What is the wave function of an atom in my table?

[Ivan Seeking]

I was once taught that we can calculate a small but non-zero probability for "quantum leaps" for things like atoms. I have tried to review this question within the context of gas molecules and for solids, but alas, I suspect my proficiency ends with very simple models.

So first is this correct: In principle, we can calculate a non-zero probability that for any atom having a reasonably well defined position, a chance still exists of finding this atom at some distant location; e.g. 1:10^50 of finding the atom 1 KM away from the classical position.

If this is true, can we estimate the real probability for such events for atoms found in gases and solids? For example, is there any way to estimate the chances that a carbon atom in my desk exists at the moon for a moment?

 

[Hydr0matic]

If the wavefunction represents what you know about the system, and you know that the atom is in your desk, then it's wavefunction must be restricted to that desk. So your calculations should give a zero probability of finding the atom anywhere else than in your desk.

 

[Ivan Seeking]

Originally posted by Hydr0matic
If the wavefunction represents what you know about the system, and you know that the atom is in your desk, then it's wavefunction must be restricted to that desk. So your calculations should give a zero probability of finding the atom anywhere else than in your desk.

This seems to contradict other claims made. According to QM, I thought that some non-zero chance exists, such as 1:10^100, for entire objects to disappear and pop up somewhere else in the universe. [?]

 

[heisenberg]

Originally posted by Ivan Seeking
This seems to contradict other claims made. According to QM, I thought that some non-zero chance exists, such as 1:10^100, for entire objects to disappear and pop up somewhere else in the universe. [?]

Thats correct, there is a probability, no matter how minute that i could disappear from my keyboard at this very moment and suddenly appear on some remote planet in the andrometer galaxy.

1 sec ...



Nope I am still here .lol

Because the probability is so small, that you'd have to wait many times longer than the current age of the universe for this to occur :(

 

[Ivan Seeking]

Originally posted by Heisenberg
Thats correct, there is a probability, no matter how minute that i could disappear from my keyboard at this very moment and suddenly appear on some remote planet in the andrometer galaxy.

1 sec ...



Nope I am still here .lol

Because the probability is so small, that you'd have to wait many times longer than the current age of the universe for this to occur :(

Thanks. I was starting to worry that I had missed something really big along the way. There are two more points that I would like to clarify:

1). The chances for such an event are astronomically bad, but it does not guarantee that I must wait does it? Somewhere in the universe, and at sometime in the lasts 12 billion years, is it not likely that events like this have happened?

2). Could time factor into this kind of event? If some atom has a 1:10^50 chance of popping up somewhere far away, could this then be interpreted as a temporal relationship? For example, could I say that the chances are nearly 1:1 that this atom could spend 10^-50 seconds out of every second somewhere else? Or, assuming this is not the same question, can Heisenberg’s statement of time and energy be used to make a similar argument?

 

[heisenberg]

Originally posted by Ivan Seeking
Thanks. I was starting to worry that I had missed something really big along the way. There are two more points that I would like to clarify:

1). The chances for such an event are astronomically bad, but it does not guarantee that I must wait does it? Somewhere in the universe, and at sometime in the lasts 12 billion years, is it not likely that events like this have happened?

2). Could time factor into this kind of event? If some atom has a 1:10^50 chance of popping up somewhere far away, could this then be interpreted as a temporal relationship? For example, could I say that the chances are nearly 1:1 that this atom could spend 10^-50 seconds out of every second somewhere else? Or, assuming this is not the same question, can Heisenberg’s statement of time and energy be used to make a similar argument?

\

Oh, man. you ,made me get my calc out now.lol :P

Say you want a particle to appear 20,000,000,000 L/Seconds away.
Then an electrons probability wavelength would be, t=d/c (Assuming it travels at C (We know it cant)) = W-length / C = 1/F

t = W-length / C so tC = w-length = 5.995849*10POWER 18 m

1 light year = 3.15576*10POWER7 secs
giving 1.8999699*10POWER11 L/Years!

But since w-length = h/mv for an electron to have this huge wavelength it would have a velocity of a meger 1.2131553*10POWER-22 m/s

This is stupendiously slow, and according to special relativity could not have a suitable reference frame for the job.\

So in retrospect, I guess this could not happen

 

[jeff]

Originally posted by Hydr0matic
If the wavefunction represents what you know about the system, and you know that the atom is in your desk, then it's wavefunction must be restricted to that desk. So your calculations should give a zero probability of finding the atom anywhere else than in your desk.

