Handbook of Solvents - George Wypych - ChemTech - Ventech!

8.2 Basic simplifications of quantum model 421
∂
∂t
i� Ψ( xt , ) HΨ( xt , )
= [8.2]
In practice, it expresses how the stateΨ(x) of the system evolves in time (i.e., it gives ∂Ψ(x)/
∂t when Ψ(x) is known).
Note that in eq. [8.1] the time t was not included among the parameters defining H
(and Ψ): with eq. [8.1] we are looking at stationary states, not depending on time. In spite of
this, eq. [8.1] may be applied to liquids, which are characterized by a continuous dynamic
exchange of energy through collisions, and by a continuous displacement of the constituting
molecules. This is not a problem for the use of the time independent formulation of the
Schrödinger equation for liquids: eq. [8.1] actually includes kinetic energy and all related
dynamic aspects. A direct use of eq. [8.1] for liquids means to treat liquids at equilibrium.
We shall not make explicit use of the second equation, even when the nonequilibrium problem
is considered; the use of the semiclassical approximation permits us to treat time dependent
phenomena at the classical level, simpler to use.
Let us come back to the expression of H(x). The number of parameters within (x) is exceedingly
large if the system is a liquid. Fortunately, things may be simplified by using
factorization techniques.
When a system contains two subsystems (say A and B) that do not interact, the
Hamiltonian can be partitioned so:
( ) ( ) ( )
H x = H x + H x
[8.3]
A A B B
This rigorously leads to a factorization of Ψ:
( x) ( x ) ( x )
Ψ = Ψ + Ψ
[8.4]
A A B B
and to a partition of the energy:
E = E A + E B
[8.5]
Equation [8.1] is in consequence transformed, without simplifications, into two simpler
equations:
( ) Ψ ( ) Ψ ( )
H x x = E x
[8.6a]
A A A A A A A
( ) Ψ ( ) Ψ ( )
H x x = E x
[8.6b]
B B B B B B B
In practice, there are always interactions between A and B. The Hamiltonian may always
be written as:
( ) ( ) ( ) ( )
H x = H x + H x + H x
[8.7]
A A B B AB
The relative magnitude of the coupling term permits us to distinguish among three notable
cases.
1) If the coupling term is very small, it may be neglected, coming back to
equations [8.6].