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					Topic 5: Schrödinger Equation
• Wave equation for Photon vs. Schrödinger equation for Electron+ • Solution to Schrödinger Equation gives wave function  – 2 gives probability of finding particle in a certain region • Square Well Potentials: Infinite and Finite walls –  oscillates inside well and is zero or decaying outside well, E  n2 • Simple Harmonic Oscillator Potential (or parabolic) –  is more complex, E  n • Step Potential of Height V0 –  is always affected by a step, even if E > V0 – For E > V0,  oscillates with different k values outside/inside step. – For E < V0,  oscillates outside step and decays inside step. • Barrier Potential of Height V0 –  oscillates outside and decays inside barrier. • Expectation Values and Operators
Phys320 - Baski Page 1

Schrödinger Equation: Applications
• Now, find the eigenfunctions  and eigenvalues E of the Schrödinger Equation for a particle interacting with different potential energy shapes.



2

  ( x)
2

2m

x 2

 V ( x) ( x)  E  ( x)

• Possible potential energies V(x) include: • Infinite and Finite square wells (bound particle). • Simple Harmonic or parabolic well (bound particle). • Step edge (free particle). • Barrier (free particle).
Phys320 - Baski Page 2

Infinite Square Well Potential: Visual Solutions
Wave and Probability Solutions n=3 Energy Solutions

n(x)

n2(x)

n=2

n=1
 2 2  k2 En   n2   2m 2mL2  
2

2  nx   n x   sin  L  L 
Phys320 - Baski

Page 3

Infinite Square Well : Probability Example
An electron is in the n = 3 excited state of a 1-D infinite square well (width L). Draw the wave function and probability distribution of the electron. Solve the probability of finding the electron at x = 0.5L in a width Dx = 0.02L.
(x)

P(x)

L/2 L

x
L/2 L

x

Because Dx is so small, use P(x)Dx =  2 (x)Dx with  for infinite well potential. Pn 3 ( x)Dx   2 3 ( x)Dx  n Pn 3 (x  L )Dx  2
Phys320 - Baski

2 2  3 x  sin   Dx L  L     0.04 
Page 4

2 2  3 L   3 sin  (0.02 L)   0.04  sin 2   L  2L   2

Simple Harmonic Well Potential: Visual Solutions
Wave and Probability Solutions n=2 n(x) n2(x)
(different well widths)

Energy Solutions

n=1

n=0

1  En   n    2 
Page 5

Phys320 - Baski

Step Potential: (x) inside step
Inside Step: V(x) = Vo



"

 x   k2   x 
2

where

(x) is oscillatory for E > Vo (x) is decaying for E < Vo

 2m  k22   2   E  Vo   

Case 1
Energy

Case 2

E > Vo
(x)

E < Vo

Scattering at Step Up: http://www.kfunigraz.ac.at/imawww/vqm/pages/samples/107_06b.html Scattering at Well - wide: http://www.kfunigraz.ac.at/imawww/vqm/pages/supplementary/107S_05d.html Scattering at Well Phys320 - Baski - various: http://www.kfunigraz.ac.at/imawww/vqm/pages/supplementary/107S_05b.html 6 Page

Barrier Potential
Outside Barrier:   x   k1   x  V(x) = 0 (x) is oscillatory
"
2

where

 2m  k12   E  2 

Inside Barrier:   x   k2   x  where V(x) = Vo (x) is decaying
"
2

k2 
2

2m
2

Vo  E   0

Energy

Transmission is Non-Zero!
(x)

Te

2 k2 a

http://www.sgi.com/fun/java/john/wave-sim.html Single Barrier: Phys320 - Baski http://www.kfunigraz.ac.at/imawww/vqm/pages/samples/107_12c.html Page 7

Barrier Potential: Example Problem
Sketch the wave function (x) corresponding to a particle with energy E in the potential shown below. Explain how and why the wavelengths and amplitudes of (x) are different in regions 1 and 3.
 (x)
Region 2

