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. Recall Requirements  All wavefunctions must be solution to TDSE  stationary state wavefunctions are solutions to TISE   must be continuous  d  /dx must be continuous  wherever V is finite   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   =0  TISE cannot be true for any non- zero  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 Ae ikx + 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   cos is zero for   sin 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  d  /dx must be continuous  wherever V is finite  d  /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 1.Find the expectation value of position for a particle in any stationary state of an infinite square well. 2.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 =0 and =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  = /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 =2  /k and a wave traveling left with wavelength =2  /k  like a standing wave in string. DEMO