QM : Transition Prob -> Decay Rate

[K.J.Healey]

Homework Statement

Its not a specific homework problem, buta general problem that a friend and I keep arguing about it.

Assume you have a system of:
N Particles.
2 Possible States, A && B
@ t=0 all N particles are in state A (such that c_a*c_a = P_a = 1)
where c_a is the coefficient for state A of a single particle.

Now the coefficient for state B is time varying. Technically it sort of goes like (1/f[t]) Sin^2 [wt], but that should be unnecessary.


If a particle is measured to be in state "B" then let's say that it is taken out of the group and the process is restarted, or it keeps going or whatever. But that particle is taken out. So N-1 particles remain.

My question is, how can I define a rate at which N is changing as a function of time.
dN/dt? How fast is the system degrading?


It seems logically I would have to come up with some way of knowing how often a measurement is taken in order to know how it evolves. (Collapsing the wavefunctions)

If I "set the removal tool on auto, with continuous monitoring", walk away for 10 minutes, is it possible to know how much smaller the group has become? Or is it only possible to know that "At some time "t" there is a probability associated with each possible value of "N" left"


Or is the rate just defined as the change in the probability as a function of time?
Like R = dP/dt = (d(c_b)^2/dt) ??

Any sort of advice would be helpful.

Thanks!

 

[Jhero]

Thank you for posting your question. I would approach this problem by first defining the system more clearly. It seems like you have a system of N particles, where each particle can exist in one of two states, A or B. At t=0, all particles are in state A.

You mention that the coefficient for state B is time-varying, but it is unclear how this coefficient is changing over time. Is it changing for all particles simultaneously, or is it changing for each particle independently? This information is important in determining the rate at which the system is degrading.

Next, you mention that when a particle is measured to be in state B, it is taken out of the group and the process is restarted. This means that the total number of particles in the system will decrease over time. In order to calculate the rate at which the system is degrading, you will need to know the frequency at which these measurements are being taken. If the measurements are taken continuously, then the rate of change of N would be dN/dt = -λN, where λ is the rate at which particles are being removed from the system.

If the measurements are not taken continuously, then the situation becomes more complicated. You will need to define the probability of a particle being measured in state B at a given time, and use that to calculate the rate of change of N.

In summary, the rate at which the system is degrading can be determined if you have a clear understanding of the system and the frequency at which measurements are being taken. I hope this helps answer your question. Best of luck in your discussions with your friend!