16.3 Oscillator Systems
209
[xm, E0] = xoscmatrix(ndim);% x-matrix:xm
...
sinx = funm(pi * xm, @sin);
% matrix operation
Hm = E0 + sinx/4;
% matrix for eigensolution
[V, D] = eig(Hm);
% eigensolution - unsorted
En2 = diag(D);
[En, enind] = sort(En2);
% eigensolution - sorted
Vn = V(:,enind);
obj = harmonoscwave(ndim-1);% harm. osc. wave functions
x = obj.z;
wavefun = obj.wave;
wavefun = wavefun.’;
Vvisu = V(:,1:nmax);
wellfunah = wavefun * Vvisu; % Matrixmultiplikation
...
2nd Example:
The Hamiltonian of the quartic anharmonic oscillator reads
ˆ
H = ¯
hω
( ˆ
a
†
ˆ
a +
1
2
) +
C
¯
hω
¯
h
2mω
3/2
1
8
√
2κ
( ˆ
a
†
+ ˆ
a)
3
+ D
¯
h
4m 2 ω 3
1
8 λ
( ˆ
a
†
+ ˆ
a)
4
,
(16.25)
with 1/2κ and 1/2λ are the prefactors to x 3 , respectively, x 4 for ¯
h = 1 = m.
ˆ
H = ˆ
H 0 +
1
2
κ x
3
+
1
2
λ x
4 λ ≥ 0.
(16.26)
In first-order perturbation theory we get for the energy
E n ≈ ¯
hω
n +
1
2
+
3
8
λ(2n
2
+ 2n + 1)
,
(16.27)
and because of ˆ
a † + ˆ
a) 3 |n = 0 the energy does not undergo an alteration due to
the cubic potential in first order. For even potentials V (x) = −V (x) eigenfunctions
of even and odd parity do not mix and the computation can be optimized by
projecting on the even, respectively, odd subspace. For vanishing fourth order term
the potential curve will undergo a maximum so that the bound states die out at
higher energy. Strictly speaking even the low lying states become quasi bound by
coupling to the continuum via tunneling. By complex coordinate rotation [4, 6] one
could take care of the 3rd order potential term. The SPECFUNPHYS function
[E,V,testwav]=anharmosc(kappa,lambda,hmany,ndim,no,s)
approximates the eigensolutions of the anharmonic oscillator by diagonalization
with the MATLAB function eigs. “kappa” and “lambda” are the potential
parameters, “hmany” are the number of requested eigensolutions, the matrix
dimension (default 50) is given by the optional input argument “ndim”, if “no”