Download presentation
Presentation is loading. Please wait.
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!
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.