B Is there a probability in QM that an event happens at time t?

[Heidi]

Hi PF,
A(x,t) is the probability amplitude at time t that a particle is at x. If it was emitted at (0,0)
the propagator gives its value. I wonder if QM can give the amplitude of time probability B(y,t) that an impact will occur (for a given y) at any t.
consider a screen behind the two slits, it one electron is emitted ar 0,0 in front
of the slits, it would be null for any y on the screen. And for y it will increase with time.
Is there a law in this case? (as we have the half life law for excited atoms)

 

[Demystifier]

Hi PF,
A(x,t) is the probability amplitude at time t that a particle is at x. If it was emitted at (0,0)
the propagator gives its value. I wonder if QM can give the amplitude of time probability B(y,t) that an impact will occur (for a given y) at any t.
consider a screen behind the two slits, it one electron is emitted ar 0,0 in front
of the slits, it would be null for any y on the screen. And for y it will increase with time.
Is there a law in this case? (as we have the half life law for excited atoms)

I'm just writing a paper about it. When I finish it (in a next couple of weeks), I'll put a link here.

 

[Heidi]

I look forward to read it. Where will it be? with which title?

 

[Demystifier]

I look forward to read it. Where will it be? with which title?

The paper is almost finished, I expect it to be finished very soon.

 

[Demystifier]

Here it is:
http://de.arxiv.org/abs/2010.07575

 

[Heidi]

Great, thanks.
Questions next time...

 

[Keith_McClary]

Here it is:
http://de.arxiv.org/abs/2010.07575

I thought you were making some subtle joke about the probability amplitude at time t of this event.

 

[Heidi]

@Demystifier,
So you may have probabilities on regions of space time.
I encountered something like that when i was reading the Wightman axioms. it was about the test functions. it was written that when a measurement is done in a laboratory it is in limited region of space and between the moments when lighr is switched on and off. these functions were defined on compact regions on space time (unlike the quantum wave functions) and could give all the details of what happened. they did not obey Klein Gordon equations or things like that. but they contained all the information.
Do you think that your model can add something to this ajpproach?

 

[Demystifier]

@Demystifier,
So you may have probabilities on regions of space time.
I encountered something like that when i was reading the Wightman axioms. it was about the test functions. it was written that when a measurement is done in a laboratory it is in limited region of space and between the moments when lighr is switched on and off. these functions were defined on compact regions on space time (unlike the quantum wave functions) and could give all the details of what happened. they did not obey Klein Gordon equations or things like that. but they contained all the information.
Do you think that your model can add something to this ajpproach?

Wightman axioms talk about correlation functions at different times, but they don't talk about the update of information induced by quantum measurements at different times (which my theory does). In this sense, Wightman correlation functions are not the correlations that are actually measured. In that regard see http://de.arxiv.org/abs/1610.03161

 

[Demystifier]

Hmm, let's say we have a system prepared as $\rho_{t_0}$ and we want to compute the probability that an event $\epsilon$ happens in a time interval $(t_1,t_2)$. We could maybe construct a Hilbert space $$\mathcal{H}_{t_1}\otimes\mathcal{H}_{t_2}$$ And compute the probability as $$\mathrm{Tr}\left[\rho_{t_0}\Pi^{\neg\epsilon}_{t_1}\Pi^{\epsilon}_{t_2}\right]$$

The following is from the book F. Laloe, Do We Really Understand Quantum Mechanics?

 

[Morbert]

The following is from the book F. Laloe, Do We Really Understand Quantum Mechanics?
View attachment 271274

Thanks. I actually deleted my message a few moments ago because I saw some holes in my thinking but will post again when I sort it. Response is appreciated regardless.

 

[Morbert]

So consider the particle decay scenario outlined in this paper https://www.fi.muni.cz/usr/buzek/zaujimave/home.pdf

The describe a particle in the state $$\Psi(t) = a_s(t)\phi_s + a_d(t)\phi_d$$ where $\phi_s$ and $\phi_d$ are states of a survived and decayed particle. If our system is prepared in the state $|\Psi(0)\rangle\langle\Psi(0)| = |\phi_s\rangle\langle\phi_s|$ then I think we could write the probability of the particle decaying in the interval $(t_1,t_2)$ as $$\mathrm{Tr}\left[\Pi_{t_2}^d\Pi_{t_1}^s|\phi_s\rangle\langle\phi_s|\Pi_{t_1}^s\Pi_{t_2}^d\right] == ||\Pi_{t_2}^d\Pi_{t_1}^s|\phi_s\rangle||^2$$
Where $\Pi_{t_2}^d$ and $\Pi_{t_1}^s$ are the relevant decayed and not decayed projectors.

