The Salpeter Equation: Nonlocality and Pseudodifferential Operators
103
The case of a linear potential for the Salpeter equation and related approximating
equations can be found in [23, 25–29] and in references therein.
After a very short introduction in where the spinless Salpeter equation has been
introduced in a physical and mathematical framework, Sect. 2 is devoted to the
definition of its solutions in the coordinate and in the momentum space, respectively.
It has been considered also the fundamental solution in its closed-analytical form.
In Sect. 3, the attention has been focused on the nonlocality of the Hamiltonian of
the spinless Salpeter equation and the consequent effects on its solution. It has been
presented an application of the theory of evolution operator to define a recursive
solution for the spinless Salpeter equation which allows to highlight the nonlocal
nature via recursive series. The result shows the presence of a space-memory, or in
other words the presence of a regular repeating spot-like structure in the light-cone
which tends to fade as time elapses. Finally a comment on another way to deal with
nonlocality has been presented exploiting the series expansion of the Hamiltonian
in the spinless Salpeter equation. This second approach is based on the so-called
Pearcey equation, a new equation introduced in [25–29] for describing what happen
between the two theories: the classical quantum mechanics ruled by the Schrödinger
equation and the relativistic quantum mechanics.
The square-modulus of the solutions in the Pearcey equation, and in particular of
its fundamental solution, illustrates the space-memory hinted in the recursive series.
2 Solutions of the Salpeter Equation
The nonlocal nature of (3) makes it difficult to deal with directly in the coordinate
space. It is then usually approached in the momentum space
i ¯
h∂ t ˜
ψ(p, t) =
m 2 c 4 + c 2 p 2 ˜
ψ(p, t), ˜
ψ(p, 0) = ˜
ψ 0 (p),
(4)
whose solution is
˜
ψ(p, t) = e
−
it
¯
h
√
m 2 c 4 +c 2 p 2 ˜
ψ 0 (p).
(5)
Here ˜
ψ(p, t) means the momentum wave function solution of the Salpeter equation.
By Fourier transform it is possible to define ψ(x, t):
ψ(x, t) =
1
√
2π ¯
h
+∞
−∞
e
ipx
¯
h e
−
it
¯
h
√
m 2 c 4 +c 2 p 2 ˜
ψ 0 (p) dp.
(6)
The same solution can be obtained considering the convolution of an initial
condition with the fundamental solution S(x, t) of the Salpeter equation that
corresponds to a δ-function input, i.e. ψ 0 (x) = δ(x):
103
The case of a linear potential for the Salpeter equation and related approximating
equations can be found in [23, 25–29] and in references therein.
After a very short introduction in where the spinless Salpeter equation has been
introduced in a physical and mathematical framework, Sect. 2 is devoted to the
definition of its solutions in the coordinate and in the momentum space, respectively.
It has been considered also the fundamental solution in its closed-analytical form.
In Sect. 3, the attention has been focused on the nonlocality of the Hamiltonian of
the spinless Salpeter equation and the consequent effects on its solution. It has been
presented an application of the theory of evolution operator to define a recursive
solution for the spinless Salpeter equation which allows to highlight the nonlocal
nature via recursive series. The result shows the presence of a space-memory, or in
other words the presence of a regular repeating spot-like structure in the light-cone
which tends to fade as time elapses. Finally a comment on another way to deal with
nonlocality has been presented exploiting the series expansion of the Hamiltonian
in the spinless Salpeter equation. This second approach is based on the so-called
Pearcey equation, a new equation introduced in [25–29] for describing what happen
between the two theories: the classical quantum mechanics ruled by the Schrödinger
equation and the relativistic quantum mechanics.
The square-modulus of the solutions in the Pearcey equation, and in particular of
its fundamental solution, illustrates the space-memory hinted in the recursive series.
2 Solutions of the Salpeter Equation
The nonlocal nature of (3) makes it difficult to deal with directly in the coordinate
space. It is then usually approached in the momentum space
i ¯
h∂ t ˜
ψ(p, t) =
m 2 c 4 + c 2 p 2 ˜
ψ(p, t), ˜
ψ(p, 0) = ˜
ψ 0 (p),
(4)
whose solution is
˜
ψ(p, t) = e
−
it
¯
h
√
m 2 c 4 +c 2 p 2 ˜
ψ 0 (p).
(5)
Here ˜
ψ(p, t) means the momentum wave function solution of the Salpeter equation.
By Fourier transform it is possible to define ψ(x, t):
ψ(x, t) =
1
√
2π ¯
h
+∞
−∞
e
ipx
¯
h e
−
it
¯
h
√
m 2 c 4 +c 2 p 2 ˜
ψ 0 (p) dp.
(6)
The same solution can be obtained considering the convolution of an initial
condition with the fundamental solution S(x, t) of the Salpeter equation that
corresponds to a δ-function input, i.e. ψ 0 (x) = δ(x):