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We propose an approach based on Floquet theorem combined with the resonating averages method (RAM), to solve the time-dependent Schr ödinger equation with a time-periodic Hamiltonian. This approach provides an alternative way to determine directly the evolution operator, and then we deduct the wave functions and the corresponding quasi-energies, of quantum systems. An application is operated for the driven cubic or/and quatric anharmonic as well as for the Morse potential. Comparisons of our results with those of other authors are discussed, and numerical evaluations are performed, to determine the dissociation energy of (HCl) and (CO) molecules.

Several physical systems have been modeled by a simple harmonic oscillator as a first approximation devoted to the construction of elegant formalism, to understand their dynamics and are dealing with the description of the nature of various physical phenomena under certain conditions. Moreover, it is possible to compare solutions derived from these studies to experimental results obtained by studying laser-matter interaction problems in many branches of physics, ranging from black body radiation to vibrations of crystal lattices [

In this regard, several approaches have been used, among them the well-known perturbation theory [

In previous works, we have applied the above mentioned approach, to the case of harmonic oscillator with time-periodic frequency and to the simple forced harmonic oscillator [

The paper is organized as follows. In Section 2, we review the basic formulation of our approach. Section 3 consists of its application to the driven cubic and quatric anharmonic oscillators, and to the Morse potentail expansion. In Section 4, some comparisons of our results with those of other published works are presented and discussed and an example of numerical evaluations was performed for the (HCl) and (CO) molecules. Concluding remarks are given in Section 5.

A quantum system that is submitted to a perturbation may be described by the following hamiltonian;

H ( t ) = H 0 + λ H ′ ( t ) (1)

where H 0 is the Hamiltonian of the unperturbed system, and H ′ ( t ) is the interaction Hamiltonian, which amplitude λ is taken as being very small.

In the case of a periodical perturbation, according to the Floquet theorem there exists a couple of operators (R, T(t)), so that the time-evolution operator can be written in the following form [

U ( t ) = T ( t ) e − i R t ℏ (2)

and U ( 0 ) = 1

where T(t) is a periodic unitary operator of the same period as H ′ ( t ) , and R is a constant hermitian operator.

In the interaction picture, U ( t ) satisfies the following differential equation

i ℏ d U I ( t ) d t = λ H I ( t ) U I ( t ) (3)

where

H I ( t ) = e i H 0 t / ℏ H ′ ( t ) e − i H 0 t / ℏ (4)

A unitary transformation T(t) may be applied to Equation (3) to obtain the so-called reduced equation of the system such as,

i ℏ d | ϕ n ( t ) 〉 d t = R | ϕ n ( t ) 〉 (5a)

| ϕ n ( t ) 〉 = e − i E n t / ℏ | n 〉 (5b)

where | ϕ n ( t ) 〉 are the eigenstates of the Floquet operator R, corresponding to the eigenvalues E n , and where | n 〉 are the states of the unperturbed system. Consequently, the Floquet states (or steady states), solutions of the time-dependent Schrödinger equation, in the Floquet representation are defined such as,

| ψ n ( t ) 〉 = T ( t ) | ϕ n ( t ) 〉 (6)

The obtained steady states form a complete set of time-dependent solutions in the extended Hilbert space, and do not depend on the choice of the couple (R, T(t)).

Search of Floquet operators is based on the resonating averages method (RAM) [

H I ( t ) = H ¯ I ( t ) + d H ˜ I ( t ) d t (7)

The application of the RAM to Equation (3) gives rise to the following solutions to first and second order in λ

U ( 1 a ) I ( t ) = [ 1 − i λ ℏ H ˜ I ( t ) ] V ( 1 ) I ( t ) (8)

U ( 2 a ) I ( t ) = [ 1 − i λ ℏ H ˜ I ( t ) + λ 2 A 2 ( t ) ] G ( t ) (9)

The determination of the solutions U ( 1 a ) I ( t ) and U ( 2 a ) I ( t ) Equations ((8), (9)) and a comparison with Floquet representation of Equation (2), enabled us to obtain the first and second order couples ( R ( 1 a ) , T ( 1 a ) ( t ) ) , ( R ( 2 a ) , T ( 2 a ) ( t ) ) , and

thence the quasi-energies E ( 1 a ) n , E ( 2 a ) n and Floquet-states | ψ ( 1 a ) n ( t ) 〉 , | ψ ( 2 a ) n ( t ) 〉 , respectively.