 

[Demystifier]

Here it is:
http://de.arxiv.org/abs/2010.07575

It's published now:
https://www.sciencedirect.com/science/article/pii/S0375960121001110

 

[Demystifier]

The work continues. We have applied the general theory of the previous post to study the arrival time distribution. https://arxiv.org/abs/2107.08777

 

[Demystifier]

Now our third paper on this topic is accepted for publication. In this paper we show, among other things, that the arrival time distribution is given by the flux of the probability current.
https://arxiv.org/abs/2207.09140

 

[vanhees71]

Hm, which journal has accepted this? I'm puzzled!

You cannot naively calculate the expectation value of the Hamiltonian with a wave function that's not in its domain as a self-adjoint (hermitean is not enough!!!) operator. That leads to paradoxes like non-real expectation values.

A nice demonstration of paradoxes, if the necessary care about the domain and co-domain of self-adjoint operators is neglected, for the case of the rigid 1D potential well just appeared in EJP:

https://arxiv.org/abs/2305.08556
https://iopscience.iop.org/article/10.1088/1361-6404/acda69

 

[Demystifier]

Hm, which journal has accepted this? I'm puzzled!

As you can see in the link, it's Fortchr. Phys. (IF=5.5).

You cannot naively calculate the expectation value of the Hamiltonian with a wave function that's not in its domain as a self-adjoint (hermitean is not enough!!!) operator. That leads to paradoxes like non-real expectation values.

You are right, you cannot calculate it naively, you must be careful about it. We spent some space in the paper to show that when everything is done carefully, no paradoxes appear.

 

[vanhees71]

Then I've to read the paper more carefully, but from the abstract I thought you make a particular point of

that expectation value of a hermitian Hamiltonian can have an imaginary part in the infinite dimensional Hilbert space

and then I've seen that you apply $\hat{\vec{p}}^2$ to a wave function which can have jumps.

 

[Demystifier]

Then I've to read the paper more carefully, but from the abstract I thought you make a particular point of

and then I've seen that you apply $\hat{\vec{p}}^2$ to a wave function which can have jumps.

Pay particular attention to the text after Eq. (9), this explains why imaginary part of the expectation value of the Hamiltonian does not contradict the fact that measured energy is real.

Note also that non-hermitian Hamiltonians are used in phenomenological descriptions of decay of resonances, what we do in our paper is quite similar.

 

[vanhees71]

Ok, but then you admit that these apparent non-real "expectation values" are unphysical and cannot be used to make a physically meaningful statement. As I said, I've to read the paper more carefully, to see what you are after. If I had to review this paper, I'd have not let through your abstract in the present form! Obviously wrong statements in the abstract usually make me just immediately ignore the paper completely.

The use of non-self-adjoint Hamiltonians to describe particle decay is of a different nature and a physically meaningful approximation in the sense of an effective theory ("optical potential" approximation). This is not due to a wrong application of the formalism.

I know that questions about the domain of the self-adjoint operators is usually ignored in the textbook literature, and that's why many physicists don't know the question to apparently simple questions like, why there are no half-integer orbital angular-momentum representation.

 

[Demystifier]

Ok, but then you admit that these apparent non-real "expectation values" are unphysical and cannot be used to make a physically meaningful statement.

No I don't.

As I said, I've to read the paper more carefully, to see what you are after.

Exactly!

