204
16 Hermite Polynomials
cases in physics, e.g., vibration of the nuclei in diamagnetic molecules, torsional
oscillations in molecules, motion of a muon inside a heavy nucleus, giant resonances
in atomic nuclei, to name only a few.
The SPECFUNPHYS Class harmonoscwave
The SPECFUNPHYS class harmonoscwave returns the eigenfunction of the harmonic oscillator in coordinate representation. The syntax is [obj, wavefun] =
harmonoscwave(n, x) with the input arguments “n”, the maximum quantum
number of the harmonic oscillator for ´evaluation, thus a positive integer value and
“x” the coordinate values at which the wave function shall be evaluated. The output
arguments are “wavefun” and the object “obj” with properties “wave”, the values
of the wave function at position “x”, “z”, the coordinate values at which the wave
function was evaluated (obj.z = x), “degree”, a column vector of the corresponding
oscillator quantum numbers, and “info” with some information about the evaluation.
The following applications are examples for the various classes and functions
mentioned above.
16.3.1 Applications
Coherent States
Coherent states [1] are minimal uncertainty states, which play an important role in
quantum optics. With the ladder operators
ˆ
a =
1
√ ¯
hω
⎛
⎝
mω 2
2
ˆ
x + i
1
2m
ˆ
p
⎞
⎠ and ˆ
a
†
=
1
√ ¯
hω
⎛
⎝
mω 2
2
ˆ
x − i
1
2m
ˆ
p
⎞
⎠
(16.15)
the Hamiltonian of the harmonic oscillator becomes in Fock space
ˆ
H = ¯
hω
ˆ
a
†
ˆ
a +
1
2
,
(16.16)
with eigenstates |n ˆ
H |n = ¯
hω(n +
1
2 )|n, and ˆ
a † |n =
√
n + 1|n + 1 (raising or
creation operator) and ˆ
a|n =
√
n|n − 1 (lowering or annihilation operator).
Coherent states |α are eigenstates of the annihilation operator ˆ
a
ˆ
a|α = α|α and α ∈ C.
(16.17)
Please note that the ladder operators are non-Hermitian operators and thus the
eigenvalue could be complex. In Fock space the coherent state is given by
|α = exp
−
1
2
|α|
2
∞
n=0
α n
√
n!
|n
(16.18)
16 Hermite Polynomials
cases in physics, e.g., vibration of the nuclei in diamagnetic molecules, torsional
oscillations in molecules, motion of a muon inside a heavy nucleus, giant resonances
in atomic nuclei, to name only a few.
The SPECFUNPHYS Class harmonoscwave
The SPECFUNPHYS class harmonoscwave returns the eigenfunction of the harmonic oscillator in coordinate representation. The syntax is [obj, wavefun] =
harmonoscwave(n, x) with the input arguments “n”, the maximum quantum
number of the harmonic oscillator for ´evaluation, thus a positive integer value and
“x” the coordinate values at which the wave function shall be evaluated. The output
arguments are “wavefun” and the object “obj” with properties “wave”, the values
of the wave function at position “x”, “z”, the coordinate values at which the wave
function was evaluated (obj.z = x), “degree”, a column vector of the corresponding
oscillator quantum numbers, and “info” with some information about the evaluation.
The following applications are examples for the various classes and functions
mentioned above.
16.3.1 Applications
Coherent States
Coherent states [1] are minimal uncertainty states, which play an important role in
quantum optics. With the ladder operators
ˆ
a =
1
√ ¯
hω
⎛
⎝
mω 2
2
ˆ
x + i
1
2m
ˆ
p
⎞
⎠ and ˆ
a
†
=
1
√ ¯
hω
⎛
⎝
mω 2
2
ˆ
x − i
1
2m
ˆ
p
⎞
⎠
(16.15)
the Hamiltonian of the harmonic oscillator becomes in Fock space
ˆ
H = ¯
hω
ˆ
a
†
ˆ
a +
1
2
,
(16.16)
with eigenstates |n ˆ
H |n = ¯
hω(n +
1
2 )|n, and ˆ
a † |n =
√
n + 1|n + 1 (raising or
creation operator) and ˆ
a|n =
√
n|n − 1 (lowering or annihilation operator).
Coherent states |α are eigenstates of the annihilation operator ˆ
a
ˆ
a|α = α|α and α ∈ C.
(16.17)
Please note that the ladder operators are non-Hermitian operators and thus the
eigenvalue could be complex. In Fock space the coherent state is given by
|α = exp
−
1
2
|α|
2
∞
n=0
α n
√
n!
|n
(16.18)