Probability density of a particle

In summary, the classical probability density describing a particle in an infinite square well of dimension L is P(x) = 1/L. This is based on the assumption that the particle bounces back and forth with constant kinetic energy and at constant speed, making it equally likely to be found at any location in the well at any given time. Additionally, the probability density function must be a constant and have an integral of 1 over the interval 0 to L. This shows that the uniform density of 1/L satisfies the criteria for being a probability density function. Furthermore, the symmetry of the potential inside the well supports this choice.
  • #1
gnome
1,041
1
I want to "show that the classical probability density describing a particle in an infinite square well of dimension L is P(x) = 1/L."

I know that classically, the particle bounces back and forth with constant kinetic energy and at constant speed, so at any given time it is equally likely to be found at any location in the well. It seems intuitively obvious that the probability that the particle will be between 0 and .25L, for example, at any particular moment, would be 1/4. Between .25L and .75L the probability must be 1/2.

But how do I show formally that the probability density function is 1/L?
 
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  • #2
gnome said:
But how do I show formally that the probability density function is 1/L?
I don't know what you mean by formally, but that won't stop me from stating the obvious. You know that the probability density must be a constant. And that its integral from 0 to L must equal 1. QED. :smile:
 
  • #3
I guess what I had in mind was something along the lines of the step-by-step proofs we've been doing in analysis of algorithms and AI (propositional & predicate logic).

The trouble was that I had only a vague notion of what a probability density function is, so I didn't know exactly what to work towards. I finally found this:

"A probability density function is a function defined on a continuous interval so that the area under the curve (and above the x-axis) described by the function is unity (meaning equal to one ( = 1))."

So, given that and your comment, is it correct to say that if one can show that the area under f(x) = 1/L from 0 to L equals 1, then it is true that f(x) is a probability density function for x in the region 0 - L?
 
  • #4
gnome said:
I guess what I had in mind was something along the lines of the step-by-step proofs we've been doing in analysis of algorithms and AI (propositional & predicate logic).
I won't be any help to you there! :eek:
...
So, given that and your comment, is it correct to say that if one can show that the area under f(x) = 1/L from 0 to L equals 1, then it is true that f(x) is a probability density function for x in the region 0 - L?
Yes, you would have shown that it meets the criteria for being a probability density function. If that's all you need to show. But that won't show that it's the correct probability density funtion. (That has to be done based on physical arguments, I would think. Our physical assumption is that the density function is a constant.)
 
  • #5
There may be a "physical" argument for a uniform density. Here's my attempt : Inside the well, the potential is a constant, so the probability density must have translational invariance inside the well. In other words, there's nothing that makes any spot inside the well special, so all points must be treated identically. So, symmetry requires that the density be constant.
 

What is the probability density of a particle?

The probability density of a particle is a mathematical concept that describes the likelihood of finding a particle at a certain position in space. It is represented by the symbol ρ and is a function of position. It is used in quantum mechanics to describe the behavior of particles on a microscopic scale.

How is the probability density of a particle calculated?

The probability density of a particle is calculated by taking the square of the particle's wave function, which is a complex-valued function that describes the probability amplitude of the particle at a given position in space. This calculation is done using mathematical equations and is a fundamental part of quantum mechanics.

What is the significance of the probability density of a particle?

The probability density of a particle is significant because it helps us understand the behavior of particles at the quantum level. It allows us to make predictions about the position and movement of particles, and helps us describe the wave-like properties of particles. It is crucial in understanding the fundamental principles of quantum mechanics.

How does the probability density of a particle change over time?

The probability density of a particle can change over time as the particle moves and interacts with its environment. This change is described by the Schrödinger equation, which is a mathematical equation that governs the time evolution of a particle's wave function. As the particle moves, its probability density will change accordingly.

Can the probability density of a particle be measured?

Yes, the probability density of a particle can be measured using various experimental techniques, such as electron microscopy or quantum tunneling. These measurements can provide information about the position and behavior of particles on a microscopic scale, and can be used to verify the predictions made by quantum mechanics.

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