Geiger counters and measurement

[jeeves]

The general principle at work is decoherence. That is not something we're going to be able to explain in detail in a single thread. You will need to spend some time learning about it (and, as I said, learning a solid understanding of basic QM first, if you don't already have that).

I have read more, and am still a bit confused.

Consider the Renninger negative-result experiment.

I place a single unstable atom, say Carbon-14, at the center of a sphere consisting of two hemispherical detectors.

First, remove one of the hemispheres and start the experiment. Suppose that the remaining detector does not signal a detection after a long period of time, long enough that for all practical purposes, we are certain the atom has decayed. Then the non-detection of the decay product on the remaining detector is logically equivalent to knowing that the particle escaped out the other hemisphere (a detection on the removed detector), and hence locates the trajectory of the particle in a subset of the original possible trajectories. In Copenhagen language, we have a partial collapse of the wave function.

Next, restore the second detector to complete the sphere, and begin the experiment anew. Suppose after some period of time neither detector registers decay. If I understand correctly, you claimed earlier that this non-detection does not qualify as a measurement, and does not collapse the wave function.

How can it be that non-detection partially collapses the wave function in the first scenario but not the second?

 

[PeroK]

How can it be that non-detection partially collapses the wave function in the first scenario but not the second?

Because they are different experiments. The implications of non detection depend on how closely you have been monitoring something.

That applies classically as well as quantum mechanically.

 

[jeeves]

Because they are different experiments. The implications of non detection depend on how closely you have been monitoring something.

What precisely is the mechanism that causes collapse in the first experiment but not the second? The detectors themselves operate the same way in both setups.

 

[PeroK]

What precisely is the mechanism that causes collapse in the first experiment but not the second? The detectors themselves operate the same way in both setups.

There's no mechanism. Wave function collapse isn't a physical thing. It's a consequence of knowledge about a system.

 

[PeroK]

If your friend is in his house and you are watching the front door, then after a certain time there is a probability he has left by the back door. Whereas, if you are monitoring all possible exits you know he's still in the house.

There's no mechanism there, only inference from knowledge about the system and its possible evolution. In this case probabilistic calculations based on your friend's likely movements.

 

[PeterDonis]

What precisely is the mechanism that causes collapse in the first experiment but not the second?

What the mechanism is, or even whether there is any mechanism at all involved, depends on which interpretation of QM you adopt. Discussion of QM interpretations is out of scope for this forum; it belongs in the interpretations subforum. All the basic math of QM can do is make predictions about the probabilities of various possible experimental results; it cannot tell you "what really happens".

 

[Nugatory]

Namely, the cat is put in the box, some time elapses, and then the box is opened. Hence, there is a single observation taken at a fixed point in time.

Opening the box is pretty much irrelevant to the quantum mechanical evolution of the system. The macroscopic elements of the quantum system in the box (detector and trigger mechanism, vial of poison, living breathing warm wiggly cat, countless air molecules drifting around inside the box, ...) have enough degrees of freedom that the quantum system consisting of the unstable nucleus entangled with all this macroscopic stuff almost immediately decoheres into the mixed "either we have a dead cat in the box or a live cat in the box" state. This happens long before and whether or not we even open the box and look to see which we have. Generally any thermodynamically irreversible interaction leading to decoherence counts as a "observation", and a system as complex as Schrodinger's cat in a box can be considered to be continuously observing itself.

It is worth noting that Schrodinger did not accept the idea that the cat was in a superposition of alive and dead until opening the box collapsed the wave function. The point of his thought experiment was that something had to be wrong with the then-current understanding of QM because it suggested that opening the box mattered.

