Probability of a Particle in a box location

In summary, the particle is most likely to be found between 0 and L/4. The expectation value for the position of the particle is (0.693x+0.305x^2). The expectation value for momentum is (2.8x+1.4x^2).
  • #1
tarletontexan
30
0

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 [tex]\psi[/tex] 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.
 
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  • #2
It looks like your wavefunction

[tex] \psi = \sin\frac{n\pi x}{L} [/tex]

is not normalized. Something like the coefficient you mention will appear if you normalize this.
 
  • #3
so how/where would i begin normalizing that??
 
  • #4
You need [itex]\int \psi^* \psi=1[/tex]
 
  • #5
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...
 
  • #6
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.
 
  • #7
That's a reasonable assumption, try the integration and see if it works out that way.
 

1. What is the "Particle in a box" model?

The "Particle in a box" model is a simplified quantum mechanical system used to study the behavior of a particle confined within a finite space.

2. How is the probability of a particle's location in a box determined?

The probability of a particle's location in a box is determined by solving the Schrödinger equation for the given potential energy function of the box.

3. What is the significance of the probability of a particle's location in a box?

The probability of a particle's location in a box provides insight into the behavior and characteristics of the particle, such as its energy levels and allowed states within the box.

4. Can the probability of a particle's location in a box be greater than 1?

No, the probability of a particle's location in a box cannot be greater than 1. This is because the total probability of finding the particle at any location within the box must equal 1.

5. How does the size of the box affect the probability of a particle's location?

The size of the box has a direct impact on the probability of a particle's location. As the size of the box increases, the probability of finding the particle at a specific location decreases. This is because the particle has more space to move around, increasing its potential locations within the box.

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