 

[lodbrok]

Interesting series of papers. Although only a side issue in the paper, I could not help but notice at the bottom of page 3 in your first paper (https://www.sciencedirect.com/science/article/pii/S0375960121001110) when you say:

The consequence is that the corresponding probability density w(t) in (29) vanishes in the limit δt → 0, in which case (28) reduces to the trivial result
P(t) = 0. (35)
In other words, if one checks with infinite frequency whether an event has happened, then the event will never happen. This seemingly paradoxical result is in fact well known as the quantum Zeno effect

The statement in bold is not accurate. For continuous probability distributions, zero probability for each point does not mean the event never happens. It simply means it is measure-zero. When you say it never happens, you imply that P(t) = 0 for all intervals of δt. As "never" implies a time interval that is not zero. This is the same error as suggesting that because every point on a line has a length of zero, therefore every line-segment has zero length, which is obviously false.

 

[Demystifier]

The statement in bold is not accurate. For continuous probability distributions, zero probability for each point does not mean the event never happens. It simply means it is measure-zero. When you say it never happens, you imply that P(t) = 0 for all intervals of δt. As "never" implies a time interval that is not zero. This is the same error as suggesting that because every point on a line has a length of zero, therefore every line-segment has zero length, which is obviously false.

It's accurate, you didn't read carefully. The P(t) is an integral of probability density from 0 to t. This integral is zero because the probability density is zero.

 

[lodbrok]

The P(t) is an integral of probability density from 0 to t. This integral is zero because the probability density is zero.

Yes, but that does not mean the event never happens if you sample at high-frequency (δt → 0).

 

[Demystifier]

Yes, but that does not mean the event never happens if you sample at high-frequency (δt → 0).

Yes it does. If probability (that is, integral of probability density over a finite interval) is zero, then the event never happens. This never happening event is called quantum Zeno effect in the literature: https://en.wikipedia.org/wiki/Quantum_Zeno_effect

 

[lodbrok]

In other words, if one checks with infinite frequency whether an event has happened, then the event will never happen.

Your statement implies that the whole event will never happen, not just that an infinitesimal interval itself has zero probability. The probability of the event happening "ever", is the integral of the full probability density function which is obviously non-zero.

 

[Haborix]

Your statement implies that the whole event will never happen, not just that an infinitesimal interval itself has zero probability. The probability of the event happening "ever", is the integral of the full probability density function which is obviously non-zero.

I can't tell if you're being pedantic about terminology or denying that the Quantum Zeno effect occurs. In your words, how would you describe the Quantum Zeno effect?

 

[Demystifier]

Your statement implies that the whole event will never happen, not just that an infinitesimal interval itself has zero probability. The probability of the event happening "ever", is the integral of the full probability density function

Exactly!

which is obviously non-zero.

It is zero, because the probability density is zero. I understand that your first reaction is that it must be wrong, my first reaction was exactly the same when I heard of quantum Zeno effect for the first time. But it's correct. I recommend you to do some reading about quantum Zeno effect to familiarize with it.

But our third paper contains another surprise, for those who already accepted the existence of quantum Zeno effect. We find that there is a way to avoid this effect, and hence turn back "common sense" into QM, by a new type of measurement that we discovered and called passive quantum measurement. This new kind of measurement plays a key role in our solution of the arrival time problem in quantum mechanics. Loosely speaking, the problem is that it looks as if a particle cannot arrive to the detector due to the quantum Zeno effect, but we solve the problem by discovering that the detector really performs a passive measurement of the particle that has not arrived yet, so the quantum Zeno effect does not really exist in this case and the particle is free to arrive.

 

[PeterDonis]

This integral is zero because the probability density is zero.

Is it actually possible, physically, to make the probability density exactly zero? Or is it only possible to make it small enough that one can reasonably expect no events over the finite duration of the experiment?

 

[Demystifier]

Is it actually possible, physically, to make the probability density exactly zero? Or is it only possible to make it small enough that one can reasonably expect no events over the finite duration of the experiment?

In practice, that is, in the real laboratory, the latter of course is true.

 

[PeterDonis]

In practice, that is, in the real laboratory, the latter of course is true.

In the theoretical model, how is this accounted for? Is it that in the actual lab one cannot make a continuous infinity of checks of the state of the system?

 

[Demystifier]

In the theoretical model, how is this accounted for? Is it that in the actual lab one cannot make a continuous infinity of checks of the state of the system?

Yes, of course. In our theory, this is modeled by small but finite $\delta t$.

