16.3 Oscillator Systems
205
and thus its coordinate representation
= exp
−
1
2
|α|
2
∞
n=0
α n
√
n!
(16.19)
with = ψ n (x) the harmonic oscillator eigenstate.
The SPECFUNPHYS function cohstates returns the expansion coefficients
of Eq. (16.18), the coherent state in coordinate representation Eq. (16.19)
and visualizes the absolute value. The syntax is [cohs, cohsx] =
cohstates(alpha, nmax, x) with the input argument “alpha” (coherent
states eigenvalue), “nmax” (number of oscillator eigenstates, optional default 100),
and “x” (optional, coordinate interval). The maximum x 0 of is given by
x 0 =
√
2|α| cos( (α)) and therefore the default values for “x” are x 0 − 5 · · · x 0 + 5.
In the Schrödinger picture it is straightforward to show that with
α(t) = exp(iωt)α(0) → ˆ
a|α(t) = α(t)|α(t)
(16.20)
the coherent states are oscillating similar to the corresponding classical system.
The SPECFUNPHYS function cohstates_animation(alpha) ( ¯
hω = 1)
uncovers this similarity. The coordinate resolution becomes insufficient for large
absolute values of α > 2π. The program is also of interest from a MATLAB
programming point of view, therefore I discuss a few aspects of this program:
function cohstates_animation(alpha)
% animation of coherent states
% alpha coherent state value
% example alpha = 1.05 + i * 0.75
%
cohstates_animation(alpha)
nmax = 100;
% # of oscillator eigenstates
nf = 1:nmax;
t = linspace(0,2 * pi,100);
% time
at = alpha * exp(i * t);
% time dependence
x0 = sqrt(2) * abs(alpha). * cos(angle(alpha));
xl = 5 + abs(imag(alpha));
% coordinate resolution:
x = linspace(-xl-abs(x0),xl+abs(x0),500);
obj = harmonoscwave(nmax,x); % harmonic osc. states
z = 0 * x; % -> will be needed for visualisation
ng = [1,nf];
k = 0;
% counter for frames
figure, shg
al = at(1);
% first frame to create surface object
alp = al * ones(size(nf));
alp = [1,alp];
cohs = alp./sqrt(ng);
cohs = cumprod(cohs);
cohs = exp(-1/2 * abs(alpha).^2). * cohs;
% coh. s.
%
Fock representation
205
and thus its coordinate representation
= exp
−
1
2
|α|
2
∞
n=0
α n
√
n!
(16.19)
with = ψ n (x) the harmonic oscillator eigenstate.
The SPECFUNPHYS function cohstates returns the expansion coefficients
of Eq. (16.18), the coherent state in coordinate representation Eq. (16.19)
and visualizes the absolute value. The syntax is [cohs, cohsx] =
cohstates(alpha, nmax, x) with the input argument “alpha” (coherent
states eigenvalue), “nmax” (number of oscillator eigenstates, optional default 100),
and “x” (optional, coordinate interval). The maximum x 0 of is given by
x 0 =
√
2|α| cos( (α)) and therefore the default values for “x” are x 0 − 5 · · · x 0 + 5.
In the Schrödinger picture it is straightforward to show that with
α(t) = exp(iωt)α(0) → ˆ
a|α(t) = α(t)|α(t)
(16.20)
the coherent states are oscillating similar to the corresponding classical system.
The SPECFUNPHYS function cohstates_animation(alpha) ( ¯
hω = 1)
uncovers this similarity. The coordinate resolution becomes insufficient for large
absolute values of α > 2π. The program is also of interest from a MATLAB
programming point of view, therefore I discuss a few aspects of this program:
function cohstates_animation(alpha)
% animation of coherent states
% alpha coherent state value
% example alpha = 1.05 + i * 0.75
%
cohstates_animation(alpha)
nmax = 100;
% # of oscillator eigenstates
nf = 1:nmax;
t = linspace(0,2 * pi,100);
% time
at = alpha * exp(i * t);
% time dependence
x0 = sqrt(2) * abs(alpha). * cos(angle(alpha));
xl = 5 + abs(imag(alpha));
% coordinate resolution:
x = linspace(-xl-abs(x0),xl+abs(x0),500);
obj = harmonoscwave(nmax,x); % harmonic osc. states
z = 0 * x; % -> will be needed for visualisation
ng = [1,nf];
k = 0;
% counter for frames
figure, shg
al = at(1);
% first frame to create surface object
alp = al * ones(size(nf));
alp = [1,alp];
cohs = alp./sqrt(ng);
cohs = cumprod(cohs);
cohs = exp(-1/2 * abs(alpha).^2). * cohs;
% coh. s.
%
Fock representation