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Let's suppose I initially have a particle with a nice and narrow wave function[1] (I will leave these unnormed): $$e^{-\frac{x^2}{a}}$$ where $a$ is some small number (to make it narrow). Let's also suppose that the wave function travels along x axis with constant speed $v$ and it smears out with constant speed $v_s$. So after time $t$ wave function looks like $$e^{-\frac{(x-vt)^2}{a+v_st}}$$

Now, if speed $v$ is close to $c$ and $v_s$ is big enough too, we could get a situation where the centre of mass moves at subluminal speed (this is the group velocity, I suppose), but the front[2] of the function moves at superluminal speed.

If I measure either the particle's position after some time $T$ or time when it reaches some point $A$ i will mostly and averagely conclude that it has travelled with speed $v$. But in some (less probable) cases it will seem that it has travelled faster, even FTL. Is this a normal thing in quantum mechanics or I understand it all wrong? Maybe there is some constraint not only on $v$, but also on $v+v_s$?

A little background

When talking about superluminal tunneling speeds I usually hear explanations like this:

That speed is apparent. In those cases only some frontal part of the wave function gets through the barrier. Although it's center of mass seems to have travelled FTL, that function would still stay under the initial wave function if it continued it's way without the barrier. The center just shifted because of dropping the rear part.

I have never actually understood why does it solve the problem, because the position of center doesn't change the fact that signal in some cases may arrive FTL. This is where my question comes from.

EDIT: Sorry, the question was not about what some classic QM models allow, but actual quantum physics. I understand that Schrodinger's equation allows me any speed, I want to know if this situation is allowed in actual quantum physics. I guess the use of term QM was wrong. Sorry, my bad.

Question reformulated

Is it possible to send a particle that's slowly spreads and with some small chance (in some rare cases) measure it arriving FTL? (provided that the mean arrival value still stays under c).

I am aware that this is possible when tunneling through a barrier (and the mean value can even move to superluminal by train dropping cars), but is the same possible without any barrier?

[1] If you say that I had some chance to measure it at any point in the very beginning and that it wasn't entirely localized, we can replace the Gauss' function with a square or triangle one that gets wider along the way.

[2] If you need, we can define the front of the wave, for example first of the points where second derivative is zero. It's position is $vt+\sqrt{\frac{a+v_st}{2}}$.

Let's suppose I initially have a particle with a nice and narrow wave function[1] (I will leave these unnormed): $$e^{-\frac{x^2}{a}}$$ where $a$ is some small number (to make it narrow). Let's also suppose that the wave function travels along x axis with constant speed $v$ and it smears out with constant speed $v_s$. So after time $t$ wave function looks like $$e^{-\frac{(x-vt)^2}{a+v_st}}$$

Now, if speed $v$ is close to $c$ and $v_s$ is big enough too, we could get a situation where the centre of mass moves at subluminal speed (this is the group velocity, I suppose), but the front[2] of the function moves at superluminal speed.

If I measure either the particle's position after some time $T$ or time when it reaches some point $A$ i will mostly and averagely conclude that it has travelled with speed $v$. But in some (less probable) cases it will seem that it has travelled faster, even FTL. Is this a normal thing in quantum mechanics or I understand it all wrong? Maybe there is some constraint not only on $v$, but also on $v+v_s$?

A little background

When talking about superluminal tunneling speeds I usually hear explanations like this:

That speed is apparent. In those cases only some frontal part of the wave function gets through the barrier. Although it's center of mass seems to have travelled FTL, that function would still stay under the initial wave function if it continued it's way without the barrier. The center just shifted because of dropping the rear part.

I have never actually understood why does it solve the problem, because the position of center doesn't change the fact that signal in some cases may arrive FTL. This is where my question comes from.

EDIT: Sorry, the question was not about what some classic QM models allow, but actual quantum physics. I understand that Schrodinger's equation allows me any speed, I want to know if this situation is allowed in actual quantum physics. I guess the use of term QM was wrong. Sorry, my bad.

Question reformulated

Is it possible to send a particle that's slowly spreads and with some small chance (in some rare cases) measure it arriving FTL? (provided that the mean arrival value still stays under $c$).

I am aware that this is possible when tunneling through a barrier (and the mean value can even move to superluminal by train dropping cars), but is the same possible without any barrier?

Maybe I could equivalently ask, if quantum uncertainties allow occasional exceeding of $c$.

