Particle in a Box: Calculate Probability at Centre

[roshan2004]

Homework Statement

A particle is moving in a one dimensional box of infinite height of width 10 Angstroms. Calculate the probability of finding the particle within an interval of 1 Angstrom at the centre of the box, when it is in its state of least energy.

Homework Equations

$$\psi _{n}=\sqrt{\frac{2}{L}}sin \frac{n\pi x}{L}$$

The Attempt at a Solution

The wave function of the particle in the ground state (n=1) is $$\psi _{1}=\sqrt{\frac{2}{L}}sin \frac{\pi x}{L}$$. Now, what should I do ?

 

[Feldoh]

How does one find the probability amplitude in QM?

 

[roshan2004]

Square of its wavefunction. I got <tex> \frac{2}{L}sin^2 \frac{\pi x}{L}</tex> Now...

 

[Feldoh]

Okay, now how do you find the probability on the interval [4 Angstroms,6 Angstroms]?

 

[roshan2004]

Why between 4 and 6 angstroms ?

 

[Feldoh]

[STRIKE]Why do you think it's between 4 and 6 angstroms?
[/STRIKE]
EDIT: Err rather I believe it should be from 4.5 to 5.5 angstroms...

 

[graphene]

probability of finding the particle between x & x+dx is $${|\Psi|}^{2}$$

probability of finding the particle between x=a and a=b is $$\int_{a}^{b}{|\Psi|}^{2}dx$$

 

[roshan2004]

What are the limits I should use for the integration?

 

[graphene]

Find a and b for "an interval of 1 Angstrom at the centre of the box".