Energy State Probability Particle in a Box

So keep at it!In summary, the conversation discusses the method for finding the probability of obtaining a specific energy in a particle in a box with a given wave function. This involves using the integral of the conjugate of the wave function and the given wave function, and solving for the probability using a polynomial and sine function. While the integral may be a bit messy, it is possible to solve and find the desired probability.
  • #1
GrantB
22
0

Homework Statement



Show that the probability of obtaining En for a particle in a box with wave function

[itex]\Psi[/itex](x) = [itex]\sqrt{\frac{30}{L^{5}}}[/itex](x)(L-x) for 0 < x < L

and [itex]\Psi[/itex](x) = 0 for everywhere else

is given by |cn|2 = 240/(n6[itex]\pi[/itex]6)[1-(-1)n]2

Homework Equations



cn = [itex]\int[/itex][itex]\psi[/itex][itex]^{*}_{n}[/itex][itex]\Psi[/itex](x)dx

The probability is cn squared.

Shouldn't have to use eigenvalues and eigenfunctions.

The Attempt at a Solution



I used the integral from (2) and used the given uppercase Psi and used the sqrt(2/L)sin(n*pi*x/L) lowercase psi (conjugate), from 0 to L.

The integral quickly turned messy with integration by parts and such.

I would like to know if I am on the right track here... If I am, I will just work through the integral until I get the right answer.

I'm hoping there is a much easier way to do this.

Thanks!

P.S. Sorry this is a repost from another section. I wasn't getting any responses from the Intro Physics section so thought I would try here.
 
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  • #2
Yep, you're on the right track. Yes, the integral takes a little bit of work, but it is of the form
[tex]\int(\text{polynomial in }u)\sin u\;\mathrm{d}u[/tex]
and those integrals are doable.
 

1. What is the "Particle in a Box" model used to study?

The "Particle in a Box" model is a simplified representation used to study the behavior of a particle confined within a potential well. It is commonly used in quantum mechanics to understand the properties of electrons in atoms or molecules.

2. What is the Energy State Probability in the Particle in a Box model?

The Energy State Probability refers to the likelihood of finding the particle in a particular energy state within the potential well. This probability is determined by solving the Schrödinger equation for the particle in the box system.

3. How does the size of the box affect the Energy State Probability?

The size of the box has a direct influence on the Energy State Probability. As the size of the box decreases, the energy levels become more closely spaced and the probability of finding the particle in a higher energy state increases.

4. What is the significance of the Energy State Probabilities in the Particle in a Box model?

The Energy State Probabilities provide information about the distribution of the particle within the potential well. They also help in understanding the stability and behavior of the system, as well as predicting the possible outcomes of experiments.

5. Can the Particle in a Box model be applied to real-life situations?

While the "Particle in a Box" model is a simplified representation, it can be applied to real-life situations such as the behavior of electrons in semiconductors or the vibrational modes of molecules. However, it is important to note that the model has limitations and may not accurately represent all physical systems.

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