handout20 - Handout 20 Quantization of Lattice Waves: From...

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1 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Handout 20 Quantization of Lattice Waves: From Lattice Waves to Phonons In this lecture you will learn: • Simple harmonic oscillator in quantum mechanics • Classical and quantum descriptions of lattice wave modes • Phonons – what are they? ECE 407 – Spring 2009 – Farhan Rana – Cornell University Quantum Simple Harmonic Oscillator Review - I 2 2 2 ˆ 2 1 2 ˆ ˆ x m m p H o x ω + = x PE () 2 2 2 ˆ 2 1 ˆ PE 2 ˆ KE x m x V m p o x = = = Consider a particle of mass m in a parabolic potential The quantum mechanical commutation relations are: [] h i p x x = ˆ , ˆ Define two new operators: x o o p m i x m a ˆ 2 1 ˆ 2 ˆ h h + = x o o p m i x m a ˆ 2 1 ˆ 2 ˆ h h = + Hamiltonian operator:
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2 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Quantum Simple Harmonic Oscillator Review - II x o o p m i x m a ˆ 2 1 ˆ 2 ˆ ω h h + = x o o p m i x m a ˆ 2 1 ˆ 2 ˆ h h = + The quantum mechanical commutation relations are: [] [ ] 1 ˆ , ˆ ˆ , ˆ = = + a a i p x x h The Hamiltonian operator can be written as: + = + = + 2 1 ˆ ˆ ˆ 2 1 2 ˆ ˆ 2 2 2 a a x m m p H o o x h The Hamiltonian operator has eigenstates that satisfy: n { ... .......... 3 , 2 , 1 , 0 ˆ ˆ = = + n n n n a a n n n a a n H o o + = + = + 2 1 2 1 ˆ ˆ ˆ h h ECE 407 – Spring 2009 – Farhan Rana – Cornell University Lattice Waves in a 1D Crystal: Classical Description A1D lattice of N atoms: x a a ˆ 1 = r x 1 a n R n r r = () ( ) ( ) ( ) ∑∑ = + = kj k j k j k j k j EQ t R u t R u R R K t R u t R u R R K V V , , , 2 1 , , , 2 1 r r r r r r r r Potential Energy: = j j dt t R du M 2 , 2 KE r Kinetic Energy: ()() EQ k j k j R u R u V R R K r r r r = 2 , Choose the zero of energy so the constant term V EQ goes away
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3 ECE 407 – Spring 2009 – Farhan Rana – Cornell University A1D lattice of N atoms: x a a ˆ 1 = r x 1 a n R n r r = () ( ) ( ) ∑∑ = kj k j k j t R u t R u R R K V , , , 2 1 r r r r Potential Energy: ( ) k j k j k j j k R R K , 1 , 1 , 2 , δ α + = + r r Nearest-neighbor interaction is always a function of only the difference ( ) k j R R K r r , k j R R r r ( ) ( ) = k j k j t R u t R u R R K V , , 2 1 r r r r Lattice Waves in a 1D Crystal: Classical Description
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handout20 - Handout 20 Quantization of Lattice Waves: From...

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