Module 6 : PHYSICS OF SEMICONDUCTOR DEVICES
Lecture 31 : Electron in a Potential Well
  Having a finite total energy the electron cannot be in the region $ x<0$ or $ x<a$. Thus the electron wavefunction in these regions vanishes. For the region $ 0\le x \le a$, the one dimensional Schrödinger equation is
$\displaystyle \frac{d^2\psi}{dx^2} + \frac{2mE}{\hbar^2}\psi = 0$
  which has a solution
 
$\displaystyle \psi(x) = A \sin kx + B\cos kx$
  where $ A$ and $ B$ are constants and the wave number $ k$ is given by
 
$\displaystyle k = \sqrt{\frac{2mE}{\hbar^2}}$
  By continuity of wavefunction at the boundaries
  $ \psi(x) =0$ at $ x=0$ which gives $ B=0$, and
 
$ \psi(x) =0$ at $ x=a$ which gives
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