Probability of finding a particle in a region

In summary: I'm trying to determine the probability of finding the particle between 0 and a4 for a t>0In summary, the author is trying to find the probability of finding the particle between 0 and a4 for a t>0.
  • #1
Gonv
4
0

Homework Statement


A particle is restrained to move in 1D between two rigid walls localized in ##x=0## and ##x=a##. For ##t=0##, it’s described by:

$$\psi(x,0) = \left[\cos^{2}\left(\frac{\pi}{a}x\right)-\cos\left(\frac{\pi}{a}x\right)\right]\sin\left(\frac{\pi}{a}x\right)+B $$

, determine the probability of finding the particle between 0 and ##\frac{a}{4}##.

Homework Equations


(1) ##\phi_{n}(x)=\sqrt{\frac{2}{a}}\sin(\frac{n\pi}{a}x)##
(2) ##B=\sum_{n} C_{n}\phi_{n}(x)##

The Attempt at a Solution



Using some trigonometry and the orthonormal base (equation 1), I can write the wave function as:

$$\psi(x,0)=\sqrt{\frac{a}{32}}\phi_{1}(x)+\sqrt{\frac{a}{8}}\phi_{2}(x)+\sqrt{\frac{a}{32}}\phi_{3}(x)+B$$

I still can’t use the evolution operator. I must find an expression to ##B##, so I can put it in terms of the base.

I used the equation 2 and after finding the value of ##C_{n}##, and noticing that only odd values of n contributes to the wave function:

$$B\rightarrow -\frac{B}{\pi}\sqrt{8a}\sum_{0}^{\infty}\frac{1}{2n+1}\phi_{2n+1}(x)$$

I can write the wave function like this:
$$\psi(x,0)=\bigg( \sqrt{\frac{a}{32}}-\frac{B}{\pi}\sqrt{8a}\bigg)\phi_{1}(x) + \sqrt{\frac{a}{8}}\phi_{2}(x) + \bigg(\sqrt{\frac{a}{32}}-\frac{B}{3\pi}\sqrt{8a}\bigg)\phi_{3}(x) - \frac{B}{\pi}\sqrt{8a}\sum_{2}^{\infty}\frac{1}{2n+1}\phi_{2n+1}(x)$$My problem is how should I continue. Am I sure this is normalized? Probably not, but how to find a constant so I can be sure it's normalized? It’s just finding the value ##B## using ##\langle\psi|\psi\rangle##? Or there is other way? Because using ##\langle\psi|\psi\rangle## I get a quadratic equation, and I’m not so sure that is this way.
 
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  • #2
What are you trying to find? Do energy eigenfunctions help you in this?
 
  • #3
Gonv said:
determine the probability of finding the particle between 0 and a4\frac{a}{4}.
To clarify, do you want this probability only at time 0 or do you want it as a function of time?
 
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  • #4
jtbell said:
To clarify, do you want this probability only at time 0 or do you want it as a function of time?

You are right, it's for t>0, I will edit it. Thank you

Edit: Well, I noticed I can't edit the original post.
 
Last edited:
  • #5
PeroK said:
What are you trying to find? Do energy eigenfunctions help you in this?

I'm trying to determine the probability of finding the particle between 0 and a/4 for a t>0
 
  • #6
Hmmm, I'm confused. You have the following initial wave function:

Gonv said:
$$\psi(x,0) = \left[\cos^{2}\left(\frac{\pi}{a}x\right)-\cos\left(\frac{\pi}{a}x\right)\right]\sin\left(\frac{\pi}{a}x\right)+B $$
When I first saw this, I thought B is a constant, maybe for normalization although it's in an odd place for a normalization constant. Then you have this:

Homework Equations


(1) ##\phi_{n}(x)=\sqrt{\frac{2}{a}}\sin(\frac{n\pi}{a}x)##
(2) ##B=\sum_{n} C_{n}\phi_{n}(x)##

OK, (1) are the energy eigenfunctions for the "particle in a box." But where does (2) come from? Is it saying that B isn't a constant after all, but instead, a function? A linear combination of all the energy eigenfunctions, with constant coefficients ##C_n## that are as yet undetermined?
 
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  • #7
jtbell said:
Hmmm, I'm confused. You have the following initial wave function:When I first saw this, I thought B is a constant, maybe for normalization although it's in an odd place for a normalization constant. Then you have this:
OK, (1) are the energy eigenfunctions for the "particle in a box." But where does (2) come from? Is it saying that B isn't a constant after all, but instead, a function? A linear combination of all the energy eigenfunctions, with constant coefficients ##C_n## that are as yet undetermined?
I understand, sorry I wasn't very clear. Yes, B it is a constant. But even if it is a known constant, when I make it act the evolution operator, I can't just make it evolve a constant. I only con aply the evolution operator to a state given by the base (1). So, what did I do so I can aply the evolution operator to the constant? I expanded in a Fourier serie, so now it's in term of the orthonormal base, basically that's the equation (2). Now it's expanded I can aply the evolution operator, and try to determine the probability asked.
I hope it is clear now.

Now I want to make sure it is normalized, and there is where I have my doubts of how to do it.
 

Related to Probability of finding a particle in a region

1. What is the probability of finding a particle in a specific region?

The probability of finding a particle in a specific region is determined by the wave function of the particle, which describes its probability distribution.

2. How is the probability of finding a particle in a region calculated?

The probability of finding a particle in a region is calculated by taking the square of the wave function at that particular point in space.

3. What factors influence the probability of finding a particle in a region?

The probability of finding a particle in a region is influenced by factors such as the shape of the potential well, the energy of the particle, and the presence of other particles or forces in the region.

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

No, the probability of finding a particle in a region cannot be greater than 1. This would violate the laws of probability, which state that the total probability of all possible outcomes must equal 1.

5. How does the probability of finding a particle in a region change over time?

The probability of finding a particle in a region can change over time as the wave function evolves according to the Schrödinger equation. This is the basis of quantum mechanics and describes the probabilistic nature of particles at the subatomic level.

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