The Hamiltonian of the considered quantum system is given by the following expression

H 1 ( t ) = p 2 2 m + 1 2 m ω 0 2 q 2 + μ 1 ℏ ω 0 q ^ 3 + μ ℏ ω 0 sin ( ν t ) q ^ (10)

where q ^ = m ω 0 ℏ q

μ 1 ℏ ω 0 q ^ 3 and μ ℏ ω 0 sin ( ν t ) q ^ are the conservative perturbation and the time-dependent perturbation due to an external force, respectively.

m and ω 0 are the mass and the frequency of the simple harmonic oscillator, respectively, ν is the driven oscillation frequency, and μ 1 , μ , are very weak amplitudes of the perturbations.

Adjusting Equation (10) with the RAM formulation needs the following variables changes,

μ 1 = λ γ 1 ; μ = λ γ and λ ≪ 1

Thus, we can write H 1 ( t ) in the form,

H 1 ( t ) = p 2 2 m + 1 2 m ω 0 2 q 2 + λ ( γ 1 ℏ ω 0 q ^ 3 + γ ℏ ω 0 sin ( ν t ) q ^ ) (11)

Introduction of the creation and annihilation operators a + and a of the unperturbed Hamiltonian [

H ′ ( t ) = ℏ ω 0 2 [ γ 1 2 ( a + 3 + a 3 + 3 a + a a + + 3 ( a + a + 1 ) a ) + γ ( a + + a ) sin ( ν t ) ] (12)

The RAM applied to the interaction picture form of H 1 ( t ) (Equations ((4), (7))) gives

H ¯ I ( t ) = 0 (13)

H ˜ I ( t ) = i γ 1 ℏ 6 2 [ − e 3 i ω 0 t a + 3 + e − 3 i ω 0 t a 3 − 9 e i ω 0 t a + a a + + 9 e − i ω 0 t ( a + a + 1 ) a ] + γ ℏ ω 0 2 ( α 0 ⋆ ( t ) e i ω 0 t a + + α 0 ( t ) e − i ω 0 t a ) (14)

where

α 0 ( t ) = − ν cos ( ν t ) + i ω 0 sin ( ν t ) ν 2 − ω 0 2 (15)

α 0 ⋆ ( t ) being the complex conjugate of α 0 (t)

From Equation (8), we obtain the evolution operator to the first ameliorated order such as

U ( 1 a ) 1 ( t ) = [ 1 + μ 1 6 2 ( − a + 3 + a 3 − 9 a + a a + + 9 ( a + a + 1 ) a ) − i μ ω 0 2 ( α 0 ⋆ ( t ) a + + α 0 ( t ) a ) ] e − i H 0 t ℏ (16)

By comparison with the formulation of Equation (2), we deduce the first ameliorate order Floquet operators (R, T(t)), such as:

R ( 1 a ) 1 = ℏ ω 0 ( a + a + 1 2 ) (17a)

T ( 1 a ) 1 ( t ) = 1 + μ 1 6 2 ( − a + 3 + a 3 − 9 a + a a + + 9 ( a + a + 1 ) a ) − i μ ω 0 2 ( α 0 ⋆ ( t ) a + + α 0 ( t ) a ) (17b)

Thus, the quasi-energies, Floquet states and wave functions developed to the first order respectively are:

E ( 1 a ) n = ℏ ω 0 ( n + 1 2 ) (18)

| ψ ( 1 a ) n ( t ) 〉 = e − i E ( 1 a ) n t ℏ [ c − 3 | n − 3 〉 + c − 1 | n − 1 〉 + | n 〉 + c + 1 | n + 1 〉 + c + 3 | n + 3 〉 ] (19)

