PH427PH427are are periodicperiodic oscillations ubiquitous oscillations ubiquitous

or merely just a paradigm?or merely just a paradigm?

Paradigm: Periodic Systems

Instructor: Matt GrahamTA: Rabindra Bajracharya

Winter 2015

PH427 …smaller goal:

full mathematical description of physically & eletronically

coupled “stuff”

BIG GOAL: show how symmetry, oscillations and quantum mechanics really describe “everyday stuff”

35% problem sets (3)

15% pick a “solid state physics” journal article, give a 10 min. talk

50% final exam (M of exam week)

Roadmap

DAYS 1- 7: •Coupled pendulum, railroad cars, atoms, etc •Coupled LC resonators (HW #1)•Atoms is crystals extend coupling to infinity and define a dispersion relation for atomic system

DAYS 7- 15: •Quantum wavestates in periodic systems•10 minute journal presentations

Translational Symmetry and Noether’s Theorem

Graphene

Any system with translational symmetry has an associated momentum conservation law.

An electron moving through a perfectly periodic crystal maintains its momentum like the electron was travelling though a vaccuum

Graphene

Draw the “Unit Cell”

Graphene

Draw the “Unit Cell”

Draw the “Unit Cell”

Celtic knot Protein/DNA

Draw the “Unit Cell”

Celtic knot Protein/DNA

PERIODIC SYSTEM IN BIOLOGY:Light Harvesting Complex II

Bacterial Light Harvesting

Bahatyrova, et al.Nature (2004) 430 1058

Hu, et al. J. Phys. Chem. B (1997) 101 3854

“Beats” in Motion of Coupled Oscillator

2sin ab

2sin ab

1

2

Chain of Many (N) Spring-Coupled Masses

m m

m n n-1 n+1

Two Masses Connected to Hooke’s Law Springs

m1 m2

Positive direction

Force on m1: 121211 F

Force on m2: 211222 F

Low frequency mode: mlow

Symmetric mode: A1 = A2

High frequency mode:mhigh

122

Antisymmetric mode: A1 = A2

From Fig. 8.3, I. G. Main,Vibrations and Waves inPhysics

symmetric mode (low frequency)

anti-symmetric mode (high frequency)

5 Mass Chain – Mode n=1 = 12a k = 2 6a

5 Mass Chain – Mode n=2 = 6a k2 = 2 3a

2k

5 Mass Chain – Mode n=3 = 4a k3 = 2 2a

3k

5 Mass Chain – Mode n=4 = 3a k4 = 2

3a4k

5 Mass Chain – Mode 5 = 12a/5 k5 =

2a5k

5 Mass Chain – Mode 6 = 2a k6 = 2a

6k

5 Mass Chain – Mode 7 = 12a/7 k7 =

2a7k

5 Mass Chain Dispersion Relation

M m

m n n-1 n+1

M M

a

Diatomic chain, 16 masses, 8 unit cells

Dispersion Relation for m =1, M = 2

0.2

0.4

0.6

0.8

1

1.2

-2 0 2

22

411112

ak

MmmMmMopt sin

22

411112

ak

MmmMmMac sin

Dispersion Relation for m = 1, M = 1.1

0.2

0.4

0.6

0.8

1

1.2

-2 0 2k

opt

opt

Amplitude (m)/Amplitude (M) (m = 1, M = 2)

-4

-2

0

2

4

-2 0 2

opt

ac

Amplitude (m)/Amplitude (M) (m = 1, M = 1.1)

-4

-2

0

2

4

a

-2 0 2k

opt

ac

Damped Wave in Forbidden Frequency Range(Evanescent Wave)

maxmax

maxcosh

12n

n e