Probability of finding a particle in a region

[Gonv]

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.

 

[PeroK]

What are you trying to find? Do energy eigenfunctions help you in this?

 

[jtbell]

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?

 

[Gonv]

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.

 

[Gonv]

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

 

[jtbell]

Hmmm, I'm confused. You have the following initial wave function:

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

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?

 

[Gonv]

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.