Infinite Square Well (Quantum Mechanics)

[ElijahRockers]

Homework Statement

An electron is trapped in an infinitely deep potential well 0.300nm in width. (a) If the electron is in its ground state, what is the probability of finding it within 0.100nm of the left-hand wall? (b) Repeat (a) for an electron in the 99th excited state (n=100). (c) Are your answers consistent with the correspondence principal?

(This question comes from Serway / Moses / Moyer Modern Physics Third edition)

Homework Equations

P(x,t)dx = \Psi^*\Psi dx

\Psi (x,t) = \psi (x) \Phi (t)

\Phi (t) = e^{-i\omega t}, \omega = \frac{E}{\hbar}

The Attempt at a Solution

Since the potential does not depend on time, I can use the time independent Schrödinger equation to solve for psi (x).

psi(x) = Asin(kx) + Bcos(kx)

E_n = \frac{n^2 \pi^2 \hbar^2}{2m(.3*10^{-9})^2}

k=\frac{\sqrt{2mE}}{\hbar} = \frac{n\pi}{L} , L=.3nmIn the regions associated with infinite potential, the eigenfunction psi = 0. Since it must be continuous along all regions, psi(0) = 0 = B.

So,

\Psi(x,t) = Asin(kx)e^{-i\omega t}

And this should be the wave function, right?

Now I could use this to get information about the particle, but I need A first. How do I find A?

 

[ElijahRockers]

normalization I suppose.