100 boxes of Length L, an application of the famous Particle in A Box

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Discussion Overview

The discussion revolves around the application of quantum mechanics to a scenario involving 100 identical boxes, each containing a particle of mass m confined within a potential well. Participants explore the probability of finding the particle in a specific region of the box, particularly between x=0 and x=L/4, and how this probability relates to measurements taken simultaneously across all boxes.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant proposes calculating the probability of finding the particle in the range [0, L/4] by integrating the square of the wave function over that interval.
  • Another participant suggests that once the probability for one box is determined, the distribution for 100 boxes can be modeled using a binomial distribution.
  • Questions arise regarding the state of the particles, with some participants confirming they are in the ground state of the potential well.
  • There is a discussion about the significance of measuring the particles "at the same time," with some arguing that it does not affect the probabilities if the particles are in stationary states.
  • One participant expresses confusion about the use of the binomial distribution and seeks an intuitive understanding of why it applies to this scenario.
  • Another participant explains that each box represents a Bernoulli trial, where the outcome is either finding the particle in the specified region or not.
  • There is a request for clarification on how to determine the actual number of boxes in which the particle is expected to be found, with references to calculating expected values in a binomial distribution.
  • Some participants acknowledge that the actual number of boxes cannot be known in advance, but the expected value can be calculated using the formula for the mean of a binomial variable.

Areas of Agreement / Disagreement

Participants generally agree on the use of the binomial distribution to model the scenario, but there is ongoing debate about the implications of measuring at the same time and the interpretation of the results. The discussion remains unresolved regarding the exact number of boxes expected to contain the particle in the specified region.

Contextual Notes

Participants express uncertainty about the assumptions underlying the probability calculations and the implications of measuring particles in stationary states. There are references to the need for further clarification on the application of the binomial distribution in this context.

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TL;DR
Particle in a box problem in infinite potential well with an ensemble of 100 boxes
Suppose I have 100 identical boxes of length L and the coordinates are x=0 at one end of the box and x=L at the other end, for each of them. Each has a particle of mass m. V=0 in [0,L], while it's equal to infinity in the rest of the regions. If I make a measurement on position of the particle on all the boxes at the *same time*, in how many boxes would the particle be expected to be found between x=0 and x=L/4 ?

My Approach: Probability of finding the mass m in a single box at a specific time is the integration of |psi(x)|^2 dx from x=0 to L/4. But how can we talk about the probability of 100 such identical boxes, measured at the same time? How does the probability of finding the mass m change in more than one box? How do I actually get a *number* of boxes in which the mass m can be expected to be found in [0,L/4]?
 
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Once you have the probability, p, for one box then the probability distribution function for 100 boxes is just the binomial distribution with ##B(n=100,p=p)##
 
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PhysicsTruth said:
Each has a particle of mass m

In what state? The ground state for the given potential well?

PhysicsTruth said:
If I make a measurement on position of the particle on all the boxes at the *same time*

Why does "at the same time" matter? If the particles are in stationary states, the probabilities for measurement results don't change with time, so it doesn't matter at what time you make the measurements on each particle.
 
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PeterDonis said:
In what state? The ground state for the given potential well?
Why does "at the same time" matter? If the particles are in stationary states, the probabilities for measurement results don't change with time, so it doesn't matter at what time you make the measurements on each particle.
Yeah, in ground state.
 
Dale said:
Once you have the probability, p, for one box then the probability distribution function for 100 boxes is just the binomial distribution with ##B(n=100,p=p)##
How can you say that it's a binomial distribution, like I don't really get it intuitively why do I use a binomial distribution😅
 
PeterDonis said:
In what state? The ground state for the given potential well?
Why does "at the same time" matter? If the particles are in stationary states, the probabilities for measurement results don't change with time, so it doesn't matter at what time you make the measurements on each particle.
Yeah, same time doesn't matter, because they are the eigenstates and the wave-function collapses to a single eigen-state on measurement. Nothing more.
 
PhysicsTruth said:
How can you say that it's a binomial distribution, like I don't really get it intuitively why do I use a binomial distribution😅
A Bernoulli trial is something that either happens (success) or does not happen (failure) with probability p of it happening and probability q = 1-p of it not happening. So in this case, each box is a Bernoulli trial. Either the particle is found in L/4 or it is not, and the probability for it being in L/4 is p. That is a Bernoulli trial.

The binomial distribution describes the number of successes in n Bernoulli trials. So you have 100 boxes, each is a Bernoulli trial, therefore the number of successes in 100 boxes is a binomial.
 
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Dale said:
A Bernoulli trial is something that either happens (success) or does not happen (failure) with probability p of it happening and probability q = 1-p of it not happening. So in this case, each box is a Bernoulli trial. Either the particle is found in L/4 or it is not, and the probability for it being in L/4 is p. That is a Bernoulli trial.

The binomial distribution describes the number of successes in n Bernoulli trials. So you have 100 boxes, each is a Bernoulli trial, therefore the number of successes in 100 boxes is a binomial.
According to my meagre knowledge, I know how to calculate the probability of *n* successes in a binomial distribution, or the *atleast* or *atmost* clauses, but in this case how do we find the actual number of boxes? Can you weigh in a bit more? Thanks.
 
PhysicsTruth said:
but in this case how do we find the actual number of boxes

It's 100. You told us that.

If you mean something else, please explain what you mean.
 
  • #10
Vanadium 50 said:
It's 100. You told us that.

If you mean something else, please explain what you mean.
No I meant the no. Of boxes where the particle would be expected to be found in [0,L/4]
 
  • #11
Ok,
PhysicsTruth said:
According to my meagre knowledge, I know how to calculate the probability of *n* successes in a binomial distribution, or the *atleast* or *atmost* clauses, but in this case how do we find the actual number of boxes? Can you weigh in a bit more? Thanks.
Ok, I think I know it😅 Is it nP, where n is the number of trials and P the probability of success in each trial?
 
  • #12
PhysicsTruth said:
According to my meagre knowledge, I know how to calculate the probability of *n* successes in a binomial distribution, or the *atleast* or *atmost* clauses, but in this case how do we find the actual number of boxes? Can you weigh in a bit more? Thanks.
It is a random number, so there is no way to know in advance the actual number. The expected value (mean) for a binomial variable ##\text{mean}(B(n,p))=np##

You can find all of this information here:
https://en.wikipedia.org/wiki/Binomial_distribution
 

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