 

[jeeves]

Opening the box is pretty much irrelevant to the quantum mechanical evolution of the system. The macroscopic elements of the quantum system in the box (detector and trigger mechanism, vial of poison, living breathing warm wiggly cat, countless air molecules drifting around inside the box, ...) have enough degrees of freedom that the quantum system consisting of the unstable nucleus entangled with all this macroscopic stuff almost immediately decoheres into the mixed "either we have a dead cat in the box or a live cat in the box" state. This happens long before and whether or not we even open the box and look to see which we have. Generally any thermodynamically irreversible interaction leading to decoherence counts as a "observation", and a system as complex as Schrodinger's cat in a box can be considered to be continuously observing itself.

Thanks. I guess I am confusing myself by thinking of "measurement" as some magic wand. It is as you and Peter said: the system rapidly decoheres into a mixed state which is a combination of "cat dies and particle has decayed" or "cat is alive and particle hasn't decayed." And if I observe the system at time ##t##, the probability that the cat is alive can be computed from the coefficient ##A(t)## of the "alive and no decay" part of the mixed state (using Peter's notation).

I think I am fine with this. Certainly this gives us a way to compute the empirical frequency of living cats if we perform the experiment repeatedly.

What I still don't understand is how decoherence explains what happens when there are multiple observations. For example, suppose I want to answer the question: "If I observe that the cat is alive at time ##t=1##, what is the probability the cat is still alive at time ##t=2##?"

To answer this question, it suffices to know the state of the cat at ##t=1##, in particular ##A(t)## and ##B(t)##. Then I just run the Schrodinger evolution and get a mixed state at ##t=2## that gives me the answer. So my question becomes: if I observe the cat is alive at ##t=1##, what is the state of the cat after the observation?

My inclination is to say that if I see the cat is alive, I'm observing a pure state, so I'm in the state with ##A(t)=1##.

But this is obviously wrong because then we could get a quantum Zeno effect by observing the cat many times in succession.

So what is the state after observation and why? Can decoherence help explain this too?

 

[PeterDonis]

what is the state after observation and why?

The "observation" you describe--"observing" that the cat is alive--does not tell you anything useful. What you would need to do is "observe" the radioactive atom whose decay will trigger the process that kills the cat, and see "how close it is to decaying". But, as others have already pointed out in this thread, there is no such thing; the probability of the atom decaying per unit time is constant (at least if we ignore the small corrections to the exponential approximation). So there is no way to "observe" anything that can give you more information than the exponential decay law about what will happen to the cat in the future.

If you could somehow "observe" the atom continuously and verify that it continued to be in the undecayed state, you would be running a quantum Zeno effect experiment on the atom; but that is not the situation you have been describing. Certainly "observing" that the cat is alive is not such an experiment.

Can decoherence help explain this too?

Decoherence can explain why there is no interference between the "atom undecayed, cat alive" and "atom decayed, cat dead" states. But it cannot, by itself, explain the "collapse" of the system to one state or the other as a result of some "observation". Basic QM does not even attempt to explain that; it just tells you what to do mathematically when you know a particular result has been recorded for some observation. It does not tell you "what really happens" or "what the actual state of the system is" or anything like that. Particular QM interpretations do make such claims, but, as has already been noted, discussion of interpretations is off limits for this forum.

 

[jeeves]

The "observation" you describe--"observing" that the cat is alive--does not tell you anything useful. What you would need to do is "observe" the radioactive atom whose decay will trigger the process that kills the cat, and see "how close it is to decaying". But, as others have already pointed out in this thread, there is no such thing; the probability of the atom decaying per unit time is constant (at least if we ignore the small corrections to the exponential approximation). So there is no way to "observe" anything that can give you more information than the exponential decay law about what will happen to the cat in the future.

If you could somehow "observe" the atom continuously and verify that it continued to be in the undecayed state, you would be running a quantum Zeno effect experiment on the atom; but that is not the situation you have been describing. Certainly "observing" that the cat is alive is not such an experiment.

I agree that if we are in the regime where the decay is almost exactly exponential, the distribution is (almost) memoryless and there is not much to be said.

So let's not assume that. Suppose we are in the small time regime where the decay dramatically deviates from exponential.