E Vo
Region 1 Region 3

x

• (x) oscillates in regions 1 and 3 because E > V(x), and decays exponentially in region 2 because E < V(x). • Frequency of (x) is higher in Region 1 vs. Region 3 because kinetic energy is higher there [Ek = E - V(x)]. • Amplitude of (x) in Regions 1 and 3 depends on the initial location of the wave packet. If we assume a bound particle in Region 1, then the amplitude is higher there and decays into Region 3 (case shown above). Phys320 - Baski Page 8

Topic 6: Atomic Physics
• Hydrogen Atom: 3D Spherical Coordinates –  = (spherical harmonics)(radial) and probability density P – E, L2, Lz operators and resulting eigenvalues • Angular momenta: Orbital L and Spin S – Addition of angular momenta – Magnetic moments and Zeeman effect – Spin-orbit coupling and Stern-Gerlach (Proof of electron spin s)

• Periodic table – Relationship to quantum numbers n, l, m – Trends in radii and ionization energies
Phys320 - Baski Page 9

Group IV

Group VI

Group III

n 1 2 3 4 5 6 7

l = 0 (s)

l = 1 (p)
l = 2 (d)

l = 3 (f)

Phys320 - Baski

Group V

Page 10

Noble Gas

Alkali

Halogen

Periodic Table

Hydrogen Atom: 3D Spherical Schrödinger Equation
“Rewritten” Schrodinger Eqn.:
ˆ p2  r , , Q   Veff r , , Q   En r , , Q  2
where

ˆ p 2   2

1   2  2 r r  r r  

Eigenfunctions:

 nlm r , ,    Rn r  Ylm  ,  
Laguerre Polynomials Spherical Harmonics

Eigenvalues:

 z E0 En  n2
2

where E0 

1  ke 
2

2

 2

   13.6eV 
Page 11

Phys320 - Baski

Hydrogen Atom: 3D Spherical Schrödinger Equation
3 Quantum Numbers (3-dimensions) n = energy level value (average radius of orbit) n = 1, 2, 3 … l = angular momentum value (shape of orbit) l = 0, 1, 2, … (n – 1) m = z component of l (orientation of orbit) m = -l,(-l + 1) ..0,1,2,..+l
How many quantum states (n, l, m) exist for n = 3? Is there a general formula?
Phys320 - Baski Page 12

Probability Density: Cross Sections

http://cwx.prenhall.com/bookbind/pubbooks/giancoli3/chapter40/multiple3/deluxe-content.html

Rank the states (1s to 3d) from smallest to largest for the electron’s most PROBABLE radial position. For which state(s) do(es) the most probable value(s) of the electron's position agree with Phys320 - Baski the Bohr model? Page 13

Orbital Momentum L: Vector Diagram
For l = 2, find the magnitude of the angular momentum L and the possible m values. Draw a vector diagram showing the orientations of L with the z axis.

l2 L  L  l (l +1) 6  2(2+1)
2h h h 2h

z

L = 6h = 2.45h m=2
55º 24º

or 2.45

m=1 m=0

m  l to l  0,  1  2 Lz  m  0,  1 ,  2

m = 1 m = 2

Can you draw the vector diagram for l = 3? For j = 3/2?
Phys320 - Baski Page 14

Angular Momentum Addition: General Rules   • General Case: J1  J 2
Vectors    J tot  J1  J 2  J tot  j  j  1 Quantum Numbers
j   j1  j2 ,  j1  j2  1,... j1  j2 m j   j , j  1,...j  1, j

Example: j1 = 3/2, j2 = 3/2

jmax  3  3  3 and jmin  2 2 j  3, 2, 1, 0

3 2

 3 0 2

m j  3,  2,  1, 0,1, 2, 3 for j  3
Phys320 - Baski Page 15

“Anomalous” Zeeman Effect: Spin-Orbit + Zeeman
Spin-Orbit

j = 3/2 l = 1 s = 1/2

Zeeman mj = 3/2 mj = 1/2 mj = –1/2 mj = –3/2
mj = 1/2 mj = –1/2

j = 1/2

mj = 1/2 l = 0 s = 1/2 j = 1/2
Phys320 - Baski

mj = –1/2
Page 16

• Quantum numbers mj (j-j coupling) for HIGHER Z elements.

