INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 9, ISSUE 02, FEBRUARY 2020 ISSN 2277-8616
760
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Bargmann Transform With Application To
Time-Dependent Schrödinger Equation
Anouar Saidi, Ahmed Yahya Mahmoud, Mohamed Vall Ould Moustapha
Abstract: This article deals with the Bargmann transform as a new method to solve the time-dependent Schrödinger equation.
Index Terms: Bargmann transform, harmonic oscillator, integral transform, intertwinning operator, linear differential equation, quantum mechanics,
Schrödinger equation.
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1. INTRODUCTION
In 1926, Erwin Schrödinger ([1], [2], [3], and [4]) proposed a
wave theory of quantum mechanics, along the lines of the de
Broglie hypothesis. He considered the evolution of waves in
time and showed that the energy levels of an atom can be
considered as eigenvalues of a Hamiltonian operator. He also
demonstrates that the wave model was equivalent to
Heisenberg’s matrix model. His formulation of the problem, later
called the Schrödinger equation, takes into account both
quantization and non-relativistic energy. The time-dependent
Schrödinger equation is given by
,
( )
( ) ( )
( )
()
()
(1.1)
where is the Hamiltonian operator of the system which
correspond to the sum of the kinetic energies and the potential
energies for all the particles in the system. In 1956, V.
Bargmann [5] present an integral transform from the space of
square integrable functions
() to the Fock space [6]. This
transform was called later Bargmann transform and a great
number of research works has been done on it [7], [8], [9], [10],
[11], [12], [13]. In this work, we use the Bargmann transform to
solve the equation (1.1) where is the Harmonic oscillator
defined by
which is one of the most famous
model of Hamiltonian operator in the Quantum mechanics. The
solution of this equation has been known for a long time as in
[14] p.145 and is based essentially on the famous Mehler’s
formula [15], but our method is new. Indeed, we use the
Bargmann transform as an intertwining operator that transports
the harmonic oscillator to a complex Euler operator. This
relationship is our main tool for solving the equation above.
This paper is outlined as follows
In section 2, some useful results about Bargmann transform are
presented.
In section 3, we compute the exact solution of the Schrödinger
equation (1.1).
In section 4, we compute explicitly the solution of heat Cauchy
problems attached to the generalized real and complex Dirac
operators by the method of section 3.
In section 5, we give some numerical results dealing with the
solution of (1.1).
1 Bargmann transform
For , we define a Gaussian measure on as follows
()
The Fock space, denoted
, is the subspace of all entire
functions in
(
) which is a Hilbert space with the inner
product
∫
()()
() ()
V. Bargmann, in [5], [16] has defined a mapping , from the
space of square integrable functions
() to the Fock space
, called Bargmann transform, that is
()
[]() (
)
∫
()
√
(
)
()
We use in this paper a parametrized form of the Bargmann
transform given by K. Zhu [17] as follows
[
]() (
)
∫
()
()
This mapping is an isometry from
() to
, its inverse is
given by
[
]() (
)
∫
()
() ()
Our aim in this section is to present some important results
involving the Bargman transform and that will be useful to prove
the principal theorems of this paper. These results are well
known in the literature for the classical Bargmann transform .
We adapt here some of their proofs for the parametrized form
.
Lemma 2.1. ([17]) Let
() we have
1. *
()+ () (
) *
+ ().
2. *
(
)+ ()(
) *
+ ().
3. *
((
) )+ () *
+ ().
4. *
((
) )+ ()
*
+ ().
Let
and
be respectively the complex Euler operator
and the quantum harmonic oscillator defined by
(2.6)
The following proposition is our principal tool in this paper, it is a
direct consequence of lemma 2.1.
Proposition 2.1. We have
*
(
)+ ()
*
+ ()
The following lemma gives the Bargmann transform of the
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Anouar Saidi, PHD,College of Arts and Sciences Gurayat, Jouf
University, Kingdom of Saudi Arabia. &Faculty of Sciences of
Monastir, 5019 Monastir-Tunisia.
E-mail address: saidi.anouar@yahoo.fr,
Ahmed Yahya Mahmoud, PHD,College of Arts and Sciences
Gurayat, Jouf University, Kingdom of Saudi Arabia.& Faculty of
science and Technology, Shendi University ,Shendi Sudan. E-mail
address: humudi999@gmail.com
Mohamed Vall Ould Moustapha, PHD, College of Arts and Science
-Gurayat, Jouf University, Kingdom of Saudi Arabia &Faculté des
Sciences et Techniques, Université de Nouakchott Al- Aasriya,
Nouakchott-Mauritanie
E-mail address: mohamedvall.ouldmoustapha230@gmail.com