Appendix F
The Harmonic Oscillator
The harmonic oscillator is one of the most prominent paradigms in physics, but is
also a system with some important peculiarities, in particular at dynamical level [1,
2, 3, 4, 5], as first noted by Galileo. Here we shall review, without deriving them,
some of the basic properties of the quantum harmonic oscillator. More complete
treatments can be found in almost any textbook on quantum mechanics, for instance
Merzbacher [6], Chaps. 5 and 10.
The harmonic oscillator Hamiltonian is
ˆ
H =
1
2m
ˆ
p
2
x +
mω
2
2
(x − x e )
2
(F.1)
where ˆ
p x = −id/dx. We replace x and ˆ
p x by the dimensionless coordinates q and
ˆ
p:
q =
mω
(x − x e ) and ˆ
p =
1
√
mω
ˆ
p x = −i
d
dq
.
(F.2)
Then ˆ
H takes a more symmetric quadratic form:
ˆ
H =
ω
2
ˆ
p
2
+ q
2
.
(F.3)
The eigenvalues of ˆ
H are
E n = ω
n +
1
2
for n = 0, 1, 2...
(F.4)
and the corresponding eigenfunctions are
χ n (q) = N n H n (q) e
−q
2 /2
= N n H n
mω
(x − x e )
e
−
mω
2 (x−x e )
2
(F.5)
© Springer International Publishing AG, part of Springer Nature 2018
M. Persico and G. Granucci, Photochemistry, Theoretical Chemistry
and Computational Modelling, https://doi.org/10.1007/978-3-319-89972-5
233