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Classical chain of harmonic oscillators: Consider a chain of 2N atoms with massesmi=mforieven andmi =M foriodd

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Theoretical Condensed Matter Physics PD A. Komnik, Universit¨at Heidelberg, SS07

1. Set of Exercises: 02.05.07

1. Classical chain of harmonic oscillators:

Consider a chain of 2N atoms with massesmi=mforieven andmi =M foriodd. They are connected to each other by springs with elastic constantsκ. With periodic boundary conditions, the corresponding Hamiltonian is given by

H =

2N

X

i=1

p2i 2mi

+ κ 2

2N−1

X

i=1

(xi−xi+1)2

2(x2N −x1)2 .

a) Find the 2 normal oscillation modes (optical and acoustical) and their respective dispersion relation.

b) Calculate the minimal energy gap between optical and acoustical branches. Why there is no dispersion for the optical modes, ifM m?

c) Show that the sound velocitycis given byc=ap2κ/(M+m), whereais the lattice constant. Is this result compatible with the prediction of the classical Laplace formula

c2= ∂P

∂ρ , whereP is the pressure andρ is the density? Why?

2. Chain of quantum oscillators:

Consider the quantum analogue

pi = −i¯h ∂

∂xi

of the Hamiltonian of problem1., where nowm =M. One can rewrite the Hamiltonian in terms of creation and annihilation operators

xi = s

¯ h 2ωm

ai+ai

pi = i s

¯ hmω

2

ai −ai

.

a) Show that this substitution, supplemented by the Fourier transformation an =

Z π

−π

dk

2πeiknak = X

k

akeikn ak = X

n

ane−ikn

(2)

leads to the Hamiltonian

H = X

k

p(k)akak+q(k)aka−k+ h.c (1)

with

p(k) = ¯hω 4 + ¯hκ

2mω(1−cosk) q(k) = −¯hω

4 + ¯hκ

2mω(1−cosk) .

b) Eq.(1) can be diagonalised by the so–called Bogolyubov rotation leading to a new set of Bosons

ak = coshλkbk+ sinhλkb−k

a−k = sinhλkbk+ coshλkb−k . (2) Show that this transformation conserves the commutation relations.

c) Transform Eq. (1) using (2) and identify theλkwhich diagonalises the Hamiltonian such that

H = X

k

ν(k)

bkbk+ 1 2

.

Findν(k).

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