User:Guy vandegrift/btvs/Effort 5

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Calculation.[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.
Example.svg

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. The classical calculation can be done using matrices and either the energy or the force law. We begin with the force law.

It is convenient to define a spring constant for the string, KS, based on the tension, T, and string length, 2Δℓ:

Force law[edit]

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:

System with 3 masses[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:

Constraint for symmetric modes:



Define:    , where

Algebra[edit]


Define:    , where


Match frequencies at zero coupling[edit]

If the solutions are and

matlab output
gamma =  0
M =
[ x - 2*sigma,    sigma,         0]
[     2*sigma, -2*sigma,         0]
[           0,        0, x - alpha]

deter=collect(-det(M),x)
deter = 
(2*sigma)*x^2 + (- 2*sigma^2 - 2*alpha*sigma)*x + 2*alpha*sigma^2

solutions = solve(deter) 
alpha
sigma

Henceforth

Eigenfrequencies[edit]

matlab code
clc;close all;clear all;
syms sigma beta gamma alpha y1 y2 y3 y4 x 
sigma = alpha
M=[x-2* sigma   ,  + sigma           ,    0      ;
   +2*sigma    ,   -2*sigma - gamma          ,  + gamma   ;
   0           ,  + gamma* beta    ,  x- alpha-gamma*beta;] 
deter=collect(-det(M),x)
solutions = solve(deter)

Results:
M =
 
[ x - 2*alpha,             alpha,                      0]
[     2*alpha, - 2*alpha - gamma,                  gamma]
[           0,        beta*gamma, x - alpha - beta*gamma]
 
deter =
 (2*alpha + gamma)*x^2 + (- 3*alpha*gamma - 4*alpha^2 - 2*alpha*beta*gamma)*x + 2*alpha^2*gamma + 2*alpha^3 + 2*alpha^2*beta*gamma
 solutions =
 
                                                              alpha
 (2*alpha*gamma + 2*alpha^2 + 2*alpha*beta*gamma)/(2*alpha + gamma)

Eigenvectors[edit]

Mode 1[edit]
Mode 2[edit]
algebra

From the eigenvalue equation above:

Mode 3[edit]