Particle in a Box: Calculating P from 0 to 0.2 nm

In summary, the probability of locating a particle between x = 0 and x = 0.2 nm in its lowest energy state in a box of length 1.0 nm is 0.05. This was calculated using the equation Probability = ∫ψ2dx, where ψ = (2/L)1/2sin(n∏x). By plugging in the numbers n=1 and L=1 and using radians instead of degrees, the correct answer was obtained. The factor highlighted in the conversation should be (L/n∏), which has the dimensions of length.
  • #1
ahhppull
62
0

Homework Statement



What is the probability, P, of locating a particle between x = 0 (the left-hand end of
a box) and x = 0.2 nm in its lowest energy state in a box of length 1.0 nm?

Homework Equations



Probability = ∫ψ2dx
ψ = (2/L)1/2sin(n∏x)

The Attempt at a Solution



ψ2 = (2/L)sin2(n∏x)
∫(2/L)sin2(n∏x)dx = 2/L [x/2 - (L/n∏x)sin(2n∏x/L)] from x = 0 to x = 0.2

I plugged in the numbers n=1 and L = 1 and got about 0.2.
The answer is 0.05.
 
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  • #2
ahhppull said:
ψ2 = (2/L)sin2(n∏x)
∫(2/L)sin2(n∏x)dx = 2/L [x/2 - (L/n∏x)sin(2n∏x/L)] from x = 0 to x = 0.2

Check the factor highlighted above. Note that this factor should have the dimensions of length.
 
  • #3
Isn't this advanced physics?
 
  • #4
lep11 said:
Isn't this advanced physics?

Not necessarily. The particle in a box is often covered in the introductory calculus-based physics course (usually in the 3rd semester of the course in the U.S.).
 
  • #5
TSny said:
Check the factor highlighted above. Note that this factor should have the dimensions of length.

I still don't understand.

I may have wrote something wrong.
The part that you highlighted should be: (L/n∏)
 
  • #6
I got the answer now, but by changing my calculator to radians when calculating sin(2n∏x/L). Am I suppose to use radian instead of degrees?
 
  • #7
Yes you are... you are working with numbers here, not with angles... in this case the sine is just a function, and want an adimensional argument. While radians are a conventional unit for angles but are not a real units (you call radians to understand that you are speaking of angles but it is still a pure number), degrees are indeed an unit, so you can't use them here
 
  • #8
Yes, well done, you're right, radians always for this sort of thing. Took me ages to get used to that.
 
  • #9
tia89 said:
Yes you are... you are working with numbers here, not with angles... in this case the sine is just a function, and want an adimensional argument. While radians are a conventional unit for angles but are not a real units (you call radians to understand that you are speaking of angles but it is still a pure number), degrees are indeed an unit, so you can't use them here

Ok...Thanks!
 

FAQ: Particle in a Box: Calculating P from 0 to 0.2 nm

1. What is a "Particle in a Box"?

A "Particle in a Box" is a simplified model used in quantum mechanics to study the behavior of a particle confined within a specific boundary. In this model, the particle is treated as a wave and its energy levels can be calculated based on the size of the box.

2. What is the significance of calculating P from 0 to 0.2 nm?

The value of P, also known as the probability density function, represents the probability of finding the particle at a specific location within the box. By calculating P from 0 to 0.2 nm, we can gain insight into the behavior and properties of the particle within this specific range.

3. How is P calculated in the "Particle in a Box" model?

In the "Particle in a Box" model, P is calculated using the Schrödinger equation, which describes the wave-like behavior of particles. The equation takes into account the size of the box, the mass of the particle, and the energy levels of the particle.

4. What are the applications of studying "Particle in a Box"?

The "Particle in a Box" model has various applications in different fields of science, such as chemistry, physics, and material science. It can be used to understand the electronic structure of atoms and molecules, as well as the behavior of electrons in semiconductors and other materials.

5. How does the size of the box affect the energy levels and P in the "Particle in a Box" model?

The size of the box directly affects the energy levels and P in the "Particle in a Box" model. As the size of the box decreases, the energy levels become more closely spaced and the value of P increases, indicating a higher probability of finding the particle within a smaller range.

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