Quantum Phenomena question: Estimating Lifetime of Unstable Particle

[superkam]

Homework Statement

Hi, I'm fairly new to these forums and I'm stuck on a problem. Just wondered if anyone could give me any hints or tips on how to get started on the following question:
An unstable particle produced in a high-energy collision has a mass of m and an uncertainty in mass that is p of the particle's mass. Estimate the liftime of the particle.

Homework Equations

I am assuming this question has something to do with Heisenbergs Uncertainty Principle. i.e
$$\Delta x*\Delta p >= \hbar$$ where p is momentum and x is some distance
From the information given I assume I have to use the above formula to get:
$$\Delta x*(p/m)*v >= \hbar$$ ...? where p/m is the uncertainty in the mass described in the question

The Attempt at a Solution

As I said at the beginning I don't really know how to get started on this question, and thus the derived relationship above is all I have managed to come up with thus far. I am beginning to think that this question requires me to calculate the energy of the particle by calculating the minimum uncertainty on the momentum (i.e (p/m)*V). If the equation that I have formulated so far is correct, what am I supposed to use for delta x? From the information given I cannot see any way to deduce this. Any indication on whether I am on the right lines here or any other help would be very much appreciated. Thanks in advance, Kam.

 

[superkam]

Can anyone please give me an idea whether I am heading in the right direction here, or indicate to me that I need to show more working before anyone will assist me further? Thanks.

 

[collinsmark]

Can anyone please give me an idea whether I am heading in the right direction here, or indicate to me that I need to show more working before anyone will assist me further? Thanks.

Okay, I'll bite. You are correct that this is related to Heisenberg's Uncertainty Principle.

You are also correct that position and momentum are canonical conjugates under the uncertainty principle. But I don't think that position and momentum are related to this problem.

Time and Energy are also canonical conjugates under the uncertainty principle. Also, from special relativity, an uncertainty in mass is the same thing as uncertainty in energy.

I don't think "p" here (originally from the problem statement, "...uncertainty in mass that is p of the particle's mass...") refers to momentum. Rather I suspect that it refers to a probability related fraction. For example, if due to uncertainty, the particle's mass can vary by 25%, interpret the problem statement as saying "...uncertainty in mass that is 0.25 of the particle's mass..."

From that, you should have the information necessary to estimate the lifetime of the particle.

[Edit: and I suspect that you are supposed to express your answer in terms of 'p' and 'm']

 

[superkam]

Okay, I'll bite. You are correct that this is related to Heisenberg's Uncertainty Principle.

You are also correct that position and momentum are canonical conjugates under the uncertainty principle. But I don't think that position and momentum are related to this problem.

Time and Energy are also canonical conjugates under the uncertainty principle. Also, from special relativity, an uncertainty in mass is the same thing as uncertainty in energy.

I don't think "p" here (originally from the problem statement, "...uncertainty in mass that is p of the particle's mass...") refers to momentum. Rather I suspect that it refers to a probability related fraction. For example, if due to uncertainty, the particle's mass can vary by 25%, interpret the problem statement as saying "...uncertainty in mass that is 0.25 of the particle's mass..."

From that, you should have the information necessary to estimate the lifetime of the particle.

[Edit: and I suspect that you are supposed to express your answer in terms of 'p' and 'm']

Ok I will take on board what you have said and I will try and solve this problem. I did not know that the uncertainty principal could be related to time and energy. Thank you for your help :)

 

[superkam]

Ok so I now have the equation $$\Delta t \Delta E = \hbar$$
I have used the equation E = mc^2 and substituted the values into the Heisenberg equation to get:
$$\Delta t = hm/(2 \pi c^2 p)$$
However mastering physics is saying that this answer is incorrect :(

 

[ehild]

Check your formula. Is m at the proper place? ehild

 

[superkam]

Check your formula. Is m at the proper place? ehild

Hi, I used the value p/m for the mass as I assumed this would represent the ratio of the uncertainty of the mass to the actual mass of particle. When I divide through by (p/m)c^2 (In the Heisenberg equation) doesn't the m go to the top? Or have I got my relationship between p and m wrong, should the uncertainty in the mass be p*m not p/m? Thanks.

[EDIT Problem solved, thanks!]

 

[Redbelly98]

[EDIT Problem solved, thanks!]

Glad you have solved it. Here is a tip for the future:

Hi, I used the value p/m for the mass as I assumed this would represent the ratio of the uncertainty of the mass to the actual mass of particle. When I divide through by (p/m)c^2 (In the Heisenberg equation) doesn't the m go to the top? Or have I got my relationship between p and m wrong, should the uncertainty in the mass be p*m not p/m? Thanks.

Units are your friend! p/m does not have units of mass; p*m does have units of mass. (As long as you realize that p is unitless.)