind o realist view of things, from which you can deterministcally infere the frog views.

I personally do not like that. Instead, you can take the view that all we have are evolving incomplete frog views, and there is no deduction from one view to another, only inductive inferences, and the interaction properties of the parts of the universe are reflected by this "undecidability".

I personally think the best view of the wave function is as a physical representation of the state of the environment, encoded by an observer. Thus the wavefunction almost become part of the identity of the observer.

The best single phrase I like to quote is from Zurek (that said I don't share Zurek's main reasoning but that's a different story):

"What the observers KNOWS is inseparable from the observer IS"

Associate here "what the observer KNOWS" with the wavefunction.

/Fredrik

 

[LeonhardEuler]

I really appreciate all of your insightful posts. It's a lot to think about, but now I see several ways to address my original concern. Thanks a lot.