In this regime, does observing the cat tell me something useful? That is, is the conditional distribution for survival conditioned on the cat surviving until ##t=1## the same as the unconditional distribution for survival, like in the memoryless case, or is it different? I would suppose it is different, because after all, we do seem to learn something about the atom after observing the living cat (it has not decayed), and because the distribution is not memoryless this affects the (conditional) frequency of decay in the future.

So in the non-exponential regime, what is the state of the cat after I observe that it is alive at ##t=1##, in terms of the ##A(t)## and ##B(t)## notation from earlier?

 

[PeroK]

I agree that if we are in the regime where the decay is almost exactly exponential ...

Let me ask you a question. I would like you to understand the nature of classical probabilities before tackling the complex probability amplitudes of QM. Which share some of the properties of classical probabilities.

We have a pack of cards and you draw a card (face down). What is the probability that it is some particular card like the six of clubs?

Now, we start looking at the other cards in the pack one by one. As each card is found not to be the six of clubs, does the probability that your card is the six of clubs change?

Do you know how to analyse a problem like that?

 

[jeeves]

Yes, I know how to analyze such problems.

 

[PeroK]

Yes, I am aware of how to analyze such problems.

Okay. Re QM. If, instead, we have 52 slits in a barrier and we carry out an experiment, then we get a 52-slit interference pattern. But, if we monitor one of the slits and run the experiment, then we get either a detection event at that slit or a 51-slit interference pattern. And, if we monitor ##n## slits, then we get either a detection event at one of the slits or a 52-##n## slit interference pattern.

The calculation in QM is different in that each slit corresponds not to a probability but a probability amplitude. It doesn't work to say that the probability that the particle went through slit 1 is 1/52 (or 1/51 or whatever). Then there would be no intereference, but simply the sum of 52 single-slit patterns Instead, there is a complex probability amplitude associated with the path through each slit. We then combine these amplitudes (which, being complex, can cancel each other out) and we get probabilitistic quantum interference (of a particle with itself).

Okay so far?

 

[jeeves]

Okay so far?

Yes, this is fine.

 

[PeroK]

Yes, this is fine.

The same logic applies to time evolution of a state. For a radioactive decay, there is a probability amplitude for the state of decay after some time ##t## and a probablity amplitude for the state of non-decay after some time ##t##. Usually the state of non-decay remains physically identical - in the sense that decay over the next time ##t## remains equally probable. In this case a definite measurement that results in "no decay" does not physically change the system.

If instead you postulate that the probability amplitude for decay changes over time (perhaps it gets less and less likely over time). Then a definite measurement of non-decay is meaningful, as you know that the state has evolved into a more stable state.

Does that make sense?

 

[jeeves]

I believe that makes sense. Let me see if I understand.

Suppose again we are in the short-time regime where the decay is not memoryless, and I observe after some time has passed that the cat has not died (atom has not decayed). Then do you agree that I have learned something nontrivial about the system? If so, how would you write the state in the alive/undecayed and dead/decayed basis arising from decoherence?

 

[PeroK]

I believe that makes sense. Let me see if I understand.

Suppose again we are in the short-time regime where the decay is not memoryless, and I observe after some time has passed that the cat has not died (atom has not decayed). Then do you agree that I have learned something nontrivial about the system?

Yes.

If so, how would you write the state in the alive/undecayed and dead/decayed basis arising from decoherence?

I guess we start in a known state ##\ket {\psi(0)}##. The system would then evolve according to something like:
$$\ket {\Psi(t)} = a(t) \ket {\psi_d} + b(t) \ket {\psi(t)}$$Where we now, hypothetically, have a superposition into decayed and undecayed states, where the undecayed state itself is a function of time. I suspect at some fundamental level this may break conservation laws, but let's not worry about that.

The rest of the calculation re entanglement would be the same, except the state "detector = no" will be entangled with the time dependent undecayed state ##\psi(t)##.