ψ ( 1 a ) n ( q , t ) = e − i E ( 1 a ) n t ℏ [ c − 3 φ n − 3 ( q ) + c − 1 φ n − 1 ( q ) + φ n ( q ) + c + 1 φ n + 1 ( q ) + c + 3 φ n + 3 ( q ) ] (20)

where the coefficients c ± 1 and c ± 3 are given by

c − 3 = μ 1 6 2 n ( n − 1 ) ( n − 2 ) (21a)

c − 1 = [ 3 μ 1 2 2 n − i μ ω 0 2 α 0 ( t ) ] n (21b)

c + 1 = − [ 3 μ 1 2 2 ( n + 1 ) + i μ ω 0 2 α 0 ⋆ ( t ) ] n + 1 (21c)

c + 3 = − μ 1 6 2 ( n + 1 ) ( n + 2 ) ( n + 3 ) (21d)

and

φ n ( q ) = ( α π ) 1 4 e − α q 2 2 2 n n ! H n ( α q ) (22)

φ n ( q ) is the wave function of the simple oscillator, the parameter α = m ω 0 ℏ and H n ( α q ) are the Hermite polynomials.

Using Equation (9) and with the help of Equations ((13), (14)) one can write the second ameliorated order evolution operator and deduce the Floquet operators R ( 2 a ) 1 in the following form

R ( 2 a ) 1 = H 0 − μ 1 2 15 ℏ ω 0 4 ( ( a + a ) 2 + a + a + 11 30 ) + μ 2 ℏ ω 0 3 4 ( ν 2 − w 0 2 ) (23)

Thus, the quasi-energies of the system up to second ameliorated order is given by

E ( 2 a ) n = ℏ ω 0 ( n + 1 2 ) − μ 1 2 15 ℏ ω 0 4 ( n 2 + n + 11 30 ) + μ 2 ℏ ω 0 3 4 ( ν 2 − w 0 2 ) (24)

We note that the correction effects on the quasi-energies to second order approximation, and the Floquet shift levels depend on the amplitudes ( μ 1 , μ ) , of the perturbations and the quantum number n.

We consider the system which Hamiltonian is given by,

H 2 ( t ) = p 2 2 m + 1 2 m ω 0 2 q 2 + μ 2 ℏ ω 0 q ^ 4 + μ ℏ ω 0 sin ( ν t ) q ^ (25)

where

μ 2 ℏ ω 0 q ^ 4 and μ ℏ ω 0 sin ( ν t ) q ^ are the quatric anharmonic perturbation with amplitude μ 2 and the external time-dependent perturbation with amplitude μ respectively.

We operate the change variable μ 2 = λ γ 2 and μ = λ γ on Equation (25), then we have,

H 2 ( t ) = p 2 2 m + 1 2 m ω 0 2 q 2 + λ ( γ 2 ℏ ω 0 q ^ 4 + γ ℏ ω 0 sin ( ν t ) q ^ ) (26)

Using the usual creation and annihilation operators yields to write H ′ ( t ) in Equation (1) as

H ′ ( t ) = γ 2 ℏ ω 0 4 [ a + 4 + a 4 + 4 a + a a + 2 − 2 a + 2 + 4 ( a + a + 1 ) a 2 + 2 a 2 + 3 ( a + a ) 2 + 3 ( a + a + 1 ) 2 ] + γ ℏ ω 0 2 ( a + + a ) sin ( ν t ) (27)

The RAM applied to the interaction picture form of H 2 ( t ) (Equations ((4), (7))), gives

H ¯ I ( t ) = 3 4 γ 2 ℏ ω 0 ( 2 ( a + a ) 2 + 2 a + a + 1 ) (28)

H ˜ I ( t ) = γ 2 i ℏ 16 [ − e 4 i ω 0 t a + 4 + e − 4 i ω 0 t a 4 − 4 e 2 i ω 0 t ( 2 a + a − 1 ) a + 2 + 4 e − 2 i ω 0 t ( 2 a + a + 3 ) a 2 ] + γ ℏ ω 0 2 ( α 0 ⋆ ( t ) e i ω 0 t a + + α 0 ( t ) e − i ω 0 a ) (29)

where α 0 ( t ) is given by Equation (15).

