200
16 Hermite Polynomials
polynomial values directly. The function hermitezero returns the roots of the
Hermite polynomial based on the corresponding Jacobi matrix.
Harmonic oscillator The SPECFUNPHYS class harmonoscwave returns the
wave functions of the harmonic oscillator in coordinate representation.
Applications As applications we discuss, e.g., the anharmonic oscillator,
graphical user interface anha01.m with anha01.fig, oscperex.m,
anharmosc.m, the animation of coherent states, cohstates.m,
cohstates_animation.m, and the visualization of Wigner functions,
wigner2.m.
16.2 Hermite Polynomials
The Hermite polynomial [2] holds the orthogonality relation
+∞
−∞
exp(−x
2 )H n (x)H m (x) =
√
π2
n n!δ nm ,
(16.1)
is, e.g., defined by its Rodrigues formula
H n (x) = (−1)
n exp
x
2
d n
dx n exp
−x
2
,
(16.2)
and the generating function reads
exp
2tx − t
2
=
∞
n=0
H n (x)
n!
t
n .
(16.3)
The following recurrence formula will be used to compute the polynomial coefficients and the polynomial values.
H n+1 (x) = 2xH n (x) − 2nH n−1 (x), with H 0 (x) = 1 , H 1 (x) = 2x .
(16.4)
By normalizing the Hermite polynomial
˜
H n (x) = π
−1/4
1
√
2 n n!
H n (x)
(16.5)
we obtain
x ˜
H n (x) =
1
2
√
2
√
n + 1 ˜
H n+1 +
1
2
√
2
√
n ˜
H n−1 (x) ,
(16.6)
16 Hermite Polynomials
polynomial values directly. The function hermitezero returns the roots of the
Hermite polynomial based on the corresponding Jacobi matrix.
Harmonic oscillator The SPECFUNPHYS class harmonoscwave returns the
wave functions of the harmonic oscillator in coordinate representation.
Applications As applications we discuss, e.g., the anharmonic oscillator,
graphical user interface anha01.m with anha01.fig, oscperex.m,
anharmosc.m, the animation of coherent states, cohstates.m,
cohstates_animation.m, and the visualization of Wigner functions,
wigner2.m.
16.2 Hermite Polynomials
The Hermite polynomial [2] holds the orthogonality relation
+∞
−∞
exp(−x
2 )H n (x)H m (x) =
√
π2
n n!δ nm ,
(16.1)
is, e.g., defined by its Rodrigues formula
H n (x) = (−1)
n exp
x
2
d n
dx n exp
−x
2
,
(16.2)
and the generating function reads
exp
2tx − t
2
=
∞
n=0
H n (x)
n!
t
n .
(16.3)
The following recurrence formula will be used to compute the polynomial coefficients and the polynomial values.
H n+1 (x) = 2xH n (x) − 2nH n−1 (x), with H 0 (x) = 1 , H 1 (x) = 2x .
(16.4)
By normalizing the Hermite polynomial
˜
H n (x) = π
−1/4
1
√
2 n n!
H n (x)
(16.5)
we obtain
x ˜
H n (x) =
1
2
√
2
√
n + 1 ˜
H n+1 +
1
2
√
2
√
n ˜
H n−1 (x) ,
(16.6)