PH 401

1. PH 401 Dr. Cecilia Vogel

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

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

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 • A2ekx+B2e-kx • where

6. Continuity waveftn corrected 11/12/11 7:40 pm • 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

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

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