Following the previous procedure given in the subsection 3.1, we obtain the time evolution operator, then the quasi-energies and Floquet states to the first ameliorated order, for this system, respectively such as,

U ( 1 a ) 2 ( t ) = [ 1 − i λ ℏ [ γ 2 i ℏ 16 [ − a + 4 + a 4 − 4 ( 2 a + a − 1 ) a + 2 + 4 ( 2 a + a + 3 ) a 2 ] + γ ℏ ω 0 2 ( α 0 ⋆ ( t ) a + + α 0 ( t ) a ) ] ] × exp ( − i t ℏ [ 3 4 μ 2 ℏ ω 0 ( 2 ( a + a ) 2 + 2 a + a + 1 ) + H 0 ] ) (30)

E ( 1 a ) n = ℏ ω 0 ( n + 1 2 ) + 3 4 μ 2 ℏ ω 0 ( 2 n 2 + 2 n + 1 ) (31)

| ψ ( 1 a ) n ( t ) 〉 = e − i E ( 1 a ) n t ℏ [ k − 4 | n − 4 〉 + k − 2 | n − 2 〉 + k − 1 | n − 1 〉 + | n 〉 + k + 1 | n + 1 〉 + k + 2 | n + 2 〉 + k + 4 | n + 4 〉 ] (32)

where

k − 4 = μ 2 16 n ( n − 1 ) ( n − 2 ) ( n − 3 ) (33a)

k − 2 = μ 2 4 ( 2 n − 1 ) n ( n − 1 ) (33b)

k − 1 = − i μ ω 0 2 α 0 ( t ) n (33c)

k + 1 = − i μ ω 0 2 α 0 ⋆ ( t ) n + 1 (33d)

k + 2 = − μ 2 4 ( 2 n + 3 ) ( n + 1 ) ( n + 2 ) (33e)

k + 4 = − μ 2 16 ( n + 1 ) ( n + 2 ) ( n + 3 ) ( n + 4 ) (33f)

Application of the RAM Equation (9) to second ameliorated order, gives the expressions of the Floquet operator and quasi-energies respectively, such as:

R ( 2 a ) 2 = ℏ ω 0 ( a + a + 1 2 ) + μ 2 3 4 ℏ ω 0 ( 2 ( a + a ) 2 + 2 a + a + 1 ) − μ 2 2 ℏ ω 0 8 ( 34 ( a + a ) 3 + 51 ( a + a ) 2 + 59 a + a + 21 ) + μ 2 ℏ ω 0 3 4 ( ν 2 − w 0 2 ) (34)

E ( 2 a ) n = ℏ ω 0 ( n + 1 2 ) + μ 2 3 4 ℏ ω 0 ( 2 n 2 + 2 n + 1 ) − μ 2 2 ℏ ω 0 8 ( 34 n 3 + 51 n 2 + 59 n + 21 ) + μ 2 ℏ ω 0 3 4 ( ν 2 − w 0 2 ) (35)

We note that the correction on quasi-energies of this system exist to first and second orders, and the Floquet shift levels depends on the parameters ( μ 2 , μ ) and the quantum number n, which means that these energies levels are not equidistants.

We also note that in the absence of the cubic and quatric anharmonic perturbations ( μ 1 = μ 2 = 0 ) we find the Floquet states and quasi-energies of the simple forced harmonic oscillator [

The Morse potential V ( q ) , is the agreed model for diatomic molecules and is given by [

V ( q ) = D e ( 1 − e − ρ q ) 2 (36)

where ρ is a parameter that controls the width of attraction (or measure the curvature at the bottom of the well), D e is a parameter that controls the depth of the attracting well (or the depth of potential well), and q is the variation of interatomic distance with respect to the equilibrium distance.

