1. PH 401 Dr. Cecilia Vogel

2. Review Outline  Particle in a box  solve TISE  stationary state wavefunctions  eigenvalues  stationary vs non-stationary states  time dependence  energy value(s)  Gaussian approaching barrier

3. FINITE Square Well  Suppose a particle is in a 1- D box  with length, L  with FINITEly strong walls  The potential energy function VoVo

4. General Solution in Box  Once again, the general solution is   where

5. General Solution outside Box  Outside box is CF, the general solution is  A 2 e  x +B 2 e  x  where

6. Continuity  Is it continuous?  at boundaries?  For finite square well, need continuous first derivative at boundaries, too  Four equations, plus normalization = 5 equations to determine how many unknowns?  A, B, C, D, and…. E! E is constrained waveftn corrected 11/12/11 7:40 pm

7. Continuity  Solving continuity equations puts constraints on the energy, E  The solution gives you a transcendental equation for k and  which in turn depend on E  These equations cannot be solved for E algebraically, but can be solved graphically or numerically waveftn corrected 11/12/11 7:40 pm

8. FSW Energy Levels  For odd n  cos ftn inside well   is even ftn of x  For even n  sin ftn inside well   is odd ftn of x  To get all solutions, you must find both even and odd-n solutions  Solve for u, from u get k, from k get E solving these eqns may be easier if you use a change of variables