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. Recall Requirements • All wavefunctions must be solution to TDSE • stationary state wavefunctions are solutions to TISE • y must be continuous • dy/dx must be continuous • wherever V is finite • y must be square integrable (normalizable) • must go to zero at +infinity

4. Discrete Energy Levels • The TISE has a solution • for every energy, E, • but most bound-state solutions are not acceptable. • Only for certain energies • will the solution obey all requirements on wavefunction. • quantized energy levels

5. Infinite “Square” Well • AKA Particle-in-a-box • Suppose a particle is in a 1-D box • with length, L • with infinitely strong walls • The potential energy function

6. Solve TISE • Outside box • y=0 • TISE cannot be true for any non-zero y where V is infinite. • Probability of finding particle outside an infinitely strong box is zero.

7. General Solution in Box • The general solution* is • where *Another general solution is Aeikx + Be-ikx , but we only need one general solution, and sin and cos are nice, ‘cause we know where they are zero

8. TISE Solution • The general solution to the TISE for infinite square well is • Are we done? • Still other requirements. • Is it square integrable? • yes

9. Continuity • Is it continuous? • at boundaries? • Only if

10. Continuity • Two equations, plus normalization = 3 equations to determine how many unknowns? • A, B, and…. E! • E is constrained • discrete energy levels for bound particle

11. Continuity Continued • Can only be true if • Don’t want both A=0 and B=0 • the particle is nowhere • Can’t have sin(kL/2)=0 and cos(kL/2)=0 • sin & cos are never both zero

12. Continuity Continued • Must be either • Or • sin is zero for • cos is zero for

13. Ground State • Ground state (n=1) wavefunction, • since • Ground state energy

14. Excited States • Odd-n wavefunctions • since • Even-n wavefunctions • since • Excited state energy

15. Final Requirement • dy/dx must be continuous • wherever V is finite • dy/dx does not need to be continuous • at the boundaries • since V is infinite

16. Normalization • A and B can be found from normalization • A=B=root(2/L)

17. PAL week 4 Friday Find the expectation value of position for a particle in any stationary state of an infinite square well. Find the expectation value of momentum for a particle in any stationary state of an infinite square well.

18. More on the ISW • PAL shows that <x>=0 and <p>=0 • for stationary state of symmetric infinite square well • In fact, it is true for all even OR odd wavefunctions • But other expectation values and uncertainties are not zero

19. ISW Kinetic Energy • <K> =<p2>/2m • also • so • and • all the energy is KE (PE =0 anywhere the particle might be)

20. ISW Momentum uncertainty • Momentum uncertainty for stationary state of ISW • also • so • and p=0+k. • The momentum of stationary state is combo of wave traveling right with wavelength l=2p/k and a wave traveling left with wavelength l=2p/k • like a standing wave in string. DEMO