 

[jeeves]

I guess we start in a known state ##\ket {\psi(0)}##. The system would then evolve according to something like:
$$\ket {\Psi(t)} = a(t) \ket {\psi_d} + b(t) \ket {\psi(t)}$$Where we now, hypothetically, have a superposition into decayed and undecayed states, where the undecayed state itself is a function of time. I suspect at some fundamental level this may break conservation laws, but let's not worry about that.

This seems reasonable, but I'm still a bit confused. Suppose I try to apply this formalism to a classic quantum Zeno experiment, where the atom is repeatedly measured at small time intervals in a way that reduces or eliminates the possibility of decay. How does the time-dependent undecayed state formalism predict the Zeno effect? (Recall such an effect is possible only because we are in the non-exponential decay regime.)

 

[PeroK]

This seems reasonable, but I'm still a bit confused. Suppose I try to apply this formalism to a classic quantum Zeno experiment, where the atom is repeatedly measured at small time intervals in a way that reduces or eliminates the possibility of decay. How does the time-dependent undecayed state formalism predict the Zeno effect? (Recall such an effect is possible only because we are in the non-exponential decay regime.)

I didn't think we were discussing the Quantum Zeno effect! I don't know anything specific about that.

 

[PeroK]

PS I think @DrChinese already gave a good summary of this:

Many things (other than radioactive decay) have a probability of occurring per unit of time. Example: when an electron drops to a lower orbital and emits a photon. I would not normally call a detection "non-event" to be equivalent to a "continuous series of observations" of the particle in question. (There might be a few cases where it is difficult to suitably define a "non-event" or a "continuous series of observations".)

 

[Morbert]

I think there is an important distinction between using QM to model if the particle has decayed in some time interval vs modelling the moment the particle decays. Let ##\Pi_\mathrm{d}, \Pi_\mathrm{nd}, \Pi_\mathrm{c}, \Pi_\mathrm{nc}## be the projectors for "decayed", "not decayed", "clicked" and "not clicked" respectively.

Seeing that the detector is in the state "not clicked" at some arbitrary time ##t## constitutes a measurement. It resolves the question of whether or not the particle has decayed in the time interval ##\left[0,t\right)## (assuming the experiment was prepared at time 0). The probability is $$\mathbf{Tr}\left[\rho \Pi_\mathrm{nd}(t)\right]$$ and the alternative (particle has decayed) trivially decoheres $$\mathbf{Tr}\left[\Pi_\mathrm{d}(t)\rho \Pi_\mathrm{nd}(t)\right] = 0$$ We can also perform a more general measurement: we can measure whether or not the the particle decays in some time interval ##\left[t,t+\Delta t\right)## by checking the detector at ##t## and at ##t+\Delta t##. The probability is $$\mathrm{Tr}\left[\Pi_\mathrm{d}(t+\Delta t)\Pi_\mathrm{nc}(t)\rho\Pi_\mathrm{nc}(t)\Pi_\mathrm{d}(t+\Delta t)\right]$$ Like before, all alternatives decohere $$\mathrm{Tr}\left[\Pi_\mathrm{nd}(t+\Delta t)\Pi_\mathrm{nc}(t)\rho\Pi_\mathrm{nc}(t)\Pi_\mathrm{d}(t+\Delta t)\right] = \mathrm{Tr}\left[\Pi_\mathrm{d}(t+\Delta t)\Pi_\mathrm{c}(t)\rho\Pi_\mathrm{nc}(t)\Pi_\mathrm{d}(t+\Delta t)\right] = 0$$ A "continuous measurement" would resolve whether or not the particle decayed precisely at time ##t##. I.e. Take the above and let ##\Delta t## go to 0. I don't think this is normaliseable so I don't think QM can model such a continuous measurement.

 

[PeterDonis]

Suppose again we are in the short-time regime where the decay is not memoryless, and I observe after some time has passed that the cat has not died (atom has not decayed). Then do you agree that I have learned something nontrivial about the system?

I don't. See below.