This is one of the fundamental axioms of quantum mechanics, so Hydr0matic is correct.

A wavefunction describing a system, one of whose observables has been experimentally determined to have some definite value (in this case the position of an atom which has been measured to lie within a desk) but remains otherwise undisturbed, must encode this information to be properly normalized. Look it up in any text on QM.

 

[quantumdude]

I don't think anyone but Heisenberg is adressing the question in the initial post. Of course you can calculate (edit: "in principle") a nonzero probability for an atom in the desk to be found outside the desk if you use the state of the desk prior to measurement[/color]. A desk in no way is a realization of an infinite potential well. When hydr0matic says, "If you know that the atom is in your desk...", he is implicitly talking about the state of the desk after measurement[/color], which is clearly not going to be the same.

 

[Ivan Seeking]

Originally posted by Tom
I don't think anyone but Heisenberg is adressing the question in the initial post. Of course you can calculate (edit: "in principle") a nonzero probability for an atom in the desk to be found outside the desk if you use the state of the desk prior to measurement[/color]. A desk in no way is a realization of an infinite potential well. When hydr0matic says, "If you know that the atom is in your desk...", he is implicitly talking about the state of the desk after measurement[/color], which is clearly not going to be the same.

Can we make some estimate of the order of magnitude for this probability? Also, what about the second part of the question about time. Can I view this probability in terms of ΔT/T - so that we speak of percentage of time spent somewhere else rather than the odds of "being" there.

Also, I think I am making another assumption: Am I correct in my understanding that an atom in my table may on occasion go away [exist somewhere outside of the table] and then come back? Of course we assume that no measurements are made on the system.

 

[Hydr0matic]

Am I correct in my understanding that an atom in my table may on occasion go away [exist somewhere outside of the table] and then come back? Of course we assume that no measurements are made on the system.

If no measurements have been made then there is no further information about your atom than what is described by the wavefunction. One cannot say anything more about the position of this atom, so statements about where the atom has went away or come back to has no meaning, since you cannot say where it is at any point in time.

 

[Ivan Seeking]

Originally posted by Hydr0matic
If no measurements have been made then there is no further information about your atom than what is described by the wavefunction. One cannot say anything more about the position of this atom, so statements about where the atom has went away or come back to has no meaning, since you cannot say where it is at any point in time.

Doesn't this assume that we know the meaning of "measurement" when we actually don't? Even though we can't say where it actually is at any moment, since we don't know what things may collapse the wave function of the atom, couldn't other things effectively measure the atom without providing us with any information? This being so, the thing does exist somewhere. Therefore it would seem that we can speak of a real location for the atom; even if we can never know it. Or, does this very scenario make impossible the "quantum dislocation" of this atom as indicated by the original question? But if this is so, then the whole proposition seems to fail and the atom was always here.

Sorry, I am trying to formulate good questions but these ideas are very difficult to word correctly.

 

[Alexander]

If to take even not an atom but just an electron in it - probability to find it far from atom drops exponentially (=fast) with distance.

Say, for main state of H atom (radial part of wave function R(r)=(2r/a^3/2)exp(-r/a), a = 53 picometers) the probability to find electron beyond some distance distance x is about (1/2)(2x/a)²exp(-2x/a) ~ exp(-2x/a), and as one can see for x ~ 1 m it becomes of the order of ~10^-1010. So, one has to take about that many measurements in order to occasinally start finding an electron at 1 m from its atom.

Probability to find whole H atom depends strongly on the exact shape of potential (=constrains) the atom is affected by. (Say, at zero potential and at room temperature this probability is quite high: about 1-exp[(-t/r)(3kT/M)^-1/2]~1-exp(-t/0.0005sec), so in 1 msec only atom is well outside of 1m sphere).

3 eV spherical potential hole of ~2 nm diameter (typical range of interatomic binding energy and of width of hole) at room temperature (~0.025 eV) results in ~ 10^-3x109 probability to find H atom 1 m away from the hole during life time of universe.

(Note here that isolated potential holes are not realistic, in practice we usually have many holes nearby of each other (=solid state) which is equivalent to a bunch of barriers).