Probability of finding a particle in a 1D box

In summary, the conversation discusses the probability of finding a particle in a one-dimensional box with a length of 1 nm between two limits: (a) x = 0 nm and x = 0.05 nm, and (b) x = 0.55 nm and x = 0.65 nm. The equation ψ = (2/L)½ sin(πx/L) is used to calculate the probability, which is ψ2. The integration of ψ2 between the given values results in a probability of 0.1796, which is deemed reasonable after cross-checking with other values.
  • #1
Lily Wright
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0

Homework Statement


If a one-dimensional box is 1 nm long, what is the probability of finding the particle between the following limits?
(a) x = 0 nm and x = 0.05 nm
(b) x = 0.55 nm and x = 0.65 nm

Homework Equations


ψ = (2/L)½ sin(πx/L)

The Attempt at a Solution


(I do chemistry and I'm really terrible at physics so apologies if this makes no sense) So that's the equation I was given. But then the probability is ψ2 which = 2/L sin2 (πx/L).
And to find the probability between x = 0.55 nm and x = 0.65 nm I need to integrate this between those values. So that's what I've done putting L = 1 (but it's in nm?)
0.650.55 ψ2(x)dx and then I get (πx−(sin(2πx)/2))π + C and then the rest is pretty difficult to type out but I basically end up with an answer of 0.1796. Any help would be much appreciated.
 
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  • #2
Lily Wright said:
So that's what I've done putting L = 1 (but it's in nm?)
You can plug in "1nm", but working in nanometers everywhere will give the correct result as well.

The answer looks reasonable. As a cross-check, you can calculate the probability between 0 nm and 0.5 nm or between 0 nm and 1 nm as those probabilities are easy to get in a direct way.
 

FAQ: Probability of finding a particle in a 1D box

1. What is the probability of finding a particle in a 1D box?

The probability of finding a particle in a 1D box is determined by the wave function of the particle, which is described by the Schrödinger equation. The probability is given by the square of the amplitude of the wave function at a specific location within the box. This means that the probability is not constant throughout the box and can be calculated for any point within the box.

2. How is the probability distribution of a particle in a 1D box determined?

The probability distribution of a particle in a 1D box is determined by the wave function of the particle, which is described by the Schrödinger equation. The square of the amplitude of the wave function at a specific location within the box gives the probability of finding the particle at that location. This distribution can be visualized as a probability density curve, where the peak of the curve represents the most probable location of the particle.

3. How does the size of the 1D box affect the probability of finding a particle?

The size of the 1D box directly affects the probability of finding a particle within it. As the size of the box decreases, the probability of finding the particle at a specific location increases. This is because the wave function is confined to a smaller space, leading to a higher probability density. Conversely, as the size of the box increases, the probability of finding the particle at a specific location decreases.

4. Can the probability of finding a particle in a 1D box be greater than 1?

No, the probability of finding a particle in a 1D box cannot be greater than 1. This is because the total probability of finding the particle within the box must be equal to 1. If the probability of finding the particle at a specific location is greater than 1, it would violate the laws of probability.

5. How does the energy level of a particle in a 1D box affect its probability of being found?

The energy level of a particle in a 1D box is directly related to its probability of being found at a specific location. As the particle's energy increases, the amplitude of its wave function also increases, leading to a higher probability of finding it at a certain location within the box. This is because the particle has a higher chance of occupying a higher energy state and therefore, a larger portion of the box.

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