About probability densities

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In summary, the conversation discusses the wave function for a free particle in a one-dimensional infinite potential well, and how the probability of finding the particle on the left and right sides are equal from the wave function. The question is raised about how the particle can go from the left to the right if the probability density is 0 at the midpoint. The concept of the uncertainty principle is brought up and its application at macroscopic scales is debated. The possibility of spontaneous particle generation in intergalactic space is also discussed.
  • #1
churi55
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It is well known that the wave function for a free particle in a one-dimentional infinite potential well is just a sinusodial wave with each end being a node.

Let me suppose that the infinite square well is v=infinity at x=0,L and the coresponding wave function is sqrt(2/L)*sin(2*Pi/L).

From the wave function, the probability of finding the particle on the left side (x=0~L/2) and the right side (x=L/2~L) are exactly the same.

The question is, if at x=L/2 (midpoint) the probability density is 0, how can the particle go from the left to the right?
 
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  • #2
Firstly the particle doesn't actually move; to see it moving you put it into a position eigenstate, and furthermore, you want to integrate the probability density function from one value of x to another to get the probability of the particle being in that region.
 
  • #3
churi55 said:
It is well known that the wave function for a free particle in a one-dimentional infinite potential well is just a sinusodial wave with each end being a node.

Let me suppose that the infinite square well is v=infinity at x=0,L and the coresponding wave function is sqrt(2/L)*sin(2*Pi/L).

From the wave function, the probability of finding the particle on the left side (x=0~L/2) and the right side (x=L/2~L) are exactly the same.

The question is, if at x=L/2 (midpoint) the probability density is 0, how can the particle go from the left to the right?

Maybe you can put the 'particle' like picture away for a minute and think of it in terms of wave. Then amplitude of wave could be zero at one point but follows continuous thereafter.
 
  • #4
The wavefunction's incorrectly written.Notice that the only relevant physical quantity is the probability.

Daniel.
 
  • #5
Quantim uncertianty and spontainous particle generation

Here is an interesting thought experiment.
An astronomical observation of the redshift of very distant objects reveals that their velocity approaches the speed of light. If Hubbels interpretation is correct regarding the expanding universe the implication is that there is a relativistic horizon where Vr (Velocity of recession) is equal to c. Nothing can be observed beyond this distance and it is probably meaningless to refer to anything "outside" this horizon.
Now here is the question. Given the uncertianty principal the more accurately we know the momentum of an object the less accurately we know its location. If we "know" the recession velocity of an object to be vanishingly close to c then we know "nothing about its location." The object, once observed, can be anywhere in the universe! Could this explain the apearant spontanious production of particles in intergalactic space? If so could this be an alternative explanation to the expansion of the universe?
 
  • #6
HUP does not apply at macroscopic scales.
 
  • #7
Following from dextercioby's comments, consider a helical wavefunction...
 
  • #8
ProfChuck said:
Here is an interesting thought experiment.
An astronomical observation of the redshift of very distant objects reveals that their velocity approaches the speed of light. If Hubbels interpretation is correct regarding the expanding universe the implication is that there is a relativistic horizon where Vr (Velocity of recession) is equal to c. Nothing can be observed beyond this distance and it is probably meaningless to refer to anything "outside" this horizon.
Now here is the question. Given the uncertianty principal the more accurately we know the momentum of an object the less accurately we know its location. If we "know" the recession velocity of an object to be vanishingly close to c then we know "nothing about its location." The object, once observed, can be anywhere in the universe! Could this explain the apearant spontanious production of particles in intergalactic space? If so could this be an alternative explanation to the expansion of the universe?

Try to think on what you really measure in order to avoid such wrong deductions from the HUP.

HUP allows you to know with a fantastic precision (with a 0 error at the limit) some values, such as the position and velocity of macroscopic objects.


Chronos said:
HUP does not apply at macroscopic scales.

What do you mean?

Seratend.
 
  • #9
seratend said:
What do you mean?

It looks like he was referring to ProfChuck just to say that our ability to measure momentum of a macroscopic object is so inexact that it removes the possibility of a significant HUP effect.
 

1. What is a probability density?

A probability density is a mathematical function that describes the likelihood of a continuous random variable taking on a specific value. It is represented by a curve on a graph, where the area under the curve represents the probability of the variable falling within a certain range of values.

2. How is a probability density different from a probability distribution?

A probability density is a continuous function that represents the probability of a continuous random variable, while a probability distribution is a function that represents the probability of a discrete random variable. In other words, a probability density is used for continuous data, while a probability distribution is used for discrete data.

3. What is the difference between a probability density function and a cumulative distribution function?

A probability density function (PDF) is a function that describes the probability of a variable falling within a certain range of values. On the other hand, a cumulative distribution function (CDF) describes the probability of a variable being less than or equal to a certain value. In other words, the PDF gives the probability density at a specific point, while the CDF gives the probability of being below that point.

4. How is the mean and variance of a probability density calculated?

The mean of a probability density is calculated by finding the weighted average of all possible values of the variable, where the weights are determined by the probabilities assigned to each value. The variance is calculated by finding the average squared difference between each value and the mean, again using the probabilities as weights.

5. What are some real-world applications of probability densities?

Probability densities are used in a variety of fields, such as statistics, physics, and finance. In statistics, they are used to model and analyze continuous data, such as height or weight. In physics, they are used to describe the distribution of particles in a given space. In finance, they are used to model and predict stock prices and other financial variables.

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