16.3 Oscillator Systems
203
The class hermitex house in addition a plot-method with syntax obj =
plot(obj, nl) and “obj” an object of the class hermitex, “nl” an optional
integer vector of the polynomial degrees which should be plotted. Default is to plot
all polynomials in quest. Example:
>> x = linspace(-3,3); obj = hermitex(3,x).plot;
>> ylim([-50,50]), shg
The SPECFUNPHYS function hermitezero returns the roots of the Hermite
polynomials. The syntax is hz = hermitezero(n) with “n” the degree of
the Hermite polynomial and “hz” a column vector with the corresponding roots.
Example:
>> hz = hermitezero(5)
hz =
-2.0202
-0.9586
0.0000
0.9586
2.0202
>> [obj, Hn] = hermitex(5, hz);
% TEST
>> Hn
Hn =
1.0000
1.0000
1.0000
1.0000
1.0000
-4.0404
-1.9171
0.0000
1.9171
4.0404
14.3246
1.6754
-2.0000
1.6754
14.3246
-41.7150
4.4565
-0.0000
-4.4565
41.7150
82.5964 -18.5964
12.0000 -18.5964
82.5964
0
0.0000
0.0000
-0.0000
0
The roots of the Hermite polynomials play an important role for estimating
quantum wave functions via discrete variable computations, see [5, 6].
16.3 Oscillator Systems
The Hamiltonian of the harmonic oscillator is given by
−
1
2
d 2
dx 2 +
1
2
x
2
− E n
ψ n (x) = 0
(16.13)
with eigenfunctions
ψ n (x) = =x|n =
1
π 1/4
√
2 n n!
exp
−
x 2
2
H n (x) ,
(16.14)
and H n (x) the Hermite polynomial of n-th order. As already mentioned above
results obtained in the study of oscillator systems are applicable to numerous