1. PH 401 Dr. Cecilia Vogel

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

3. Recall: Step barrier • STATIONARY STATE with energy E>Vo incident from the left • Solutions to TISE: k1>k2 l1<l2 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> = En • DE=0

7. Characteristics of Stationary States • Time dependence is exp(-iEnt/hbar) • Y(x,t)=y(x)e-iEnt/hbar • Probability density =Y*Y • =|y(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 Y(x,0)=ay1(x) +by2(x) • where y1(x) is stationary state with energy E1 • where y2(x) is stationary state with energy E2 • 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 Y(x,0)=ay1(x) +by2(x) • How does it develop with time? • there isn’t just one E for e-iEnt/hbar • each term develops according to its own energy • Y(x,t)=ay1(x)e-iE1t/hbar+by2(x)e-iE2t/hbar

11. Non-Stationary States • Example • Y(x,t)=ay1(x)e-iE1t/hbar+by2(x)e-iE2t/hbar • Wavefunction has time dependence, but what about probability density? • Probability density =Y*Y • |ay1(x)|2 +|by2(x)|2 +(ay1(x))*by2(x) e-i(E2-E1)t/hbar +(by2(x))*ay1(x) e-i(E1-E2)t/hbar • depends on time!

12. Non-Stationary States • generally, non-stationary state’s • probability density depends on time! • averages can change • <x> can change – object moves! • <p> 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/Sternin/teaching/mirrors/qm/packet/wave-map.html • the wavepacket moves toward the barrier • the wavepacket partially reflects • partially tunnels!