 

[vanhees71]

I'm still pretty skeptical about the math in the paper. It is very clear that in QT hermitecity is NOT enough for an operator to represent observables. It should be self-adjoint. Of course the Hamiltonian of a free particle, $\hat{H}=\vec{p}^2/(2m)$ is self-adjoint. The important point is exactly the issue with the "apparent paradox", i.e., by definition a self-adjoint operator on the Hilbert space must be defined on a dense subspace (domain) and this dense subspace must be mapped by it to itself. In the text below Eq. (17) one should add that with $\psi,\varphi \in \mathcal{D}(H)$ also $H \psi,H \varphi \in \mathcal{D}(H)$. Already for the momentum operator a wave function which is not smooth is not in the domain, and thus you cannot naively calculate expectation values or matrix elements with such functions. The way out is to use a sequence of wave functions that approximate the wave function in question and take the limit for the expectation values/matrix elements.

I have to read a bit further to come to a conclusion, but I'm still skeptical!

 

[weirdoguy]

I have to read a bit further to come to a conclusion, but I'm still skeptical!

I guess it would take less time to read the paper properly than to write out all these scepticisms

 

[Demystifier]

It is very clear that in QT hermitecity is NOT enough for an operator to represent observables. It should be self-adjoint.

Note that the operator $\overline{H}$ is not an observable. It is a non-hermitin and non-self-adjoint operator that generates a non-unitary evolution.

 

[Demystifier]

Already for the momentum operator a wave function which is not smooth is not in the domain, and thus you cannot naively calculate expectation values or matrix elements with such functions.

Sure, one cannot calculate naively, but it doesn't mean that one cannot calculate at all. To calculate such things, one has to precisely define how one calculates it. Eq. (9) is exactly the precise definition of our calculation.

The way out is to use a sequence of wave functions that approximate the wave function in question and take the limit for the expectation values/matrix elements.

No, it is a way out, not the way out. In this paper we use a different way out. Our alternative definition turns out to have physical consequences, which in the end solve physical problems that appear with standard mathematical definitions. So we solve a physical problem by thinking more carefully how certain formal mathematical entities should really be defined, in order to get sensible physics. From this point of view, one can say that it turns out that the standard (not ours) definition was "naive".

 

[vanhees71]

Sure, one cannot calculate naively, but it doesn't mean that one cannot calculate at all. To calculate such things, one has to precisely define how one calculates it. Eq. (9) is exactly the precise definition of our calculation.

As I said, you need to use the correct limiting procedures.

No, it is a way out, not the way out. In this paper we use a different way out. Our alternative definition turns out to have physical consequences, which in the end solve physical problems that appear with standard mathematical definitions. So we solve a physical problem by thinking more carefully how certain formal mathematical entities should really be defined, in order to get sensible physics. From this point of view, one can say that it turns out that the standard (not ours) definition was "naive".

I don't understand, what problem you want to solve to begin with. No matter what, I'm not convinced that starting from obvious wrong maths leads to anything useful. Self-adjoint operators don't have non-real expectation values, as soon as you properly define them!

 

[vanhees71]

Note that the operator $\overline{H}$ is not an observable. It is a non-hermitin and non-self-adjoint operator that generates a non-unitary evolution.

If $H$ is self-adjoint then also $\pi H \pi$ is self-adjoint.

 

[Demystifier]

If $H$ is self-adjoint then also $\pi H \pi$ is self-adjoint.

In finite-dimensional Hilbert space, yes. In infinite-dimensional, not necessarily. We have explained it in detail in the cited paper.

 

[Demystifier]

Self-adjoint operators don't have non-real expectation values, as soon as you properly define them!

Definitions are not written in stones, they can be changed.

 

[vanhees71]

Then use another word. In the natural sciences it's good practice not to change the meaning of words!

 

[Demystifier]

Then use another word. In the natural sciences it's good practice not to change the meaning of words!

I'm thinking what word would better convey the idea. Perhaps "extended Hamilton operator", because we are extending the domain on which the operator $H\propto\nabla^2$ is allowed to act.

 

[Demystifier]

I don't understand, what problem you want to solve to begin with.

The arrival time problem, as a central problem of the more general problem indicated by the title of this thread.

