104
A. Lattanzi
ψ(x, t) =
+∞
−∞
mc 2 t
π ¯
h
K 1
imc
¯
h
c 2 t 2 − (x − x ) 2
c 2 t 2 − (x − x ) 2
ψ 0 (x
)dx
.
(7)
In the above equation, K 1 is the modified Bessel function of the second kind of first
order, also known as the McDonald function [30].
3 A Recursive Solution: Nonlocality via Laplace Transform
The Hamiltonian (2) for V (x, t) = 0 is a nonlocal operator whose natural scale
is inversely proportional to the mass and it is given by the (reduced) Compton
wavelength λ C = ¯
h
mc . The Compton wavelength represents the cutoff between the
quantum and the quantum field theory: below its value, the concept of single particle
is no more applicable. This justifies the normalization used to define the following
dimensionless variables:
ξ =
x
λ C
,
τ =
ct
λ C
and κ =
p
¯
h
λ C ,
(8)
which allows not only to simplify the analysis but also to bridge formally quantum
mechanics and optics [25–29]. Accordingly, in dimensionless variables (8) the
initial value problem (3) writes as
i∂ τ ψ(ξ, τ ) =
1 − ∂ 2
ξ ψ(ξ, τ ), ψ(ξ, 0) = ψ 0 (ξ ),
(9)
and its formal solution is then
ψ(ξ, τ ) =
iτ
π
+∞
−∞
K 1 (
(ξ − ξ ) 2 − τ 2 )
(ξ − ξ ) 2 − τ 2
ψ 0 (ξ
)dξ
.
(10)
To illustrate the nonlocal nature of the Hamiltonian operator in (9) and in
particular to emphasize its influence on the evolution of an initial input from a
mathematical point of view, it is interesting to deal with the initial value problem
using another mathematical approach based on the Laplace transform method
[31–35]. This technique is an effective alternative method that allows us to treat
fractional operators as the Hamiltonian of the Salpeter equation revealing its
nonlocal nature via recursive series.
To this end, a key notion is the following Laplace-like identity [30]
1
√
A 2 + 1
=
∞
0
e
−tA J 0 (t)dt.
(11)
A. Lattanzi
ψ(x, t) =
+∞
−∞
mc 2 t
π ¯
h
K 1
imc
¯
h
c 2 t 2 − (x − x ) 2
c 2 t 2 − (x − x ) 2
ψ 0 (x
)dx
.
(7)
In the above equation, K 1 is the modified Bessel function of the second kind of first
order, also known as the McDonald function [30].
3 A Recursive Solution: Nonlocality via Laplace Transform
The Hamiltonian (2) for V (x, t) = 0 is a nonlocal operator whose natural scale
is inversely proportional to the mass and it is given by the (reduced) Compton
wavelength λ C = ¯
h
mc . The Compton wavelength represents the cutoff between the
quantum and the quantum field theory: below its value, the concept of single particle
is no more applicable. This justifies the normalization used to define the following
dimensionless variables:
ξ =
x
λ C
,
τ =
ct
λ C
and κ =
p
¯
h
λ C ,
(8)
which allows not only to simplify the analysis but also to bridge formally quantum
mechanics and optics [25–29]. Accordingly, in dimensionless variables (8) the
initial value problem (3) writes as
i∂ τ ψ(ξ, τ ) =
1 − ∂ 2
ξ ψ(ξ, τ ), ψ(ξ, 0) = ψ 0 (ξ ),
(9)
and its formal solution is then
ψ(ξ, τ ) =
iτ
π
+∞
−∞
K 1 (
(ξ − ξ ) 2 − τ 2 )
(ξ − ξ ) 2 − τ 2
ψ 0 (ξ
)dξ
.
(10)
To illustrate the nonlocal nature of the Hamiltonian operator in (9) and in
particular to emphasize its influence on the evolution of an initial input from a
mathematical point of view, it is interesting to deal with the initial value problem
using another mathematical approach based on the Laplace transform method
[31–35]. This technique is an effective alternative method that allows us to treat
fractional operators as the Hamiltonian of the Salpeter equation revealing its
nonlocal nature via recursive series.
To this end, a key notion is the following Laplace-like identity [30]
1
√
A 2 + 1
=
∞
0
e
−tA J 0 (t)dt.
(11)