[1] If you say that I had some chance to measure it at any point in the very beginning and that it wasn't entirely localized, we can replace the Gauss' function with a square or triangle one that gets wider along the way.

[2] If you need, we can define the front of the wave, for example first of the points where second derivative is zero. It's position is $vt+\sqrt{\frac{a+v_st}{2}}$.

Let's suppose I initially have a particle with a nice and narrow wave function[1] (I will leave these unnormed): $$e^{-\frac{x^2}{a}}$$ where $a$ is some small number (to make it narrow). Let's also suppose that the wave function travels along x axis with constant speed $v$ and it smears out with constant speed $v_s$. So after time $t$ wave function looks like $$e^{-\frac{(x-vt)^2}{a+v_st}}$$

Now, if speed $v$ is close to $c$ and $v_s$ is big enough too, we could get a situation where the centre of mass moves at subluminal speed (this is the group velocity, I suppose), but the front[2] of the function moves at superluminal speed.

If I measure either the particle's position after some time $T$ or time when it reaches some point $A$ i will mostly and averagely conclude that it has travelled with speed $v$. But in some (less probable) cases it will seem that it has travelled faster, even FTL. Is this a normal thing in quantum mechanics or I understand it all wrong? Maybe there is some constraint not only on $v$, but also on $v+v_s$?

A little background

When talking about superluminal tunneling speeds I usually hear explanations like this:

That speed is apparent. In those cases only some frontal part of the wave function gets through the barrier. Although it's center of mass seems to have travelled FTL, that function would still stay under the initial wave function if it continued it's way without the barrier. The center just shifted because of dropping the rear part.

I have never actually understood why does it solve the problem, because the position of center doesn't change the fact that signal in some cases may arrive FTL. This is where my question comes from.

EDIT: Sorry, the question was not about what some classic QM models allow, but actual quantum physics.

Question reformulated

Is it possible to send a particle that's slowly spreads and with some small chance (in some rare cases) measure it arriving FTL? (provided that the mean arrival value still stays under c).

I am aware that this is possible when tunneling through a barrier (and the mean value can even move to superluminal by train dropping cars), but is the same possible without any barrier?

[1] If you say that I had some chance to measure it at any point in the very beginning and that it wasn't entirely localized, we can replace the Gauss' function with a square or triangle one that gets wider along the way.

[2] If you need, we can define the front of the wave, for example first of the points where second derivative is zero. It's position is $vt+\sqrt{\frac{a+v_st}{2}}$.

Let's suppose I initially have a particle with a nice and narrow wave function[1] (I will leave these unnormed): $$e^{-\frac{x^2}{a}}$$ where $a$ is some small number (to make it narrow). Let's also suppose that the wave function travels along x axis with constant speed $v$ and it smears out with constant speed $v_s$. So after time $t$ wave function looks like $$e^{-\frac{(x-vt)^2}{a+v_st}}$$

Now, if speed $v$ is close to $c$ and $v_s$ is big enough too, we could get a situation where the centre of mass moves at subluminal speed (this is the group velocity, I suppose), but the front[2] of the function moves at superluminal speed.

If I measure either the particle's position after some time $T$ or time when it reaches some point $A$ i will mostly and averagely conclude that it has travelled with speed $v$. But in some (less probable) cases it will seem that it has travelled faster, even FTL. Is this a normal thing in quantum mechanics or I understand it all wrong? Maybe there is some constraint not only on $v$, but also on $v+v_s$?

A little background

When talking about superluminal tunneling speeds I usually hear explanations like this:

That speed is apparent. In those cases only some frontal part of the wave function gets through the barrier. Although it's center of mass seems to have travelled FTL, that function would still stay under the initial wave function if it continued it's way without the barrier. The center just shifted because of dropping the rear part.

I have never actually understood why does it solve the problem, because the position of center doesn't change the fact that signal in some cases may arrive FTL. This is where my question comes from.

EDIT: Sorry, the question was not about what some classic QM models allow, but actual quantum physics. I understand that Schrodinger's equation allows me any speed, I want to know if this situation is allowed in actual quantum physics. I guess the use of term QM was wrong. Sorry, my bad.

Question reformulated

Is it possible to send a particle that's slowly spreads and with some small chance (in some rare cases) measure it arriving FTL? (provided that the mean arrival value still stays under c).