Where we now, hypothetically, have a superposition into decayed and undecayed states, where the undecayed state itself is a function of time.

Even if you do this, you still haven't changed the fundamental fact about the "observation" the OP is describing: it's an observation of the cat, not an observation of the atom. And all you're observing about the cat is that it's alive. You can deduce from this that the atom has not decayed, but, if you are including multiple "undecayed states" in your model for the atom, observing that the cat is alive does not tell you which "undecayed state" the atom is in, and therefore does not tell you anything useful about the probability of decay. The only way to know anything useful about the atom's state is either to observe that the cat died--which tells you the atom decayed--or to prepare the atom in a known "undecayed" state (which amounts to observing the atom directly and obtaining the result that the atom is in that particular state). A "quantum Zeno effect" experiment would amount to doing the latter; but just observing that the cat is alive does not.

 

[vanhees71]

The entire question boils down to the question, whether the Geiger counter affects the decaying particle sufficiently before it decays. In more physical terms that means the question is, whether the presence of the Geiger counter leads to interactions with the decaying nucleus in such a way that it affects the dynamics leading to the decay of the nucleus. This is, FAPP, not the case, because the nuclear forces holding the nucleus together are very strong against the long-range interactions with the material of the Geiger counter (i.e., electromagnetic interactions and, though totally academic, in principle gravitation). So the presence of the Geiger counter does not affect the dynamics of the nucleus before the decay and thus also not its mean lifetime. What interacts with the Geiger counter is the decay product (He nuclei, electrons, or ##\gamma##'s for ##\alpha##, ##\beta##, or ##\gamma## decay), and there is indeed always some propability that the Geiger counter does not register this decay product, but no matter whether it does or not, it doesn't affect the lifetime of the unstable nucleus.

This changes for other systems, where the sheer existence of a measurement device interacts with the observed unstable system in such a way that it affects the dynamics of this system and thus may change the transition probability/aka its lifetime tremendously. An example is to put an atom in some cavity such that a photon of a transition does not "fit" with its frequency in the cavity. Then this transition is suppressed and the lifetime of the corresponding excited state can be very much longer than for an atom in free space.

All this has nothing to do with "collapse" but just with interaction between measurement devices/or any other stuff around an observed quantum object. Imho, "Collapse" should only be discussed in the interpretation subforum, because whether or not you assume a collapse, depends on your personal interpretation of quantum theory. I strongly plead against introducing the collapse at all since it's a kind of Pandora's Box in the context of relativity and causality.

 

[jeeves]

The entire question boils down to the question, whether the Geiger counter affects the decaying particle sufficiently before it decays. In more physical terms that means the question is, whether the presence of the Geiger counter leads to interactions with the decaying nucleus in such a way that it affects the dynamics leading to the decay of the nucleus.

Thank you, vanhees. Based on your answer, I have the following understanding. Is it correct?

Suppose we have a Schrodinger's cat setup in a transparent box. I use ##A(t)## as the coefficient of the "undecayed" atom state, and ##B(t)## as the coefficient of the "decayed" atom state.

At ##t=0##, we have ##A(0) = 1##.

At ##t=1##, I look at the box at see the cat is alive. My observation of the cat (or the cat itself, or me staring directly at the location of the atom, etc.) has no effect on the evolution of the decaying atom. So the atom is in the same superposition as it would've been had I not looked. That is, we have coefficients ##A(1)## and ##B(1)##.

I keep observing the box. At ##t=2## I see the cat is still alive. The coefficients are now ##A(2)## and ##B(2)##. It is still the case that nothing has interacted with the atom in a meaningful way.

I keep observing the box. At ##t=3##, the atom decays, the poison is released, and I observe the cat die. We now have ##A(3) = 0## and ##B(3) = 1##.

Is this correct so far?

Regarding how I learn of the state of the atom: Should I think of this as the cat being entangled with the decay product, which is in turn entangled with the atom? So, after the particle decays, the cat becomes highly entangled with the atom and knowledge of the cat's status is equivalent to knowledge of the atom's status? And crucially (to avoid quantum Zeno), this entanglement does not appear prior to the decay?

