Delayed choice in the Copenhagen interpretation

[hellfire]

Could someone explain to me how the delayed choice experiment is described within the framework of the Copenhagen interpretation? (Double slit experiment with two slits open but a late decission to measure the path of the particle). I do not understand how the deterministic evolution of the wavefunction passing through both slits, fits with the fact that no interference is observed after the decision to measure the path. Can we say that the wavefunction is $\psi = \psi_1 + \psi_2$ (subindices for the two slits) and obeys the Schrödinger equation all the time until the measurement is done? In that case, how is it that the measurement can be only explained if $\psi = \psi_1$? I am sorry to bring up this very often discussed topic, but I found only heuristic explanations to that and I would like to see something more formal.

 

[Demystifier]

I would like to emphasize that the so-called which-way measurements are actually weak measurements, which, in a sense, are not true measurements. To make it more clear, in these experiments there is no detector at the slit, so one does not really determine the slit through which the particle passes. It is possible in principle that the actual (but not directly measured) path that particle took differs from that obtained by the which-way experiment. Indeed, this is exactly what occurs in the Bohmian interpretation. For more details see the paper of Aharonov and Vaidman
http://lanl.arxiv.org/abs/quant-ph/9511005 [published in the book "Bohmian Mechanics and Quantum Theory: An Appraisal," Kluwer]

 

[hellfire]

What does weak measurement mean? I was searching in the web and found some information but I do not understand the relation to this topic. For example, in Wheeler's astronomical delayed choice experiment, isn't the measurement a usual QM measurement?

 

[Demystifier]

What does weak measurement mean? I was searching in the web and found some information but I do not understand the relation to this topic. For example, in Wheeler's astronomical delayed choice experiment, isn't the measurement a usual QM measurement?

Why don't you download the paper above? Everything is explained there.

 

[cesiumfrog]

(Demystifier, that paper seems off topic from this thread, ie. focussed on the Bohmian interpretation.)

Hellfire, the Copenhagen interpretation is basically that experiments can predictively be modeled as evolving according to SE until "measurement", at which point the quantum system "collapses" into a measurement eigenstate. So Schroedinger's cat stays super-positioned for an hour, then instantly collapses into the state of having already been dead for an hour (or not) the moment that the box is opened. (Wheeler's DC exp. differs only in that there also exists an orthogonal measurement basis option..)

In Kim's DCQE exp., the choice of measurement for one particle decides the basis into which the wavefunction of *two particles* collapses, and the measurement of the second particle turns out consistently (eg. contributing to an interference pattern if, and only if, the measurement of the first particle was orthogonal to determining which paths both took). (It gets a bit more mathematical if you switch the chronology of measurements.)

 

[hellfire]

Hellfire, the Copenhagen interpretation is basically that experiments can predictively be modeled as evolving according to SE until "measurement", at which point the quantum system "collapses" into a measurement eigenstate. So Schroedinger's cat stays super-positioned for an hour, then instantly collapses into the state of having already been dead for an hour (or not) the moment that the box is opened. (Wheeler's DC exp. differs only in that there also exists an orthogonal measurement basis option..)

I still have a trouble to understand how the system "knows" that it has to collapse the state $\psi = \psi_1$, and how does this fit with a time evolution that was the evolution of $\psi = \psi_1 + \psi_2$ according to the Schrödinger equation during all the propagation time (until the end when the measurement is done and $\psi = \psi_1$ is selected). The "two-state formalism" that is mentioned in the reference Demystifier provided might explain things, but I would like to understand first the answer to this question in view of the usual formalism and the Copenhagen interpretation.

 

[Demystifier]

(Demystifier, that paper seems off topic from this thread, ie. focussed on the Bohmian interpretation.)

That is true, but this paper is written by authors who are not actually adherents of the Bohmian interpretation and contains a lot of useful explanations that do not depend on the interpretation, including further references on weak measurements (which, I believe, are essential to properly understand the so-called "which-way" experiments).

 

[cesiumfrog]

I still have a trouble to understand how the system "knows" that it has to collapse [to] the state $\psi = \psi_1$, and how does this fit with [..] evolution [..] according to the Schrödinger equation[?]

Firstly, in my understanding (which admittedly may conflate Copenhagen with Feynman's "shut up and calculate"), Copenhagen does not claim to "understand how the system knows" anything. Rather, Copenhagen proscribes a systematic procedure for monkeys to correctly predict the results of certain experiments, no more or less. (It can offer nothing deeper conceptually, because of the inherent inconsistency in treating measurement as distinguished despite that the macroscopic apparatus is itself just a system of many quantum particles.)

