User:Guy vandegrift/btvs/Effort 3

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General case[edit]

Longitudinal modes in a lattice of springs and masses near the ground state form a quantized phonon field that interacts with the atom via a weak linear coupling spring at the center.

Place springs at ℓ = 0 ±Δℓ, ±2Δℓ, ... ±L, so that the length of the string is 2L. Often we shall count from the left, with the center at (N+1)Δℓ, so that there are 2N-1 little mass that make up the string, plus one "atomic" mass. Counting from the left:

where . For atoms not touching the wall or the coupling spring connecting to the "atom":

The N-th mass is attached to the atom, and obeys:

and the atom obeys:

Simple system[edit]


One atom. Two masses plus one massless object on the string. N=2 and 2N-1=3 masses on the string. There are 4 masses and we need 4 equations:

We don't need the third equation for modes symmetric about the atom.

Establish two nearly degenerate modes[edit]

Use the symmetry of modes that are symmetric about the atom, to obtain a matrix equation:

a bit of algebra

Divide rows 1 and 2 by MS and row 3 by MA:

,    ,    ,    ,    where , to obtain:

The result is:

where ,    ,    ,    ,    and

Finding the normal modes[edit]

Let: , , , and

The determinant equals A-B-C, where:


Calculate term A in determnant[edit]



. ..........all times alpha

......all times epsilon


Group according to power of x


Combine like terms and group with small terms last


Switch all signs and factor

Normal mode frequencies[edit]



Place the B and C stuff at end, spaced with \qquad

All the new terms combine with terms already present

Normal modes at , where:

Check with matlab[edit]

Degeneracy at zero coupling[edit]

If , the modes are decoupled, and we have,

,     ,    

WLOG[1] we may set and consider:

,     , and     ,

so that: 

if .

check with matlab[edit]

If σ=1, then we have:

Simple system σ=2, α=3= x0[edit]

Under construction. I think x=x0+x1 where x0=3 and


0 = 4*epsilon - 4*x_1 + beta*epsilon - beta*epsilon*x_1


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