1. PH 401 Dr. Cecilia Vogel

2. Review Outline  stationary vs non-stationary states  time dependence  energy value(s)  Gaussian approaching barrier  unbound state wavefunctions  tunneling probability

3. Recall: Step barrier   STATIONARY STATE with energy E>Vo incident from the left  Solutions to TISE: k1>k2 1< 2 sketch wavefunction

4. Recall: Tunneling   STATIONARY STATE with energy E<Vo incident from the left  Solutions to TISE: sketch wavefunction

5. Characteristics of Stationary States  When we solve the TISE,  we get stationary states  What are stationary states?  Characteristics of stationary states:  Eigenstates of energy  Has definite energy  call it En  measurement of E will yield En with 100% prob

6. Characteristics of Stationary States  Characteristics of stationary states:  Eigenstates of energy  Has definite energy  call it En  = E n   E=0

7. Characteristics of Stationary States  Time dependence is exp(-iE n t/hbar)   (x,t)=  (x)e -iE n t/hbar  Probability density =     =|  (x)| 2  does NOT depend on time  All probabilities, expectation values, uncertainties are constant, independent of time  hence “stationary”

8. Non-Stationary States  Stationary states are kinda boring  What if we want something to happen?  We need a non-stationary state  one that does NOT have a definite energy  Non-stationary states are linear combinations of stationary states of different energy

9. Non-Stationary States  Example  (x,0)=a   (x) +b   (x)  where   (x) is stationary state with energy E   where   (x) is stationary state with energy E   a is the amplitude for energy E1  probability of finding energy E1 is |a| 2  similarly for b  |a| 2 +|b| 2 =1

10. Non-Stationary States  Example  (x,0)=a   (x) +b   (x)  How does it develop with time?  there isn’t just one E for e -iE n t/hbar  each term develops according to its own energy   (x,t)=a   (x)e -iE 1 t/hbar +b   (x)e -iE 2 t/hbar

11. Non-Stationary States  Example   (x,t)=a   (x)e -iE 1 t/hbar +b   (x)e -iE 2 t/hbar  Wavefunction has time dependence, but what about probability density?  Probability density =     |a   (x)| 2 +|b   (x)| 2 +(a   (x))*b   (x) e -i(E 2 -E 1 )t/hbar +(b   (x))*a   (x) e -i(E 1 -E 2 )t/hbar  depends on time!

12. Non-Stationary States  generally, non-stationary state’s  probability density depends on time!  averages can change  can change – object moves!  can change – object accelerates!  wow!

13. Gaussian tunneling  If you combine infinitely many stationary states, you can make a Gaussian wavepacket approaching the tunneling barrier  http://www.physics.brocku.ca/faculty/S ternin/teaching/mirrors/qm/packet/wav e-map.html http://www.physics.brocku.ca/faculty/S ternin/teaching/mirrors/qm/packet/wav e-map.html  the wavepacket moves toward the barrier  the wavepacket partially reflects  partially tunnels!