 

[vanhees71]

The cat's state is entangled with the nucleus + decay products after the decay. Without looking at a time ##t## all you know is the probability for the cat to be dead or alive. It's given by the radioactive decay law (to very good accuracy). The survival probability of the nucleus is ##P(t)=\exp(-t/\tau)##, where ##\tau## is the mean lifetime of the mother nucleus.

I'm a follower of the minimal statistical interpretation since that's all needed to use quantum theory as a physical theory. For me everything beyond this (including the collapse hypothesis) is metaphysics (for some people it seems to take the status of a kind of religion) and thus beyond the aim of physics as a natural science, which is to find mathematical descriptions of phenomena to be observable (meaning measurable and quantifiable).

A probability has an epistemic meaning. It is a measure for the expectation of the cat's state some time ##t## after a given preparation of the system at time ##t=0##. The probabilities are described by quantum theory, i.e., by the time-evolution equation for the statistical operator or (if you deal with pure states) the Schrödinger equation for the corresponding state ket. If you take notice of the state of the cat nothing specific need to happen with the cat. All you do is to update your (probabilistic) description of the situation, gaining new information about the state of the ket being either dead or alive when looking. There is no mysterious collapse.

Note that this posting (and imho the entire thread) does not belong to the quantum mechanics forum but to the interpretation subforum ;-)).

 

[WernerQH]

Suppose we have a Schrodinger's cat setup in a transparent box. I use ##A(t)## as the coefficient of the "undecayed" atom state, and ##B(t)## as the coefficient of the "decayed" atom state.

You are oversimplifying. You need more than two coefficients to describe the situation using the Schrödinger equation. In the description of the decay you need to include also the decay products (typically an escaping electron and anti-neutrino). Whether or not a Geiger counter is present, in the derivation of Fermi's Golden Rule we take the squared modulus of the coefficients of the final states (Born rule) and add them up. This would lead to a decay probability increasing with time like ## t^2 ##, were it not for the density of final states. As time progresses, the contributing final states decrease, their distribution in energy becoming ever sharper (the width decreasing proportional to ## t^{-1} ##). In this way we arrive at a constant decay rate. And for all we know, radioactive decay happens even without Geiger counters present.

What really happens, is that a radioactive nucleus just sits there for a long time, and suddenly decays. A slowly (over the course of minutes, days, or years) evolving wave function exists only in the minds of theoreticians. You should not think that your two coefficients ## A(t), B(t) ## have some direct physical meaning.

 

[jeeves]

Note that this posting (and imho the entire thread) does not belong to the quantum mechanics forum but to the interpretation subforum ;-)).

I object to this characterization. I do not believe I am asking anything about interpretation. I am simply asking: What is the state of the atom at various times? This is a question with empirically testable consequences. In principle, the atom could be measured, and its state determined. Interpretation has nothing to do with it. (Also, I did not mention collapse in that most recent post.)

What really happens, is that a radioactive nucleus just sits there for a long time, and suddenly decays. A slowly (over the course of minutes, days, or years) evolving wave function exists only in the minds of theoreticians. You should not think that your two coefficients ## A(t), B(t) ## have some direct physical meaning.

For similar reasons, I hold that am not imbuing my coefficients with any physical meaning. I just want to know what the coefficients are at ##t=1,2,3## in the situation I described, and why they are that way. By "why" I mean I am asking for strictly empirical explanations (such as decoherence, lack of interactions, etc.), not anything to do with interpretation.

 

[Morbert]

I do not believe I am asking anything about interpretation. I am simply asking: What is the state of the atom at various times?

The problem is the answer to this question is interpretation-dependent (e.g. Some interpretations ascribe no objective character to the state).