Topic #7: Solid State Physics
• Types of Solids – Ionic, Covalent, and Metallic. • Classical Theory of Conduction – Current density j, drift velocity vd, resistivity . • Band Theory and Band Diagrams – Energy levels of separated atoms form energy “band” when brought close together in a crystal. – Fermi Function for how to “fill” bands. – Metal, Insulator, and Semiconductor band diagrams. – Donor and Acceptor dopants (Hall Effect). • Devices – pn junction, diode, LED, solar cell, laser.
Phys320 - Baski Page 17

Classical Theory of Conduction: Resistivity vs. Temp.
• Temperature dependence of resistivity.

E e  e  m  1   J ne vd ne (a ) n ne2
• Metal: Resistance increases with Temperature.

FE

ma

• Why? Temp  , n same (same # conduction electrons)  
• Semiconductor: Resistance decreases with Temperature.

• Why? Temp  , n (“free-up” carriers to conduct)  

Phys320 - Baski

Page 18

Band Theory: “Bound” Electron Approach
• For the total number N of atoms in a solid (1023 cm–3), N energy levels split apart within a width DE. – Leads to a band of energies for each initial atomic energy level (e.g. 1s energy band for 1s energy level).

Two atoms

Six atoms

Solid of N atoms

Electrons must occupy different energies due to Phys320 - Pauli Exclusion principle. Baski

Page 19

Band Diagram: Metal
“Fill” the energy band with electrons.

EC,V

Fermi “filling” function

Energy band to be “filled”

EC,V

EF
T=0K Moderate T

EF

• At T = 0, energy levels below EF are filled with electrons, while all levels above EF are empty. • Electrons are free to move into “empty” states of conduction band with only a small electric field E, leading to high electrical conductivity! • At T > 0, electrons have a probability to be thermally “excited” from below the Fermi energy to above it.
Phys320 - Baski Page 20

Band Diagram: Semiconductor with moderate Egap
T>0
Conduction band
(Partially Filled)

EF
Valence band
(Partially Empty)

EC EV

• At T = 0, valence band is filled with electrons and conduction band is empty, leading to zero conductivity. • At T > 0, electrons thermally “excited” from valence to conduction band, leading to partially empty valence and partially filled conduction bands.

What happens to the conductivity for T > 0? How - Baski Phys320 would the band diagram look for lower & higher temperatures?
Page 21

Band Diagram: Donor Dopant in Semiconductor
• Increase the conductivity of a semiconductor by adding a small amount of another material called a dopant (instead of heating it!) • For group IV Si, add a group V element to “donate” an electron and make n-type Si (more negative electrons!) • “Extra” electrons donated from donor energy level ED just below EC.
– Resultant electrons in conduction band increase conductivity by increasing free carrier density n.

n-type Si

EC EF EV

ED

• Fermi level EF moves up because there are more carriers.

Fermi Baski Phys320 -Function & Doping: http://jas.eng.buffalo.edu/education/semicon/fermi/bandAndLevel/fermi.html Page 22

pn Junction: Band Diagram
• At equilibrium, Fermi levels (or charge carrier densities) must equalize. • Hence, electrons move from n to p-side (diffusion process). • Depletion zone occurs at junction where immobile charged ion cores remain. • Results in a built-in electric field (103 to 105 V/cm), which opposes further diffusion.
pn regions “touch” & free carriers move

EC EF EV

n-type electrons EF p-type pn regions in equilibrium

EC EF
EV

–– – ++–– – + + – + ++–– – + ++–– ++ Depletion Zone
Page 23

PIN junction: Phys320 - Baski http://jas.eng.buffalo.edu/education/pin/pin/#

Semiconductor: Dopant Density via Hall Effect
• Why Useful? Determines carrier type (electron vs. hole) and carrier density n for a semiconductor. • How? Place semiconductor into external B field, push current along one axis, and measure induced Hall voltage VH along perpendicular axis.

Carrier density n = (current I) (magnetic field B) (carrier charge q) (thickness t)(Hall voltage VH) • Derived from Lorentz equation FE (qE) = FB (qvB).

Hole
+ charge

FB  qv  B

Electron
– charge

Phys320 - Baski

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