Chapter 2
Quantum-Mechanical Harmonic Oscillator
Quantum-mechanical treatment of a harmonic oscillator has been a well-studied
topic from the beginning of the history of quantum mechanics. This topic is a
standard subject in classical mechanics as well. In this chapter, first we briefly
survey characteristics of a classical harmonic oscillator. From a quantum-mechanical
point of view, we deal with features of a harmonic oscillator through matrix
representation. We define creation and annihilation operators using position and
momentum operators. A Hamiltonian of the oscillator is described in terms of the
creation and annihilation operators. This enables us to easily determine energy
eigenvalues of the oscillator. As a result, energy eigenvalues are found to be positive
definite. Meanwhile, we express the Schrödinger equation by the coordinate representation. We compare the results with those of the matrix representation and show
that the two representations are mathematically equivalent. Thus, the treatment of the
quantum-mechanical harmonic oscillator supplies us with a firm ground for studying
basic concepts of the quantum mechanics.
2.1 Classical Harmonic Oscillator
Classical Newtonian equation of a one-dimensional harmonic oscillator is expressed
as
m
d
2 x t
ð Þ
dt
2
¼ Àsx t
ð Þ,
ð2:1Þ
where m is a mass of an oscillator and s is a spring constant. Putting s/m ¼ ω
2 , we
have
© Springer Nature Singapore Pte Ltd. 2020
S. Hotta, Mathematical Physical Chemistry,
https://doi.org/10.1007/978-981-15-2225-3_2
31