Probability of a Particle in a box location

[tarletontexan]

Homework Statement

A particle of mass m is located in a box of length L and found to be in its ground state

A) what is the probability of finding the particle between x=0 and x=L/4
B) What is the expectation value for the position of the particle?
C)What is the expectation value <x^2>?
D)What is the kinetic energy of the particle?
E) What are the 2 possible momentum values for the particle
F)What is the expectation value for momentum
G)what is the expectation value <p^2>?

Homework Equations

Probability= the integral from 0 to L/4 of \psi squared, dx
sin^2((pi)x/L)=1-cos(2(pi)x/L)

The Attempt at a Solution

My attempt at this solution doesn't have a coefficient in front of the integral like my book does in the example it has. I don't know where that one came from and don't know what to put in front of mine for a the book has (2/L) is that like the portion of the box that is being evaluated over? All of the questions above need the question before last to answer starting with the probability. I just need some help getting started and maybe a little more along the way.

 

[fzero]

It looks like your wavefunction

\psi = \sin\frac{n\pi x}{L}

is not normalized. Something like the coefficient you mention will appear if you normalize this.

 

[tarletontexan]

so how/where would i begin normalizing that??

 

[Mindscrape]

You need \int \psi^* \psi=1[/tex]

 

[tarletontexan]

ok so if the particle is supposed to be in between 0 and L/4 then I'm going to get 4/L because it is 1/4 of the total length of the box...

 

[fzero]

Normalization is the statement that the probability of finding the particle anywhere in the box is 1. That means the normalization integral is over the whole box.

 

[Mindscrape]

That's a reasonable assumption, try the integration and see if it works out that way.