46
2 The Quantum Approach to the Two-Body Problem
of differential equations and are based on higher level algebraic treatments. Among
the simplest (and of widespread use) are the special functions Gamma, (z), (a
generalization of the factorial) defined by the integral
(z) =
∞
0
t
z−1 e
−t dt
(2.30)
although most often calculated from the relationship
(z + 1) = z(z)
(2.31)
The so-called mother of all special functions is the hypergeometric function which
we will discuss shortly.
2.1.3 The Harmonic Oscillator
The radial part of the solution, in general, is obtained by using numerical techniques
except in some cases where for specific model problems the solution can be obtained
in closed form. One such case, which is commonly used, is that of a particle moving
under the influence of a linear restoring force F(x) = −kx (i.e., subject to a potential
V (x) =
1
2
kx
2 ) is the harmonic oscillator (HO). The harmonic oscillator is used to
study the quantum nature of light (photons) and is also important in chemical physics,
because it is often used as an initial approximation in the calculation of the vibrational
frequencies of polyatomic molecules and crystal lattices. Such a potential (V (x) →
∞ for x → ±∞) admits only bound solutions (no dispersion) and represents the
extreme case of total energy E always lower than the potential asymptote.
From the classical treatment, we know that a particle of mass m which is subject
to this potential
5 has an angular frequency ω (or angular displacement per unit time)
ω =
k/m = 2πν
where ν is the frequency of rotation usually measured in hertz.
Then expressing the potential as V (x) = (m/2)w
2 x
2 the Schrödinger equation
describing the one-dimensional system is
−
2
2m
d
2
dx 2 +
mω
2 x
2
2
φ(x) = Eφ(x)
(2.32)
5 The discussion attains of course also to the relative motion of two particles of mass m 1 and m 2
interacting with this potential because, as already pointed earlier, the problem of the motion of two
particles is isomorphous with that of a single particle having a mass equal to the reduced mass
(μ = m 1 m 2 /(m 1 + m 2 )) of the two particle one.
2 The Quantum Approach to the Two-Body Problem
of differential equations and are based on higher level algebraic treatments. Among
the simplest (and of widespread use) are the special functions Gamma, (z), (a
generalization of the factorial) defined by the integral
(z) =
∞
0
t
z−1 e
−t dt
(2.30)
although most often calculated from the relationship
(z + 1) = z(z)
(2.31)
The so-called mother of all special functions is the hypergeometric function which
we will discuss shortly.
2.1.3 The Harmonic Oscillator
The radial part of the solution, in general, is obtained by using numerical techniques
except in some cases where for specific model problems the solution can be obtained
in closed form. One such case, which is commonly used, is that of a particle moving
under the influence of a linear restoring force F(x) = −kx (i.e., subject to a potential
V (x) =
1
2
kx
2 ) is the harmonic oscillator (HO). The harmonic oscillator is used to
study the quantum nature of light (photons) and is also important in chemical physics,
because it is often used as an initial approximation in the calculation of the vibrational
frequencies of polyatomic molecules and crystal lattices. Such a potential (V (x) →
∞ for x → ±∞) admits only bound solutions (no dispersion) and represents the
extreme case of total energy E always lower than the potential asymptote.
From the classical treatment, we know that a particle of mass m which is subject
to this potential
5 has an angular frequency ω (or angular displacement per unit time)
ω =
k/m = 2πν
where ν is the frequency of rotation usually measured in hertz.
Then expressing the potential as V (x) = (m/2)w
2 x
2 the Schrödinger equation
describing the one-dimensional system is
−
2
2m
d
2
dx 2 +
mω
2 x
2
2
φ(x) = Eφ(x)
(2.32)
5 The discussion attains of course also to the relative motion of two particles of mass m 1 and m 2
interacting with this potential because, as already pointed earlier, the problem of the motion of two
particles is isomorphous with that of a single particle having a mass equal to the reduced mass
(μ = m 1 m 2 /(m 1 + m 2 )) of the two particle one.