Probability of finding a particle in a box

In summary, the conversation discusses the calculation of the probability of finding a particle in the middle half of a box using the given solution for a particle in a box. It also mentions the relationship between this probability and the classical solution of 0.5 as n approaches infinity. The speaker has integrated and obtained probabilities for n=1,2,3 but is unsure about the behavior of the probabilities as n increases. They suggest evaluating the integral for a general value of n and looking at the result as n approaches infinity. Drawing a picture may also provide insight into the expected answer.
  • #1
ahhppull
62
0

Homework Statement



Consider ψ (x) for a particle in a box:

ψn(x) = (2/L)1/2sin(n∏x/L)

Calculate the probability of finding the particle in the middle half of the box (i.e., L/4 ≤ x ≤ 3L/4). Also, using this solution show that as ''n'' goes to infinity you get the classical solution of 0.5.


Homework Equations





The Attempt at a Solution



I integrated and figured out the probability for n=1,2,3. For n=1 I got 1/2 + 1/∏ which is about 0.818. For n = 1 I got 1/2 and for n=3, I got 0.430.

I don't understand where the problem asks "Also, using this solution show that as ''n'' goes to infinity you get the classical solution of 0.5." From my calculations, as n goes to infinity, it does not approach a value of 0.5.
 
Physics news on Phys.org
  • #2
It will help if you can evaluate the integral for a general (unspecified) values of n and then look at the result as n goes to infinity.

From the 3 values you have obtained, you can't tell whether or not the probability is approaching any specific value as n gets large. (By the way, I agree with your answers for n = 1 and 2, but not for n = 3.)
 
  • #3
Drawing a picture of some of the solutions might give you some insight into what kind of answer you are looking for.
 

FAQ: Probability of finding a particle in a box

1. What is the "particle in a box" experiment?

The "particle in a box" experiment is a theoretical model used in quantum mechanics to study the behavior of a particle confined to a one-dimensional space, such as a box. It is often used as a simplified representation of more complex systems, such as atoms or molecules.

2. How is the probability of finding a particle in a box calculated?

The probability of finding a particle in a box is calculated using the Schrödinger equation, which describes the wave function of the particle. The square of the wave function, known as the probability density, gives the probability of finding the particle at a specific location within the box.

3. What is the relationship between the size of the box and the probability of finding the particle?

The size of the box has a direct impact on the probability of finding the particle. As the box becomes smaller, the probability of finding the particle at any given location increases. Conversely, as the box becomes larger, the probability of finding the particle decreases.

4. How does the energy level of the particle affect its probability of being found in the box?

The energy level of the particle is directly proportional to its probability of being found in the box. As the energy level increases, the probability of finding the particle at any given location also increases. This is because higher energy levels correspond to larger wavelengths, which increases the probability density of the particle.

5. What is the significance of the "particle in a box" experiment in quantum mechanics?

The "particle in a box" experiment is significant because it provides insight into the behavior of particles in confined spaces, which is relevant in many areas of physics, chemistry, and materials science. It also serves as a fundamental model for understanding more complex quantum systems and their properties.

Back
Top