Let us consider the Taylor development of exponential term in V ( q ) to forth order, and collecting terms in this development gives,

V ( q ) = D e ρ 2 q 2 − D e ρ 3 q 3 + 7 12 D e ρ 4 q 4 (37)

Then we make the following changes to the notations:

ω 0 = ρ 2 D e m (38a)

μ 1 = − D e ρ 3 m ω 0 2 ℏ m w 0 (38b)

μ 2 = 7 12 D e ρ 4 ℏ m 2 ω 0 3 (38c)

where ω 0 is the vibrational constant with the reduced mass m of the diatomic molecule.

With the help of Equations ((36), (37)), we obtain the similar situation given, by the cubic together with quatric anharmonic oscillator, in the absence of the time dependent perturbation ( μ = 0 ), for which the Hamiltonian is given by,

H 3 ( t ) = p 2 2 m + D e ρ 2 q 2 − D e ρ 3 q 3 + 7 12 D e ρ 4 q 4 (39)

Using Equations (24) and (35), one can obtain the full quasi-energy to second order as:

E ( 2 a ) n = ℏ ω 0 ( n + 1 2 ) + 7 16 D e ρ 4 ℏ 2 m 2 w 0 2 ( 2 n 2 + 2 n + 1 ) − 7 16 D e 2 ρ 6 ℏ 2 m 3 w 0 4 ( n 2 + n + 11 30 ) − 49 1152 D e 2 ρ 8 ℏ 3 m 4 ω 0 5 ( 34 n 3 + 51 n 2 + 59 n + 21 ) (40)

The difference between two adjacent Floquet levels for the cubic and quatric anharmonic oscillators are given by,

Δ E = E ( 2 a ) n + 1 − E ( 2 a ) n = ℏ ω 0 [ 1 + 3 μ 2 ( n + 1 ) − μ 1 2 15 2 ( n + 1 ) − μ 2 2 51 4 ( ( n + 1 ) 2 + 7 17 ) ] (41)

We note that when n increases Δ E decreases, until becoming equal to zero when the energy level reached the dissociation energy of diatomic molecular system, then the quantum number takes the maximum value n max .

In the previous paragraphs we have developed calculations to first and second orders, and presented a number of results of the quantum anharmonic oscillator (Floquet states, quasi-energies).

The coefficients A ( n ) , B ( n ) , C ( n ) , D ( n ) , E ( n ) , F ( n ) , G ( n ) , H ( n ) , A ( 2 ) ( n ) , B ( 1 ) ( n ) and B ( 2 ) ( n ) , are given by Equation (15) in page (77) of the reference [

We observe in

We express E ( 2 a ) n of the Equation (40) as a function of ( n + 1 2 ) and since the experimental data show that the terms in ( n + 1 2 ) 3 are negligible [

E ( 2 a ) n ≈ ℏ ρ 2 D e m ( 1 − 3283 18432 ℏ 2 ρ 2 m D e ) ( n + 1 2 ) − ℏ 2 ρ 2 2 m ( n + 1 2 ) 2 (42)

Our method | Wang et al. [ | |
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The states | ψ n 〉 | | ψ ( 1 a ) n ( t ) 〉 = e − i E ( 1 a ) n t ℏ [ k − 4 | n − 4 〉 + c − 3 | n − 3 〉 + k − 2 | n − 2 〉 + c − 1 | n − 1 〉 + c + 1 | n + 1 〉 + k + 2 | n + 2 〉 + c + 3 | n + 3 〉 + k + 4 | n + 4 〉 + | n 〉 ] | | φ n 〉 = | n 〉 + A ( n ) | n − 1 〉 + B ( n ) | n + 1 〉 + C ( n ) | n − 2 〉 + D ( n ) | n + 2 〉 + E ( n ) | n − 3 〉 + F ( n ) | n + 3 〉 + G ( n ) | n − 4 〉 + H ( n ) | n + 4 〉 |

