32
2 Molecular States
However, neglecting ˆ
V s one is left with the electrostatic Hamiltonian
ˆ
H mol,SF = ˆ
T n + ˆ
T e + V el
(2.35)
where the subscript S F stands for “spin free.” Even with this simplification, to
solve the eigenvalue equation for the molecular Hamiltonian represents a formidable
task, which may be undertaken as such only for a very small number of particles.
It is therefore necessary to introduce some simplifications, which are based on the
concept of separation of variables. As we shall see in this chapter, this concept is
particularly important because of the physical representation of a molecular system
and of its dynamics that it provides.
2.2.1 Independent Variables
We consider a Hamiltonian operator which is dependent on two groups of variables
ˆ
H (x, y). If we have ˆ
H (x, y) = ˆ
H A (x) + ˆ
H B (y), the variables x and y are separable:
the probability density to find the system in x is independent on y and vice versa.
In fact, if ψ A,i are the eigenstates of ˆ
H A , so that ˆ
H A (x)ψ A,i (x) = E A,i ψ A,i (x), and
similarly for ˆ
H B (y), the stationary states ψ i j are given by the products ψ A,i ψ B, j
ˆ
H (x, y)ψ i j (x, y) = ( ˆ
H A (x) + ˆ
H B (y))ψ A,i (x)ψ B, j (y)
= (E A,i + E B, j )ψ i j (x, y) .
(2.36)
In this way, in the stationary state ψ i j the probability density for finding the system
in x, y is the product of the probabilities for x and y
ψ i j (x, y)
2 =
ψ A,i (x)
2
ψ B, j (y)
2
(2.37)
which confirms that the two events are mutually independent.
The same is true for a time-dependent state, if the wavefunction at a given time
t = 0 is the product of two factors, one dependent on x and the other on y:
Ψ (x, y, 0) = Ψ A (x)Ψ B (y)
=
i
A i ψ A,i (x)
⎡
⎣
j
B j ψ B, j (y)
⎤
⎦
(2.38)
where X i =
ψ X,i |Ψ X
, with X = A, B. By applying the time evolution operator
(2.26) we see that the separation of the variables is preserved in time
2 Molecular States
However, neglecting ˆ
V s one is left with the electrostatic Hamiltonian
ˆ
H mol,SF = ˆ
T n + ˆ
T e + V el
(2.35)
where the subscript S F stands for “spin free.” Even with this simplification, to
solve the eigenvalue equation for the molecular Hamiltonian represents a formidable
task, which may be undertaken as such only for a very small number of particles.
It is therefore necessary to introduce some simplifications, which are based on the
concept of separation of variables. As we shall see in this chapter, this concept is
particularly important because of the physical representation of a molecular system
and of its dynamics that it provides.
2.2.1 Independent Variables
We consider a Hamiltonian operator which is dependent on two groups of variables
ˆ
H (x, y). If we have ˆ
H (x, y) = ˆ
H A (x) + ˆ
H B (y), the variables x and y are separable:
the probability density to find the system in x is independent on y and vice versa.
In fact, if ψ A,i are the eigenstates of ˆ
H A , so that ˆ
H A (x)ψ A,i (x) = E A,i ψ A,i (x), and
similarly for ˆ
H B (y), the stationary states ψ i j are given by the products ψ A,i ψ B, j
ˆ
H (x, y)ψ i j (x, y) = ( ˆ
H A (x) + ˆ
H B (y))ψ A,i (x)ψ B, j (y)
= (E A,i + E B, j )ψ i j (x, y) .
(2.36)
In this way, in the stationary state ψ i j the probability density for finding the system
in x, y is the product of the probabilities for x and y
ψ i j (x, y)
2 =
ψ A,i (x)
2
ψ B, j (y)
2
(2.37)
which confirms that the two events are mutually independent.
The same is true for a time-dependent state, if the wavefunction at a given time
t = 0 is the product of two factors, one dependent on x and the other on y:
Ψ (x, y, 0) = Ψ A (x)Ψ B (y)
=
i
A i ψ A,i (x)
⎡
⎣
j
B j ψ B, j (y)
⎤
⎦
(2.38)
where X i =
ψ X,i |Ψ X
, with X = A, B. By applying the time evolution operator
(2.26) we see that the separation of the variables is preserved in time