Today: Finite box wavefunctions

Transcription

1 Toay: Finite bo wavefunctions 1.Think outsie the bo!.solving the S.E. 3.Unerstaning the results. HWK1 ue We. 1AM. Reaing for Monay.: TZ&D Chap. 8

2 Reaing Quiz Classically forbien regions are where A. a particle s total energy is less than its kinetic energy B. a particle s total energy is greater than its kinetic energy C. a particle s total energy is less than its potential energy D. a particle s total energy is greater than its potential energy E. None of the above. Answer is c.

3 Ψ(, t) = L nπ sin( ) e L iet / n= Quantize: k=nπ/l π ml E = n = Quantize: 1 n E What you epect classically: What you get quantum mech.: Electron can have any energy Lowest energy in the bo has zero KE. Particle at rest. Electron can only have specific energies. (quantize) Lowest energy in the bo still has KE! ZERO POINT MOTION Aitional new physics appears when the bo sies are not infinitely high (all real cases!).

4 V() Goo Approimation: Electrons never got out of (< or >L) =. (OK when << work function) L What happens if electron bigger? Eact Potential curve (V): small chance electrons get out of (< or >L)~, but not eactly! What if two s very close to each other? E_total Then whether leaks out a little or not, is very important. How much coupling to other?

5 Nee to solve for eact Potential curve: V(): small chance electrons get out of (< or >L)~, but not eactly! V() L Important for thinking about Quantum tunneling : Raioactive ecay Scanning tunneling microscope to stuy surfaces Bining of molecules Behavior or soli state materials really important ieas. Finite Square Well Work function

6 In Region II, when the wavefunction is negative, it: (A) curves up. (B) curves own. (C) can curve either way (D) Not enough info 4.7eV V() Nee to solve Schroinger Eqn: m + V = E L Region I Region II Region III In Region II total energy E > potential energy V m = ( V E) = k Negative number k is real

7 4.7eV V() Nee to solve Schroinger Eqn: m + V = E L Region I Region II Region III In Region II total energy E > potential energy V m = ( V E) = k Negative number When E>V: Solutions = sin(k), cos(k), e ik. Always epect sinusoial functions k is real

8 4.7eV V() Nee to solve Schroinger Eqn: m + V = E Region II L Region II Region III In Region I an III total energy E < potential energy V m = ( V E) Positive = α α is real What functional forms of () work? a. e iα b. sin(α) c. e α. more than one of these

9 4.7eV V() L Region I Region II Region III III = Ae α + Be α In Region III total energy E < potential energy V m = ( V E) Positive Answer is C: e α coul also be e -α. Eponential ecay or growth = α α is real Why not e iα? LHS RHS α α

10 Back to case of with workfunction of 4.7 ev 4.7 ev = V E particle ev L Positive number m = ( E V ) = α Answer is c coul also be e -α. Eponential ecay or growth = Ae α + Be α > < > < (curves upwar) (curves ownwar)

11 4.7eV V() = I L Region I Region II Region III Ee α + Fe α = C sin( k) D cos( k) II + III = Ae α + Be What will wave function in Region III look like? What makes sense for constants A an B? a. A must be b. B must be c. A an B must be equal. A= an B= e. A an B can be anything, nee more info. α

12 What will wave function in Region III look like? What makes sense for constants A an B? Answer is a. A must be.. otherwise blows up as gets bigger. This oesn t make sense! an probability shoul go to at large! Nee to be able to normalize 4.7eV V() L Region I Region II Region III I = Ee α + Fe α = C sin( k) D cos( k) II + III = Ae α + Be α

13 Outsie well (E<V): (Region I) On HW net week 4.7 ev Insie well (E>V): (Region II) II = k II = C sin( k) D cos( k) II + Outsie well (E<V): (Region III) III = α III = Be III α V= ev L Bounary Conitions: ( L) = continuous ( L) = continuous ( L) = ( L) II III ( L) III ( L = II ) as

14 m = ( V E) 4.7 ev Insie well (E>V): Outsie well (E<V): V= ev L Electron is elocalize sprea out. Some small part of wave is where Total is less than potential energy! Classically forbien regions. The particle tunnels into the forbien regions. TUNNELING is a new quantum effect.

15 L If very very long gets closer an closer, what will happen? a. electron is share between s, with fraction in each constant over time b. the electron will flow away through c. electron will jump back an forth between 1 an. electron stays in 1. e. something else happens.

16 L wi If very very long gets closer an closer, what will happen? a. electron is share between s, with fraction in each constant over time b. the electron will flow away through c. electron will jump back an forth between 1 an. electron stays in 1. e. something else happens.

17 L If gets closer an closer, what will happen? a. electron is share between s, with fraction in each constant over time b. the electron will flow away through c. electron will jump back an forth between 1 an. electron stays in 1. e. something else happens.

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