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$$\int_a ^b \Pi(t) \text{dt} = \text{Probability that the particle arrives at time } t \in [a,b] $$

There is no lack of proposals for this, actually the problem is that there are very many proposals which do not agree with one another. You can see a non-exhaustive summary of some of those proposals in this review paper by Muga (2000). It contains about half of the proposals I am aware of today.

Having gone through many of the existing proposals in detail, I will give my opinion: they are, for the most part, grotesquely unscientific. Problems with some these proposals (in peer-reviewed papers!) include:

• Not normalizable even for reasonable $\psi $ like gaussian wave packets
• Predicts negative probabilities
• Only works in 1 dimension
• Only works when $V(x)=0$

In recent months, an effort has accumulated to actually do experiments to rule out many of these proposals. An experiment is planned for the near future. Until the results come out, any conclusions on which proposal is best are subject to being proven wrong. That being said, some proposals are clearly very ad-hoc and inspire little confidence, while I cannot find objective flaws in others. According to my own, always-possibly-flawed understanding after working in this field, the best proposals we have today are

However, the quantity $\vec{J}\cdot \hat{n}$, and therefore the entire integral, may be negative. In this case that the flux clearly does not work as a probability density, and it has been shown that it is exactly in this case (negativity for some point on the detector) that the Bohmian Mechanics prediction differs from the flux. The prediction made by Bohmian Mechanics, obtained by averaging over many trajectories, is always nonnegative. Negative flux corresponds to Bohmian Trajectories which loop around and leave the detector region.

3.2. The Kijowski Distribution

Arrival time proposals are a dime a dozen at the moment, and even having done research in this field it is infeasible to rigorously go through every approach anyone has used in the literature. In addition, an experiment has not yet been done, so in some sense, science does not have an answer for you yet. To remedy this I have given what I can, namely my own understanding of the state of things after having spent a fair amount of time on the subject. If things go as I hope they do, there will be a scientific answer to this question in the coming years. In addition to the aforementioned experiment, there is for example an experimental proposal, possible to implement with modern-day technology, which could test arrival times in the most "juicy" regime: where the flux is negative. To be clear about potential biases, I know the authors of this paper. My thesis was not on the Bohmian approach.

$$\int_a ^b \Pi(t) \text{dt} = \text{Probability that the particle arrives at a time } t \in [a,b] $$

There is no lack of proposals for how to derive $\Pi(t)$, actually the problem is that there are very many proposals which do not agree with one another. You can see a non-exhaustive summary of some of those proposals in this review paper by Muga (2000). It contains about half of the proposals I am aware of today.

Although there are many proposals, in my opinion good ideas are still very welcome. Having gone through many of the existing proposals in detail, I will give my opinion: they are, for the most part, low-effort / quite unscientific. Problems with some these proposals (in peer-reviewed papers!) include:

• Not normalizable even for straightforward Schrödinger solutions $\psi(\vec{r},t) $ like gaussian wave packets
• Predicts negative probabilities
• Only works in 1 dimension
• Only works when $V(x)=0$
• Not enough details are specified to actually calculate $\Pi(t)$ due to not much time spent thinking about the proposal in-depth

However, there are some proposals with which I have found no such issues to date. I will share the ones I am aware of below.

There has been ongoing conversation in recent years, trying to match with experimental groups to figure out which theoretical proposals might give the correct experimental distribution of $\Pi(t)$. Distinguishing between each of the different calculable proposals is within technological capability today. However such an experiment has not yet been done (all that remains from the 1997 Kurtsiefer data, which was originally taken for other purposes, is the graphic in the paper). Thus there is a significant potential for experiment to advance this field. 

Until the experimental results are out, any conclusions on which proposal is best are subject to being proven wrong (Aside: If you you know of a group that has the equipment for an arrival time experiment, I can put you in contact with (imo) capable/interested people on the theory side). According to my own, always-possibly-flawed understanding after working in this field, the best proposals we have today are

However, the quantity $\vec{J}\cdot \hat{n}$, and therefore the entire integral, may be negative. In this case that the flux clearly does not work as a probability density, and it has been shown that it is exactly in this case (negativity for some point on the detector) that the Bohmian Mechanics prediction differs from the flux. The prediction made by Bohmian Mechanics, obtained by averaging over many trajectories, is always nonnegative. Negative flux corresponds to Bohmian trajectories which loop around and leave the detector region.

Note: there is a camp among researchers in Bohmian Mechanics which disagrees with this treatment [yes, such researchers exist on our earth, albeit in small number]. They argue that this distribution should only be calculated by modeling the distribution of positions of a macroscopic pointer that is correlated to the arrival time, rather than the distribution of the particle itself. For the sake of brevity of this answer I will not elaborate but feel free to ask.