Quasi-energies | E ( 2 a ) n = ℏ ω 0 ( n + 1 2 ) + μ 2 3 4 ℏ ω 0 ( 2 n 2 + 2 n + 1 ) − μ 1 2 15 ℏ ω 0 4 ( n 2 + n + 11 30 ) − μ 2 2 ℏ ω 0 8 ( 34 n 3 + 51 n 2 + 59 n + 21 ) | E n = ℏ ω 0 [ ( n + 1 2 ) − ( α w ℏ ω 0 ) 2 A n ( 2 ) + β w ℏ ω 0 B n ( 1 ) − ( β w ℏ ω 0 ) 2 B n ( 2 ) ] |

Notations: m → m ⋆ , μ = 0 , μ 1 = α w ℏ ω 0 , μ 2 = β w ℏ ω 0 , q ^ = z L .

Molecule | D e ( eV ) | m ( amu ) | ρ ( m − 1 ) |
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HCl [ | 4.61907 | 0.9801045 | 1.86769967 × 10^{10} |

CO [ | 11.2256 | 6.8606719 | 2.29939732 × 10^{10} |

Molecule | n max | D e t h ( eV ) | E 0 ( eV ) | D 0 ( eV ) |
---|---|---|---|---|

HCl | 24 | 4.6126 | 0.1835 | 4.4291 |

CO | 83 | 11.2244 | 0.1341 | 11.0903 |

Let us consider the values of the parameters corresponding of (HCl) and (CO) diatomic molecules given in

The derivative of Equation (42) with respect to n and setting it equal to zero gives the vibrational level associated with the dissociation limit n max , which allowed us to find the values of the theoretical dissociation energy corresponding to the equilibrium point ( D e t h ) and the dissociation energy with respect to the zero-point level ( D 0 = D e t h − E 0 ) which are given in

The Floquet theory is one of the most useful tools which provides an alternative way for solving the Schrödinger equation of quantum systems with time-periodic Hamiltonian. In this paper, this approach was applied to driven quantum anharmonic oscillators. We have given the Floquet operators, solutions of the Schrödinger evolution equation, with the help of the RAM applied to first and to second ameliorated orders approximation and then we have calculated the Floquet states and the corresponding quasi-energies as well as the wave functions. Indeed, the approach used in our study determined, in a natural way, the explicit expressions for the time-dependent states of the anharmonic potential systems. It can be noticed that when we switch off the time-perturbation, we obtain the conservative energies of the cubic and the quatric anharmonic potential, and that the energy levels spacing decrease with increasing values of n and allows us to estimate the dissociation energy of the molecule.

The comparisons of our expressions with published works by other authors, which have used different methods [

The authors declare no conflicts of interest regarding the publication of this paper.

Idrissi, M.J., Fedoul, A., Sayouri, S. and Amila, I. (2020) Anharmonic Potentials Analysis through the Floquet Representation. Journal of Applied Mathematics and Physics, 8, 184-195. https://doi.org/10.4236/jamp.2020.81014

H ( t ) : Time-dependent Hamiltonian.

H 0 : Unperturbed Hamiltonian.

H ′ ( t ) : Interaction Hamiltonian.

λ : Amplitude of the perturbation.

ℏ : Reduced Planck's constant.

| ϕ n ( t ) 〉 : Eigenstates of the operator R.

E n : Eigenvalues of R (Quasi-energies).

| n 〉 : Sates of the unperturbed system.

| ψ n ( t ) 〉 : Floquet states.

U ( 2 a ) ( t ) : Second order ameliorated evolution operator.

p: Impulsion operator.

q: Position operator.

m: Mass of the system

ω 0 : Unperturbed oscillator frequency.

μ 1 : Amplitude of the cubic anharmonic oscillator.

μ : Amplitude of the time-dependent perturbation.

ν : Frequency of the time-dependent perturbation.

a: Annihilation operator.

a + : Creation operator.

μ 2 : Amplitude of the quatric anharmonic oscillator.

D e : Depth of potential well.

ρ : Parameter that controls the width of attraction.

D e t h : Theoretical dissociation energy corresponding to the equilibrium point.

E 0 : Zero-point energy (n = 0).

D 0 : Dissociation energy with respect to the zero-point level.