Hellfire, am I wrong in assuming you've learned the mechanical details of doing first-year undergrad QM? Yes, prior to the measurement the systems evolve according to SE, but upon each measurement they are assumed to collapse irreversibly (in a process not described by the actual SE) into a probabilistic distribution of the states that the apparatus is able to measure. So if we start in state $\psi$, which say is a fairly equal superposition of states $\psi_1$ and $\psi_2$ (and hence can equivalently be expressed as a perhaps unequal superposition of states $\xi_1=\psi_1+\psi_2$ and $\xi_2=\psi_1-\psi_2$, or can also equivalently be expressed in a spectrum of other bases), the chosen nature of the measurement will determine whether it collapses into $\psi_1$ and $\psi_2$ or into $\xi_1$ and $\xi_2$.

But you could learn that much from Griffith's introductory textbook, so I must not have understood your question properly?

 

[hellfire]

Ok, thanks for your help. I will try to reformulate my question.

Consider the usual double slit experiment. You can have two slits open having a wavefunction $\psi = \psi_1 + \psi_2$. Here the subindices mean the two parts of the wavefunction corresponding to the different paths through both slits. When a position measurement is done with a photographic plate, the system collapses into a eigenstate of position. Since both slits were open, interference between $\psi_1$ and $\psi_2$ occurs. This leads to the usual interference pattern when performed with many particles. You may also have only one single slit open having a wavefunction $\psi = \psi_1$. In that case the position measurement selects an eigenstate of $\psi = \psi_1$ only. This leads to the usual pattern around one single point in the photographic plate.

I have no problem to understand both of these situations: you start with some boundary conditions, either both slits open or only one single slit open, and you are able to solve the Schrödinger equation obtaining either $\psi = \psi_1 + \psi_2$ or $\psi = \psi_1$. Afterwards, you simply apply the collapse postulate at the time of measurement to the first or to the second scenario. Now, consider for example Wheelers astronomical delayed choice. Both slits are open and therefore I would guess that I should solve the Schrödinger equation to obtain $\psi = \psi_1 + \psi_2$. If I do so, then applying the collapse postulate at the time of measurement will lead to the wrong answer, because the system behaves as if $\psi = \psi_1$. Does the shape of the wavefunction depend on my decision how to measure? If yes, how does this fit with a deterministic time evolution of the wavefunction according to the Schrödinger equation? May be I have simply misunderstood some obvious issue, but I hope that the question is clear now and you can comment on it.

 

[cesiumfrog]

Here's a simplified system: connect optical fibres to each slit, and place a photon detector at the other end of each fibre. Now, let the experimenter make a choice (delayed until the light is already traveling in the fibres) whether or not to splice a beam-splitter carefully between the fibre ends and the detectors.

Say the initial wave-function is w=a.w1+b.w2 (where w1 is the wave-function when slit B is blocked). If the photon is allowed to directly reach the detectors (this position measurement, like a which-slit measurement, causes collapse into the basis of w1 or w2), so detection A occurs with probability a^2 etc.

If the beam-splitter is inserted (so that this position measurement, equivalent to measuring the interference or phase difference between the paths, causes collapse into the basis of .7 w1+w2 or .7 w1-w2) then the probability of detection A is .5(a+b)^2, and so forth.

In both cases we've effectively calculated the probabilities by summing the amplitudes of each possible photon-path leading to the detector's position. Of course, when we look at it this way, we actually haven't solved the Schroedinger equation. Perhaps this isn't Copenhagenny enough, so here is an alternative proceedure to apply to Wheeler's experiment:

---

Start with the initial assumed wave-function, which is a function of amplitude across the transverse plane (ie. a double top-hat function, representing the double-slit). Next, solve the SE for the evolution of the wave-function. For propagation a long distance through free space, the SE is equivalent to a Fourier transform (giving a sinc function). (For optional propagation through a thin lens, the SE reduces to ... a Fourier untransform.) So we again know the amplitude at each point where we might choose to make a measurement. (Note this detailed procedure is important to confirm the predictions; I think it's incorrect application of the shorthand notation which has given you wrong answers.)

But this hasn't really discussed collapse. Plainly, I think collapse is irrelevant unless we make a second measurement. For example, if I wanted to add additional aperture screens, I would collapse the wave-function at those axial-positions by zeroing the wave-function at transverse-positions blocked by the aperture (then renormalising, all assuming we're only interested in photons that do get through). So understanding collapse is useful if we have a sequence of inter-spaced lenses and apertures in front of our detector (or if we have a multiple entangled particles to measure), but just doesn't enter into Wheeler's experiment. Collapse tells the shape of the wave-function after a particular measurement result, but is not used directly to determine the probability of that result occurring.

 

[Demystifier]

I am currently reading the popular-science book "Fabric of the Cosmos" by Brian Greene. One of the best popular science books I ever seen. In particular, I have never seen a better popular explanation of the foundational aspects of quantum mechanics. This includes an excellent explanation of the delayed choice experiments.