3.2. The Kijowski Distribution

Arrival time proposals are a dime a dozen at the moment. An experiment has not yet been done, so in some sense, science does not have an answer for you yet. To remedy this I have given what I can, namely my own understanding of the state of things after having spent a fair amount of time on the subject. If things go as I hope they do, there will be a scientific answer to this question through experiment in the coming years. There exists an experimental proposal, possible to implement with modern-day technology, which could test arrival times in the most "juicy" regime: where the flux is negative. To be transparent about any potential biases, I know the authors of this paper. However I have not worked on the Bohmian or Kijowski-based approaches which I mentioned in this post and my career does not benefit from public opinion on this in any way.

Edit: I have removed my answer to (hopefully) prevent its use for training AI models. I believe that making an AI which is smarter than humans is a decision that humanity will come to regret.

I just finished a thesis on this subject and I'm happy to share. None of the linked papers are my own.

The time of arrival in quantum mechanics is actually a subject of ongoing research. It is certainly a question which begs for an answer, as experiments have been able to measure the distribution of arrival times for decades (see for example Fig. 3 of this 1997 paper by Kurtsiefer et. al). Note: If you do not have access to journals let me know and I will see if I can include the figure in this answer.

Part 1 of this answer describes why there is a problem with arrival time in quantum mechanics.

Part 2 outlines the modern situation in regards to this problem.

Part 3 gives, in my view, the best answers we currently have, which still need experimental verification.

1. New Ideas are Here Needed: The observable-operator formalism seems not to work for arrival times

Normally in QM you have operators $A$ corresponding to the variables used in classical mechanics. This lets you define a basis of eigenfunctions of that operator, which are found through the equation $A|a\rangle = a |a\rangle$. With such a basis in hand, the probability of finding the value $a$ in an experiment on a particle in state $|\psi\rangle $is $|\langle a|\psi\rangle|^2$.

Though the probability distribution of arrival times can be measured in experiment, predicting it in theory is less straightforward. There are two theorems I am aware of which indicate that the textbook observable formalism above will not work for arrival times:

1. Pauli's Theorem: In 1933, Wolfgang Pauli published a book on Quantum Mechanics called The General Principles of Wave Mechanics. In a footnote of this book, Pauli notes that if you have the commutation relation $[T,H]=i\hbar$ for some supposed self-adjoint time operator $T$, then $H$ would have to have all eigenvalues $[-\infty, \infty]$, which is not possible because systems could not have a ground state. His is an early variant of the theorem which has since been made more precise (modern proofs can be found in section 2 of this 1981 paper).
2. Allcock's Theorem: In 1969, Allcock gave another proof that the usual formalism won't work with time. He shows that it is impossible to have a complete set of orthonormal arrival time eigenstates which transform properly under change of coordinates $(t,\vec{r}) \to (t+\Delta t,\vec{r})$ - and thus that there cannot be an adequate self-adjoint time operator, since this would result in such eigenstates. The proof begins just before Equation 2.18 with "The reader...".

A number of authors have tried to define a time operator anyway, yet none of the variants I have seen were able to subvert both of the above theorems, rendering them unphysical.

2. Arrival time approaches outside of the textbook formalism

Because of the issues in Part 1 of this answer, many authors have tried to come up with ways to derive a distribution for the arrival time of a particle outside of the usual formalism. The distribution we seek is usually notated $\Pi(t)$ and should of course have the property that

$$\int_a ^b \Pi(t) \text{dt} = \text{Probability that the particle arrives at time } t \in [a,b] $$

There is no lack of proposals for this, actually the problem is that there are very many proposals which do not agree with one another. You can see a non-exhaustive summary of some of those proposals in this review paper by Muga (2000). It contains about half of the proposals I am aware of today.

Having gone through many of the existing proposals in detail, I will give my opinion: they are, for the most part, grotesquely unscientific. Problems with some these proposals (in peer-reviewed papers!) include:

• Not normalizable even for reasonable $\psi $ like gaussian wave packets
• Predicts negative probabilities
• Only works in 1 dimension
• Only works when $V(x)=0$

3. The best answers we have today

In recent months, an effort has accumulated to actually do experiments to rule out many of these proposals. An experiment is planned for the near future. Until the results come out, any conclusions on which proposal is best are subject to being proven wrong. That being said, some proposals are clearly very ad-hoc and inspire little confidence, while I cannot find objective flaws in others. According to my own, always-possibly-flawed understanding after working in this field, the best proposals we have today are

3.1 Bohmian Mechanics / The Quantum Flux

Bohmian Mechanics is a quantum theory in which particles follow definite trajectories (see the double slit trajectories for example). The predictions of Bohmian Mechanics agree with standard QM for position measurements. For each individual trajectory the arrival time is the moment when it first hits the detector. Since the initial position is unknown, many different trajectories are possible, and this defines a distribution of different possible arrival times.

