next up [*]
Next: 4.2 A2 = 0 Up: 4. Bound states of Previous: 4. Bound states of

4.1 B2 = 0

In the first case we read off that A1 = B3, and we find that k and $ \kappa$ are related by

ka = $\displaystyle \kappa$atan$\displaystyle \kappa$a. (4.17)

This equation can be solved graphically. Use k = $ \sqrt{-\kappa^2
+\kappa_0^2}$, with $ \kappa_{0}^{2}$ = $ {\frac{2m}{\hbar^2}}$V0, and find that there is always at least one solution of this kind, no matter how small V0!

  
Figure 4.2: The graphical solution for the even states of the square well.
\includegraphics[width=10cm]{Figures/sq_well_even.ps}

In the middle region all these solutions behave like sines, and you will be asked to show that the solutions are invariant when x goes to - x. (We say that these functions are even.)



© 1998 Niels Walet, UMIST
Email Niels.Walet@umist.ac.uk