This is the problem:

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:

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

For $t>0$, determine the probability of finding the particle between 0 and $\frac{a}{4}$.

So, using some trigonometry and the orthonormal base $\phi_{n}(x)=\sqrt{\frac{2}{a}}\sin(\frac{n\pi}{a}x)$, I can write the wave function as:


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 use: $B=\sum_{n} C_{n}\phi_{n}(x)$ where, after finding the value of $C_{n}$, and noticing that only odd values of n contributes to the wave function:

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

So now, how could I add a constant to the wave function so 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, and I’m not sure that is the way.


1 Answer 1


In a $1D$ box your wave function must vanish at $x=0,a$. In particular



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