26
2 Molecular States
The Hamiltonian operator is linear, as most of the other operators that appear
in this book. That is: ˆ
H (c 1 Ψ 1 + c 2 Ψ 2 ) = c 1 ˆ
H Ψ 1 + c 2 ˆ
H Ψ 2 , where c 1 and c 2 are
complex coefficients. Therefore, the TDSE is also linear, which has a very important
consequence: a linear combination of distinct solutions of the TDSE is still a solution.
This is the superposition principle.
The quantity |Ψ (x, t)|
2 dx is proportional to the probability of finding the system
in the interval of coordinates dx. More precisely, for each particle i we consider
an infinitesimal volume dr
3
i and a given value of the z spin component s i . We can
integrate |Ψ (x, t)|
2 over the whole range of each space coordinate and sum over all
the allowed spin values. This operation can be for simplicity indicated as an integral
over the vector x or, in Dirac’s notation (see Appendix B), as the scalar product of
state |Ψ with itself:
I =
all space
|Ψ (x, t)|
2 dx ≡ Ψ |Ψ
(2.2)
If I is a finite quantity, Ψ (x, t) can be divided by the factor
√
I (thanks to the linearity
of the TDSE), obtaining a normalized wavefunction Ψ
= I
−1/2
Ψ . In that case,
Ψ
Ψ
= I
−1
Ψ |Ψ = 1
(2.3)
and ρ(x, t) =
Ψ
(x, t)
2 exactly corresponds to the probability density of finding
the system in x.
Contrary to what happens in classical mechanics, the TDSE is first order in the
time derivative. Therefore, once the wavefunction has been specified at some time
t 0 , the physical state of the system is univocally determined at any t. In particular,
Ψ (x, t 0 ) is easily propagated to an infinitesimally close time t 0 + dt
Ψ (x, t 0 + dt) = Ψ (x, t 0 ) +
dΨ (x, t 0 )
dt
dt
= Ψ (x, t 0 ) −
idt
ˆ
H (t 0 )Ψ (x, t 0 )
= ˆ
U (t 0 , t 0 + dt)Ψ (x, t 0 )
(2.4)
where only the first-order terms in dt have been retained and
ˆ
U (t 0 , t 0 + dt) = 1 −
idt
ˆ
H (t 0 )
(2.5)
is the infinitesimal time evolution operator from time t 0 to t 0 + dt. The above expression for ˆ
U (t 0 , t 0 + dt) is generally valid, even for a time-dependent Hamiltonian.
Note that the wavefunction Ψ (x, t) is inherently a complex-valued quantity: In fact,
as ˆ
U (t 0 , t 0 + dt) contains the imaginary unit, even if we start with a real Ψ at t 0 , it
will become complex at later (or previous) times.
2 Molecular States
The Hamiltonian operator is linear, as most of the other operators that appear
in this book. That is: ˆ
H (c 1 Ψ 1 + c 2 Ψ 2 ) = c 1 ˆ
H Ψ 1 + c 2 ˆ
H Ψ 2 , where c 1 and c 2 are
complex coefficients. Therefore, the TDSE is also linear, which has a very important
consequence: a linear combination of distinct solutions of the TDSE is still a solution.
This is the superposition principle.
The quantity |Ψ (x, t)|
2 dx is proportional to the probability of finding the system
in the interval of coordinates dx. More precisely, for each particle i we consider
an infinitesimal volume dr
3
i and a given value of the z spin component s i . We can
integrate |Ψ (x, t)|
2 over the whole range of each space coordinate and sum over all
the allowed spin values. This operation can be for simplicity indicated as an integral
over the vector x or, in Dirac’s notation (see Appendix B), as the scalar product of
state |Ψ with itself:
I =
all space
|Ψ (x, t)|
2 dx ≡ Ψ |Ψ
(2.2)
If I is a finite quantity, Ψ (x, t) can be divided by the factor
√
I (thanks to the linearity
of the TDSE), obtaining a normalized wavefunction Ψ
= I
−1/2
Ψ . In that case,
Ψ
Ψ
= I
−1
Ψ |Ψ = 1
(2.3)
and ρ(x, t) =
Ψ
(x, t)
2 exactly corresponds to the probability density of finding
the system in x.
Contrary to what happens in classical mechanics, the TDSE is first order in the
time derivative. Therefore, once the wavefunction has been specified at some time
t 0 , the physical state of the system is univocally determined at any t. In particular,
Ψ (x, t 0 ) is easily propagated to an infinitesimally close time t 0 + dt
Ψ (x, t 0 + dt) = Ψ (x, t 0 ) +
dΨ (x, t 0 )
dt
dt
= Ψ (x, t 0 ) −
idt
ˆ
H (t 0 )Ψ (x, t 0 )
= ˆ
U (t 0 , t 0 + dt)Ψ (x, t 0 )
(2.4)
where only the first-order terms in dt have been retained and
ˆ
U (t 0 , t 0 + dt) = 1 −
idt
ˆ
H (t 0 )
(2.5)
is the infinitesimal time evolution operator from time t 0 to t 0 + dt. The above expression for ˆ
U (t 0 , t 0 + dt) is generally valid, even for a time-dependent Hamiltonian.
Note that the wavefunction Ψ (x, t) is inherently a complex-valued quantity: In fact,
as ˆ
U (t 0 , t 0 + dt) contains the imaginary unit, even if we start with a real Ψ at t 0 , it
will become complex at later (or previous) times.