- #1

Gonv

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- 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.