 

[Morbert]

iirc from a previous thread, it's concluded that the Zeno effect isn't due to measurement per se, but interaction between the measurement device and the measured system. I.e. the effect is present in the dynamics (even for seemingly, but not actually, indirect measurement scenarios) regardless of when we build "collapsed states" of measurement outcomes from some projective decomposition. https://www.physicsforums.com/threads/geiger-counters-and-measurement.1015428/

 

[.Scott]

I will not claim that I have read your article "properly", but I have read through a lot of it and scanned through the rest.

My questions are:

In your setup, you sample at t0,t1, ... until you get a hit and you calculate P(0), P(1), ...
Let's say that we performed these experiments two ways. In the first case (case 'a'), I sample exactly as described in your paper (t0 on until I get a hit). I will call these Pa(x) for x=0, 1, ....

In the second case (case 'b'), I start my sampling at t(n-1). I will call these Pb(x) for x=n-1, n, n+1, ....

I am guessing that by performing measurements more often, I will get fewer hits (a la Zeno). So by the time I reach t(n-1), we will be less likely to have reached the end in cases 'a' than with cases 'b'. Ie, Sum(Pa(0) to Pa(n-1)) < Pb(n-1)
Is that correct?

Next, I will "normalize" the result "tails" (sample n and on) for both cases as follows:
1) Compute the sums Sa=sum of Pa(x) and Sb=sum of Pb(x) for x=n, n+1, n+2, ...
2) Compute Ta(x)=Pa(x+1)/Sa and Tb(x)=Pb(x+n)/Sb for x=0,1,2,...
Lets call Ta and Tb the tails.
Will the tails always the same?
My guess would be that the different sampling history from t0 to tn-1 would (in some cases) have an effect than cannot be completely erased by that Sa and Sb "normalization" that we applied. But it would take me some time with you math to verify this.

Thanks.

 

[Demystifier]

iirc from a previous thread, it's concluded that the Zeno effect isn't due to measurement per se, but interaction between the measurement device and the measured system. I.e. the effect is present in the dynamics (even for seemingly, but not actually, indirect measurement scenarios) regardless of when we build "collapsed states" of measurement outcomes from some projective decomposition. https://www.physicsforums.com/threads/geiger-counters-and-measurement.1015428/

I don't have tome to read the whole thread. Can you tell which post concludes it?

 

[.Scott]

More questions:
The quasi-spontaneous is interesting. I assume it would apply to isotope decay.
For example, Lithium 8 has a half life under a second. But left in an empty universe, would it ever decay?
It sounds to me that an empty universe (or one with only a single lithium 8 atom) would be "timeless".

 

[Demystifier]

More questions:
The quasi-spontaneous is interesting. I assume it would apply to isotope decay.
For example, Lithium 8 has a half life under a second. But left in an empty universe, would it ever decay?

We claim that it wouldn't.

It sounds to me that an empty universe (or one with only a single lithium 8 atom) would be "timeless".

Not necessarily, it can be oscillating if it is in a superposition of two energies.

 

[Demystifier]

In the second case (case 'b'), I start my sampling at t(n-1). I will call these Pb(x) for x=n-1, n, n+1, ....

It doesn't really matter when the sampling starts, i.e. what time is chosen as "initial".

I am guessing that by performing measurements more often, I will get fewer hits (a la Zeno). So by the time I reach t(n-1), we will be less likely to have reached the end in cases 'a' than with cases 'b'.

I don't understand. What performing measurements more often has to do with the choice of time of the first measurement?

 

[.Scott]

It doesn't really matter when the sampling starts, i.e. what time is chosen as "initial".

That would suggest that your were discussing (in your paper) strictly logarithmic decay - So P(n)/P(n-1) is a constant.
Is that really the case?
If it isn't, then knowledge of what measurements were attempted before t0 could be used to tweak the P(n)'s.

I don't understand. What performing measurements more often has to do with the choice of time of the first measurement?

So you have two labs (a and b), each making observations at the same delta t (say 100msec) and each one sets up for a series of measurement at the start of each minute. By 2 seconds into the minute, lab a has already made 21 measurements but lab b is just making it's first measurement. My thought is that Pb(20) will be greater than the sum of Pa(0) to Pa(20) only because the rapid samplings at 'a' would have a Zeno-type effect - keeping the experiment at the initial setup state.