An interpretation-independent question that might still serve your purposes might be something like: "Given preparation of the particle X and sequence of observations Y that occur at times {t1, t2, t3, ...} what are the likelihoods of the possible outcomes"

 

[jeeves]

Sure, we can gloss "What is the state?" as asking about statistics of identical ensembles. And my question "Why is the state that way?" can be glossed as "How does one derive the empirically observed statistics a priori by reasoning about Schrodinger evolution and entanglement?" .

The key point is: We all know what the empirically observed statistics are going to be. I would like to know how to correctly reason about the formalism of quantum mechanics to derive those statistics. (It's easy to reason incorrectly and predict a quantum Zeno effect if, for example, one views watching the cat as a "measurement" of the atom.) I presented some reasoning on the last page and am curious if it is right.

 

[Morbert]

Sure, we can gloss "What is the state?" as asking about statistics of identical ensembles. And my question "Why is the state that way?" can be glossed as "How does one derive the empirically observed statistics a priori by reasoning about Schrodinger evolution and entanglement?" .

The key point is: We all know what the empirically observed statistics are going to be. I would like to know how to correctly reason about the formalism of quantum mechanics to derive those statistics. (It's easy to reason incorrectly and predict a quantum Zeno effect if, for example, one views watching the cat as a "measurement" of the atom.) I presented some reasoning on the last page and am curious if it right.

So here his how I would model a decaying particle ##\phi## and detector ##\psi##:

The particle and detector at some initial time ##t_i## are prepared as $$|\Psi_0\rangle = |\phi_s\rangle|\psi_\text{ready}\rangle$$ with unitary evolution $$U(t)|\Psi_0\rangle = a(t)|\phi_s\rangle|\psi_\text{not clicked}\rangle + b(t)|\phi_d\rangle|\psi_\text{clicked}\rangle$$ where ##s## and ##d## are for "survived" and "decayed" respectively. We can model the behaviour of the detector with a projective decomposition of the identity operator on the detector's state space ##I = E_\text{clicked} + E_\text{not clicked}## . We construct a chain of projectors for whatever temporal resolution we demand to represent the various possible outcomes. A simple example: we can model an outcome like "the detector clicks at time ##T##" as $$E_\text{not clicked}(T-\delta t)\otimes E_\text{clicked}(T+\delta t)$$ which has a probability $$p_a = \mathrm{Tr}\left[E_\text{clicked}(T+\delta t) \otimes E_\text{not clicked}(T-\delta t) |\Psi_0\rangle\langle\Psi_0|E_\text{not clicked}(T-\delta t)\otimes E_\text{clicked}(T+\delta t) \right]$$

 

[PeroK]

(It's easy to reason incorrectly and predict a quantum Zeno effect if, for example, one views watching the cat as a "measurement" of the atom.) I presented some reasoning on the last page and am curious if it is right.

IMO, that's why it's not a good idea to try to relate QM directly to macroscopic behaviour. One thing you must do is deconstruct your macroscopic ideas before you can assume they are directly related to underlying QM phenomena. In other words, an "observation" at the macroscopic level may not map very well to measurements of a QM system, as generally understood. For example:

When you are watching a cat, what is actually happening? If the cat is poisoned, how long does it take to react? How long before you know it's dying rather than yawning or whatever?

These are not silly questions once you've taken the step of assuming that watching a cat is equivalent to a continuous measurement of a microscopic system on whose state its live depends!

In answer to my own question, watching a cat is an enormously fuzzy and imprecise set of measurements compared to the sort of precise measurement required in a QM experiment.

We could compare this with the recent experiments regarding the anomalous dipole moment of the muon. The mean lifetime of a muon is about ##2.2## microseconds. You have no ability in watching a cat to pinpoint changes in its macroscopic state over those timescales.

If you rigged up a system where the muon's decay was linked to the death of a cat by some mechanism, then you simply cannot claim that by "continuously" watching the cat you are continuously aware of whether the muon has decayed or not. The muon will have been created and have decayed in a timescale shorter than the timescales of any macroscopic phenomenon. Even if the cat reacted to poison in ##1## second, that is still 500,000 times longer than the muon's lifetime. And your observations of the cat and its reactions do not pin down the lifetime of the muon in any meaningful way.