It has been proven that typically, the arrival time distribution in Bohmian Mechanics is exactly equal to the (integrated) flux of probability across the detector $D$:

$$\Pi_{BM}(t) = \int_{\partial D} \vec{J}(\vec{r},t)\cdot \hat{n} \text{ dA}$$

where $\vec{J}$ is the flux as described in any QM textbook, and $\hat{n}$ is a unit vector pointing into the detector surface. This is the rate at which probability enters the detector, and so it very nicely correlates the arrival time statistics with position statistics.

However, the quantity $\vec{J}\cdot \hat{n}$, and therefore the entire integral, may be negative. In this case that the flux clearly does not work as a probability density, and it has been shown that it is exactly in this case (negativity for some point on the detector) that the Bohmian Mechanics prediction differs from the flux. The prediction made by Bohmian Mechanics, obtained by averaging over many trajectories, is always nonnegative. Negative flux corresponds to Bohmian Trajectories which loop around and leave the detector region.

3.2. The Kijowski Distribution

The second-most reasonable candidate I have seen is the Kijowski distribution. In this 1974 paper, Kijowski postulated it for the free particle by declaring a series of axioms. These axioms yield nicely a unique distribution, but as Kijowski notes,

Our construction is set up for free particles in both the non-relativistic and relativistic case and cannot be generalized for the non-free wave equation

Nonetheless the approach is well-liked as it yields a priori reasonable results and has a tendency to resemble the quantum flux. For this reason, Muga began calling it & its generalizations the "standard distribution".

By abandoning the axiomatic approach, a variant inspired by Kijowski's distribution has been created which works for other potentials, see paper here (2000). However there is a spacial nonlocality to this distribution, i.e. the position statistics don't correspond to the arrival time statistics. Basically it predicts that a particle can be found after a finite time at a location where, according to standard quantum mechanics, there is a 0% chance of finding it - this seems unphysical. A critique is given by Leavens in this paper (2002).

Final Remarks

Arrival time proposals are a dime a dozen at the moment, and even having done research in this field it is infeasible to rigorously go through every approach anyone has used in the literature. In addition, an experiment has not yet been done, so in some sense, science does not have an answer for you yet. To remedy this I have given what I can, namely my own understanding of the state of things after having spent a fair amount of time on the subject. If things go as I hope they do, there will be a scientific answer to this question in the coming years. In addition to the aforementioned experiment, there is for example an experimental proposal, possible to implement with modern-day technology, which could test arrival times in the most "juicy" regime: where the flux is negative. To be clear about potential biases, I know the authors of this paper. My thesis was not on the Bohmian approach.

I just finished a thesis on this subject and I'm happy to share. None of the linked papers are my own.

The time of arrival in quantum mechanics is actually a subject of ongoing research. It is certainly a question which begs for an answer, as experiments have been able to measure the distribution of arrival times for decades (see for example Fig. 3 of this 1997 paper by Kurtsiefer et. al). Note: If you do not have access to journals let me know and I will see if I can include the figure in this answer.

Part 1 of this answer describes why there is a problem with arrival time in quantum mechanics.

Part 2 outlines the modern situation in regards to this problem.

Part 3 gives, in my view, the best answers we currently have, which still need experimental verification.

1. New Ideas are Here Needed: The observable-operator formalism seems not to work for arrival times

Normally in QM you have operators $A$ corresponding to the variables used in classical mechanics. This lets you define a basis of eigenfunctions of that operator, which are found through the equation $A|a\rangle = a |a\rangle$. With such a basis in hand, the probability of finding the value $a$ in an experiment on a particle in state $|\psi\rangle $is $|\langle a|\psi\rangle|^2$.

Though the probability distribution of arrival times can be measured in experiment, predicting it in theory is less straightforward. There are two theorems I am aware of which indicate that the textbook observable formalism above will not work for arrival times:

1. Pauli's Theorem: In 1933, Wolfgang Pauli published a book on Quantum Mechanics called The General Principles of Wave Mechanics. In a footnote of this book, Pauli notes that if you have the commutation relation $[T,H]=i\hbar$ for some supposed self-adjoint time operator $T$, then $H$ would have to have all eigenvalues $[-\infty, \infty]$, which is not possible because systems could not have a ground state. His is an early variant of the theorem which has since been made more precise (modern proofs can be found in section 2 of this 1981 paper).
2. Allcock's Theorem: In 1969, Allcock gave another proof that the usual formalism won't work with time. He shows that it is impossible to have a complete set of orthonormal arrival time eigenstates which transform properly under change of coordinates $(t,\vec{r}) \to (t+\Delta t,\vec{r})$ - and thus that there cannot be an adequate self-adjoint time operator, since this would result in such eigenstates. The proof begins just before Equation 2.18 with "The reader...".

