I Geiger counters and measurement

[gentzen]

Summary: Do Geiger counters cause collapse when they don't click?

Even if your interpretation would claim that they do, this would not lead any problems (like the quantum zeno effect). Radioactive decay is described by the exponential distribution, which is memoryless. This does not contradict radioactive decay being a quantum mechanical process:

In the early 20th century, radioactive materials were known to have characteristic exponential decay rates, or half-lives. At the same time, radiation emissions were known to have certain characteristic energies. By 1928, Gamow in Göttingen had solved the theory of the alpha decay of a nucleus via tunnelling, with mathematical help from Nikolai Kochin...

... In quantum mechanics, however, there is a probability the particle can "tunnel through" the wall of the potential well and escape. Gamow solved a model potential for the nucleus and derived from first principles a relationship between the half-life of the alpha-decay event process and the energy of the emission, which had been previously discovered empirically and was known as the Geiger–Nuttall law.

 

[vanhees71]

The exponential-decay law is an approximation, called the Wigner-Weisskopf approximation. Quantum mechanically it cannot be exact. See, e.g.,

J.J. Sakurai, Modern Quantum Mechanics, extended edition, Addison-Wesley (1994)

 

[jeeves]

Thanks, Peter and vanhees, for the references. I was not able to locate a section on decoherence in Ballentine's book (it does not appear in the index or table of contents). I read Zureck's arxiv paper 2107.03378, since this was the only survey of his that I could find, but it was more of a remembrance than an expository article. I have yet to read the Schlosshauer book.

One reference I found helpful was "The Metaphysics of Decoherence" by Vassallo and Ramano. My understanding now is that decoherence explains why we see mixed states and not superpositions in real life. That is, using the example of that paper, decoherence theory explains why we see that Schrodinger's cat is alive or dead, but we never see a superposition of an alive and dead cat. This is great, however it seems not directly relevant to my question of what constitutes a measurement, since opening the box to view the cat is obviously a "measurement."

In particular, there is a difference with the Geiger counter case from my the first post in this thread that I do not know how to accommodate. 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. While listening to the Geiger counter, it seems that I am taking a continuous series of observations of the atom. How precisely does decoherence theory show that these observations do not act like Copenhagen-esque collapsing measurements?

 

[DrChinese]

In particular, there is difference with the Geiger counter case my the first post in this thread that I do not know how to accommodate. Namely, with the cat, 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. While listening to the Geiger counter, it seems that I am taking a continuous series of observations of the atom. How precisely does decoherence theory show that these observations do not act like Copenhagen-esque collapsing measurements?

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".)

Also, in case this was not already clear: There is no known difference in the state of a radioactive particle at T=0 and T=100 in the sense that it is no more (or less) likely to decay at T=100 than at any other time. That likelihood remains constant.

 

[WernerQH]

1) It is in state a(100)*N + b(100)*D. This is because the particle has evolved until time t=100, and the Geiger counter has not interacted with the particle or done anything to affect its state.

2) It is in state a(1)*N + b(1)*D. At t=100, we know there has been no decay product formed until at least t=99, so at t=99 we know the particle did not decay yet. Then the state at t=99 is N, so at t=100 the particle has evolved into the superposition one second after starting at N.

Which of these is correct, and why?

The short answer is neither. You've probably been led to believe that the wave function accurately represents an individual system. But it is really only a statistical description. Already Schrödinger struggled to make sense of the wave function for non-stationary states.

But you hit an important point relating to the time scales of the problem. The lifetime of a neutron, for example, is several minutes, whereas the actual decay occurs in less than a microsecond. (That's an upper limit on the time it takes the created anti-neutrino to leave the laboratory.) As has been explained already by gentzen and DrChinese, a "newly prepared" neutron is not discernible in any way from a neutron that has "aged" in the apparatus for 100 seconds. For the experimenter they will be in the same state (if it hasn't decayed in the meantime). An individual neutron decays at a rather well defined (measurable!) but random time. Only the average number of neutrons varies in a deterministic and continuous way. Continuous and deterministic evolution according to the time-dependent Schrödinger equation simply doesn't square with the discontinuous and random character of the events that we observe in the real world! The wave function is but a piece in a bigger mathematical apparatus, and there's more to quantum theory than Schrödinger's equation.

 

[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 ;-)).