 

[PeterDonis]

I am simply asking: What is the state of the atom at various times?

And the best answer that basic QM, with no interpretation involved, can give you is that the QM math is not telling you "the state of the atom" in the sense you appear to be using the term. It's just telling you how to predict probabilities for possible observations. There is a mathematical thingie in the machinery used to predict probabilities that is called "the state of the atom" (more precisely the "wave function" or "state vector" of the atom), but basic QM, without adopting any particular interpretation, does not make any claim whatsoever about that mathematical thingie's physical meaning. All basic QM says is that you can use the mathematical thingie to predict probabilities.

This is a question with empirically testable consequences.

No, it isn't. You can empirically test the predictions for probabilities of possible observations without bringing in any particular QM interpretation, but you cannot empirically test what "the state of the atom" is without bringing in some particular QM interpretation, because different interpretations make different claims about what "the state of the atom" is in any particular situation, even though they all agree on the predictions of probabilities.

 

[PeroK]

PS watching a cat is very different from continually subjecting an atom to a series of ultraviolet pulses to suppress its evolution to an excited state - as described in a quantum Zeno experiment:

https://en.wikipedia.org/wiki/Quantum_Zeno_effect#Experiments_and_discussion

Fuzzy, imprecise macroscopic measurements cannot be used to interrogate microscopic QM systems in the way you are imagining.

 

[jeeves]

And the best answer that basic QM, with no interpretation involved, can give you is that the QM math is not telling you "the state of the atom" in the sense you appear to be using the term. It's just telling you how to predict probabilities for possible observations.

Sure. As I said in post #64, I just want to know how to apply the math appropriately to correctly predict the empirically observed outcomes. We make predictions in QM by constructing a wave function and reading off the desired probabilities. I am asking only how to construct the wave function appropriately. I do not intend to give it any special ontological status, or interpret it metaphysically in any way. When I ask "What is the state of the atom?" I am only asking about the appropriate wave function to assign. I will endeavor to be more precise about this distinction.

PS watching a cat is very different from continually subjecting an atom to a series of ultraviolet pulses to suppress its evolution to an excited state - as described in a quantum Zeno experiment:

https://en.wikipedia.org/wiki/Quantum_Zeno_effect#Experiments_and_discussion

Fuzzy, imprecise macroscopic measurements cannot be used to interrogate microscopic QM systems in the way you are imagining.

I completely agree. I would just like to know how to correctly account for this (for all practical purposes, etc.) when writing down wave functions to predict things.

What you say in post #66 seems consistent with the description sketched in my post #59. (In particular, "You have no ability in watching a cat to pinpoint changes in its macroscopic state over those timescales.") Do you find the description I give there reasonable? (Again, please read "state" in an ontologically minimal way.)

 

[PeterDonis]

I just want to know how to apply the math appropriately to correctly predict the empirically observed outcomes.

You use the known probability of decay per unit time for whatever atom you have in your apparatus, and integrate that over the time since you prepared the atom to get a cumulative probability. In other words, you use the familiar math of radioactive decay that was already known even before QM was discovered (unless you are using a setup that involves QM corrections to the exponential decay law, in which case you have to use the decay probability math that includes those corrections).

You can, of course, write things in terms of a QM wave function with coefficients ##A(t)## and ##B(t)## in front of the "non-decayed" and "decayed" terms, where ##B(t)## is determined by the radioactive decay law and ##A(t)## is determined by normalization (the squared norm of the wave function as a whole must be 1). But this doesn't tell you anything new as far as probabilities go that the radioactive decay law doesn't already tell you. And if you're not picking any particular QM interpretation, you have no reason to even bother writing down the wave function in the first place since you're not assigning it any physical meaning and it doesn't add any ability to predict probabilities that you don't already have.