A number of authors have tried to define a time operator anyway, yet none of the variants I have seen were able to subvert both of the above theorems, rendering them unphysical.

2. Arrival time approaches outside of the textbook formalism

Because of the issues in Part 1 of this answer, many authors have tried to come up with ways to derive a distribution for the arrival time of a particle outside of the usual formalism. The distribution we seek is usually notated $\Pi(t)$ and should of course have the property that

$$\int_a ^b \Pi(t) \text{dt} = \text{Probability that the particle arrives at time } t \in [a,b] $$

There is no lack of proposals for this, actually the problem is that there are very many proposals which do not agree with one another. You can see a non-exhaustive summary of some of those proposals in this review paper by Muga (2000). It contains about half of the proposals I am aware of today.

Having gone through many of the existing proposals in detail, I will give my opinion: they are, for the most part, grotesquely unscientific. Problems with some these proposals (in peer-reviewed papers!) include:

• Not normalizable even for reasonable $\psi $ like gaussian wave packets
• Predicts negative probabilities
• Only works in 1 dimension
• Only works when $V(x)=0$

3. The best answers we have today

In recent months, an effort has accumulated to actually do experiments to rule out many of these proposals. An experiment is planned for the near future. Until the results come out, any conclusions on which proposal is best are subject to being proven wrong. That being said, some proposals are clearly very ad-hoc and inspire little confidence, while I cannot find objective flaws in others. According to my own, always-possibly-flawed understanding after working in this field, the best proposals we have today are

3.1 Bohmian Mechanics / The Quantum Flux

Bohmian Mechanics is a quantum theory in which particles follow definite trajectories (see the double slit trajectories for example). The predictions of Bohmian Mechanics agree with standard QM for position measurements. For each individual trajectory the arrival time is the moment when it first hits the detector. Since the initial position is unknown, many different trajectories are possible, and this defines a distribution of different possible arrival times.

It has been proven that typically, the arrival time distribution in Bohmian Mechanics is exactly equal to the (integrated) flux of probability across the detector $D$:

$$\Pi_{BM}(t) = \int_{\partial D} \vec{J}(\vec{r},t)\cdot \hat{n} \text{ dA}$$

where $\vec{J}$ is the flux as described in any QM textbook, and $\hat{n}$ is a unit vector pointing into the detector surface. This is the rate at which probability enters the detector, and so it very nicely correlates the arrival time statistics with position statistics.

However, the quantity $\vec{J}\cdot \hat{n}$, and therefore the entire integral, may be negative. In this case that the flux clearly does not work as a probability density, and it has been shown that it is exactly in this case (negativity for some point on the detector) that the Bohmian Mechanics prediction differs from the flux. The prediction made by Bohmian Mechanics, obtained by averaging over many trajectories, is always nonnegative. Negative flux corresponds to Bohmian Trajectories which loop around and leave the detector region.

3.2. The Kijowski Distribution

The second-most reasonable candidate I have seen is the Kijowski distribution. In this 1974 paper, Kijowski postulated it for the free particle by declaring a series of axioms. These axioms yield nicely a unique distribution, but as Kijowski notes,

Our construction is set up for free particles in both the non-relativistic and relativistic case and cannot be generalized for the non-free wave equation

Nonetheless the approach is well-liked as it yields a priori reasonable results and has a tendency to resemble the quantum flux. For this reason, Muga began calling it & its generalizations the "standard distribution".

By abandoning the axiomatic approach, a variant inspired by Kijowski's distribution has been created which works for other potentials, see paper here (2000). However there is a spacial nonlocality to this distribution, i.e. the position statistics don't correspond to the arrival time statistics. Basically it predicts that a particle can be found after a finite time at a location where, according to standard quantum mechanics, there is a 0% chance of finding it - this seems unphysical. A critique is given by Leavens in this paper (2002).

Final Remarks

Arrival time proposals are a dime a dozen at the moment, and even having done research in this field it is infeasible to rigorously go through every approach anyone has used in the literature. In addition, an experiment has not yet been done, so in some sense, science does not have an answer for you yet. To remedy this I have given what I can, namely my own understanding of the state of things after having spent a fair amount of time on the subject. If things go as I hope they do, there will be a scientific answer to this question in the coming years. In addition to the aforementioned experiment, there is for example an experimental proposal, possible to implement with modern-day technology, which could test arrival times in the most "juicy" regime: where the flux is negative. To be clear about potential biases, I know the authors of this paper. My thesis was not on the Bohmian approach.

Edit: I have removed my answer to (hopefully) prevent its use for training AI models. I believe that making an AI which is smarter than humans is a decision that humanity will come to regret.