1. 1 atom many atoms ++++++++++ Potential energy of electrons in a metal Potential energy  = work function  “Finite square well.”

2. Need to solve the finite square well wire 0 Energy x 0 a V(x) 4.7 eV Work function This will be used to understand quantum tunneling which provides the basis for understanding Radioactive decay Scanning Tunneling Microscope which is used to study surfaces Binding of molecules x < 0: V(x) = 4.7 eV x > a: V(x) = 4.7 eV 0 < x < a: V(x) =0 Need to solve TISE:

3. Region II: Analyzing the finite square well wire 0 Energy x 0a V(x) 4.7 eV Region IRegion IIRegion III E particle TISE: Consider three regions We rewrite the TISE as In Region II: total energy E > potential energy V so V − E < 0 Replace with −k 2 to get(k is real) Guess form of  (x) : sin(kx) and cos(kx) or e ikx and e -ikx

4. Analyzing the finite square well wire 0 Energy x 0a V(x) 4.7 eV Region IRegion IIRegion III E particle TISE: In Region I & III: E 0 Replace with  2 to get (  is real) Which functional forms of  (x) work? A. B. C. D. E. More than one A,B,D give a minus sign so This is not what we want. Both and give us what we want Is there another solution as well?

5. wire 0 Energy x 0 a V(x) 4.7 eV Region IRegion IIRegion III E particle What will the wave function in Region III look like? What can we say about the constants C and D (assuming  >0)? A. C = 0 B. D = 0 C. C = D D. C = D = 0 E. C & D can be anything; need more information Analyzing the finite square well If C ≠ 0 then as Makes it impossible to normalize For D ≠ 0 as so it is OK.

6. wire 0 Energy x 0 a V(x) 4.7 eV Region IRegion IIRegion III E particle What will the wave function in Region I look like? What can we say about the constants E and F (assuming  >0)? A. E = 0 B. F = 0 C. E = F D. E = F = 0 E. E & F can be anything; need more information Analyzing the finite square well

7. wire Energy x 0 a V(x) 4.7 eV Region IRegion IIRegion III Matching boundary conditions Matching boundary conditions at x=0 and x=a requires:  (x) is continuous so and is continuous so and

8. wire Energy x 0 a V(x) 4.7 eV Region IRegion IIRegion III Matching boundary conditions

9. Energy x 0 a V(x) 4.7 eV Evaluating results Outside well: E < V Inside well: E > V Outside well: E < V E particle Is it possible to find the particle outside the well, i.e. x> a or x<0? A)Yes B)No Classically this is totally IMPOSSIBLE. In the classically forbidden regions, the particle has total energy less than the potential energy!

10. Comparison of infinite and finite potential wells Infinite potential well (a = 2 nm and V = ∞) Electron in finite square well (a=2 nm and V=1.0 eV)

11. Particle in finite square well can be found in classically forbidden regions Penetration depth  measures how far particle can be found in the classically forbidden region. Note that energy level n has n antinodes

12. How does the level spacing depend on the length of the wire? A)It is independent of the length of the wire B)Larger spacing for a longer wire C)Smaller spacing for a longer wire A typical macroscopic wire has extremely closely spaced energy levels. Closely spaced compared to what? Thermal energy k B T 300K ~ 25 meV

13. Quantum bound state simulation http://phet.colorado.edu/simulations/sims.php?sim=Quantum_Bound_States The quantum bound state simulation can be used to figure out and visualize wave functions and probabilities for various potential curves.