I am aware that this is possible when tunneling through a barrier (and the mean value can even move to superluminal by train dropping cars), but is the same possible without any barrier?

[1] If you say that I had some chance to measure it at any point in the very beginning and that it wasn't entirely localized, we can replace the Gauss' function with a square or triangle one that gets wider along the way.

[2] If you need, we can define the front of the wave, for example first of the points where second derivative is zero. It's position is $vt+\sqrt{\frac{a+v_st}{2}}$.

Let's suppose I initially have a particle with a nice and narrow wave function[1] (I will leave these unnormed): $$e^{-\frac{x^2}{a}}$$ where $a$ is some small number (to make it narrow). Let's also suppose that the wave function travels along x axis with constant speed $v$ and it smears out with constant speed $v_s$. So after time $t$ wave function looks like $$e^{-\frac{(x-vt)^2}{a+v_st}}$$

Now, if speed $v$ is close to $c$ and $v_s$ is big enough too, we could get a situation where the centre of mass moves at subluminal speed (this is the group velocity, I suppose), but the front[2] of the function moves at superluminal speed.

If I measure either the particle's position after some time $T$ or time when it reaches some point $A$ i will mostly and averagely conclude that it has travelled with speed $v$. But in some (less probable) cases it will seem that it has travelled faster, even FTL. Is this a normal thing in quantum mechanics or I understand it all wrong? Maybe there is some constraint not only on $v$, but also on $v+v_s$?

A little background

When talking about superluminal tunneling speeds I usually hear explanations like this:

That speed is apparent. In those cases only some frontal part of the wave function gets through the barrier. Although it's center of mass seems to have travelled FTL, that function would still stay under the initial wave function if it continued it's way without the barrier. The center just shifted because of dropping the rear part.

I have never actually understood why does it solve the problem, because the position of center doesn't change the fact that signal in some cases may arrive FTL. This is where my question comes from.

[1] If you say that I had some chance to measure it at any point in the very beginning and that it wasn't entirely localized, we can replace the Gauss' function with a square or triangle one that gets wider along the way.

[2] If you need, we can define the front of the wave, for example first of the points where second derivative is zero. It's position is $vt+\sqrt{\frac{a+v_st}{2}}$.

Let's suppose I initially have a particle with a nice and narrow wave function[1] (I will leave these unnormed): $$e^{-\frac{x^2}{a}}$$ where $a$ is some small number (to make it narrow). Let's also suppose that the wave function travels along x axis with constant speed $v$ and it smears out with constant speed $v_s$. So after time $t$ wave function looks like $$e^{-\frac{(x-vt)^2}{a+v_st}}$$

Now, if speed $v$ is close to $c$ and $v_s$ is big enough too, we could get a situation where the centre of mass moves at subluminal speed (this is the group velocity, I suppose), but the front[2] of the function moves at superluminal speed.

If I measure either the particle's position after some time $T$ or time when it reaches some point $A$ i will mostly and averagely conclude that it has travelled with speed $v$. But in some (less probable) cases it will seem that it has travelled faster, even FTL. Is this a normal thing in quantum mechanics or I understand it all wrong? Maybe there is some constraint not only on $v$, but also on $v+v_s$?

A little background

When talking about superluminal tunneling speeds I usually hear explanations like this:

That speed is apparent. In those cases only some frontal part of the wave function gets through the barrier. Although it's center of mass seems to have travelled FTL, that function would still stay under the initial wave function if it continued it's way without the barrier. The center just shifted because of dropping the rear part.

I have never actually understood why does it solve the problem, because the position of center doesn't change the fact that signal in some cases may arrive FTL. This is where my question comes from.

EDIT: Sorry, the question was not about what some classic QM models allow, but actual quantum physics.

Question reformulated

Is it possible to send a particle that's slowly spreads and with some small chance (in some rare cases) measure it arriving FTL? (provided that the mean arrival value still stays under c).

I am aware that this is possible when tunneling through a barrier (and the mean value can even move to superluminal by train dropping cars), but is the same possible without any barrier?

[1] If you say that I had some chance to measure it at any point in the very beginning and that it wasn't entirely localized, we can replace the Gauss' function with a square or triangle one that gets wider along the way.

[2] If you need, we can define the front of the wave, for example first of the points where second derivative is zero. It's position is $vt+\sqrt{\frac{a+v_st}{2}}$.