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Exact numerical solutions of the Schrödinger equation for a two-dimensional exciton in a constant magnetic field of arbitrary strength
Physica B 423 (2013) 31–37
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Exact numerical solutions of the Schrödinger equation for a two-dimensional exciton in a constant magnetic field of arbitrary strength Ngoc-Tram Hoang-Do a, Dang-Lan Pham b, Van-Hoang Le a,n a b
Department of Physics, Ho Chi Minh City University of Pedagogy 280, An Duong Vuong Street, District 5, Ho Chi Minh City, Vietnam Institute for Computational Science and Technology, Quang Trung Software Town, District 12, Ho Chi Minh City, Vietnam
art ic l e i nf o
a b s t r a c t
Article history: Received 22 March 2013 Received in revised form 10 April 2013 Accepted 11 April 2013 Available online 30 April 2013
Exact numerical solutions of the Schrödinger equation for a two-dimensional exciton in a constant magnetic field of arbitrary strength are obtained for not only the ground state but also high excited states. Toward this goal, the operator method is developed by combining with the Levi-Civita transformation which transforms the problem under investigation into that of a two-dimensional anharmonic oscillator. This development of the non-perturbation method is significant because it can be applied to other problems of two-dimensional atomic systems. The obtained energies and wave functions set a new record for their precision of up to 20 decimal places. Analyzing the obtained data we also find an interesting result that exact analytical solutions exist at some values of magnetic field intensity. & 2013 Elsevier B.V. All rights reserved.
Keywords: Two-dimensional exciton Exact solution Schrödinger equation Magnetic field
1. Introduction A two-dimensional exciton in a magnetic field has been of great interest to both theoretical and experimental researchers for many years [1–5] and continues to be after several new and interesting physical effects were discovered in recent years [6–10]. The energy spectrum and wavefunction of an exciton in magnetic field, therefore, need to be calculated with increasing precision. Since the 1990s, the perturbation method, the variational method and some other numerical methods have been employed to calculate the energy of this system in weak and strong magnetic field [2–4]. The solution of the problem for a medium magnetic field was calculated using extrapolation (see [3] and references therein). Note that there are some powerful methods which were developed for finding the numerical solutions of the problems with interaction potentials of general shape, see for example Refs. [11,12] for the potential morphing method. In the last decade, however, solving the problem with higher precision was preferred. In the work [13], the problem was solved using the mixed-basis variational method in combination with the shifted 1=N method, while in the work [14], the asymptotic iteration method was employed. Both of these methods provided solutions with a precision of up to seven decimal places for the − ground state 1s and the excited states 2p− and 3d only. Numerical results for higher excited states have not been obtained up till
n
Corresponding author. Tel.: +84 8 38352020-109; fax: +84 8 38398946. E-mail address: [email protected] (V.-H. Le).
0921-4526/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physb.2013.04.040
now. As per our understanding, increasing the precision and applying these methods to higher excited states are not easy and are inefficient in terms of computing resources. Therefore, in our point of view, developing a new method to calculate energy and wavefunctions for not only the ground state but also other high excited states with any given precision is of interest. In this work, we will use the operator method originated in Refs. [15,16] to obtain exact numerical solutions of the Schrödinger equation for a two-dimensional exciton in a constant magnetic field of arbitrary strength. The operator method was constructed by Komarov and Feranchuk in the 1980s and was employed to solve the Schrödinger equation for several different physical systems (for an overview, see [17] and references therein; see also [18,19] for more recent works). To distinguish this method from many other operator methods, in this paper we will refer to it as the FK operator method. This method is similar to the perturbation theory method in the splitting of the Hamiltonian into two components: (i) the major component whose exact analytical solutions can be derived and (ii) the perturbation component which is taken into account through corrections to energies and wavefunctions. However, the splitting of the Hamiltonian in the operator method simply bases on the representation of the creation and annihilation operators. In addition, by changing the value of a free parameter we can regularize the comparison between the major component and the perturbation. That makes the FK operator method applicable to non-perturbation problems. Furthermore, schemes of calculating high-order terms of corrections allow finding exact numerical solutions with a given precision with a high convergence rate [20–23].
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Unfortunately, the FK operator method can not be applied directly to atomic systems because the expression of Coulomb interaction which contains coordinates in the denominator can not be calculated using algebraic transformations of the creation and annihilation operators. We will overcome this difficulty of the FK operator method by using the Levi-Civita transformation [24]. According to the work [25], a two-dimensional exciton problem through the Levi-Civita transformation is equivalent to a harmonic oscillator problem. This connection makes it possible to obtain the wavefunctions of a two-dimensional exciton in an algebraic form which is very convenient for calculation. This leads us to using the Levi-Civita transformation to transform the problem of a twodimensional exciton in magnetic field into that of an anharmonic oscillator in order to apply the FK operator method. The obtained results are interesting in the meaning of developing a nonperturbation method for two-dimensional atomic systems, which can be applied to other specific problems of interest in recent years, such as those mentioned in Refs. [26,27]. In this work, new results are obtained while applying the developed FK operator method to the problem of a twodimensional exciton in a magnetic field. In principle, we can obtain both energies and wave functions with any given precision (exact numerical solutions) for any given state. Particularly, we develop an algorithm to get solutions with precision of up to 20 decimal places due to the high convergence rate of this method. − For the ground state 1s and the low excited states 2p− and 3d , whose results were presented with a precision of seven decimal places in Refs. [13,14], our results include these seven decimal places. Moreover, the computing program can be used to calculate energies and wavefunctions for any high excited states with the principal quantum number of up to 150. This is a new record. While analyzing the obtained data, we also find an interesting result: the values of magnetic field intensity at which the exact analytical solutions of the problem exist. The structure of this paper is as follows. After this introduction comes the section on theoretical and computational methods. We first introduce the Schrödinger equation of the system through the Levi-Civita transformation. Then we present the basic steps and the general principles of the FK operator method and derive the necessary formula for applying the FK operator method to the problem. In Section 3, we present the numerical results for the ground state as well as high excited states. Some discussions are given in this section too. Finally, the conclusion section summarizes the new results of this work and proposes some ideas for follow-up research.
2. Theoretical and computational methods 2.1. The Schrödinger equation through the Levi-Civita transformation In this section, the Levi-Civita transformation [24] is used to transform the Schrödinger equation of a two-dimensional exciton in a magnetic field into that of a two-dimensional anharmonic oscillator. The idea for this relation comes from our previous work [25] and the relation is originated in this paper. For a two-dimensional exciton in a magnetic field we consider the following Schrödinger equation written in atomic unit: ^ ðrÞ ¼ EΨ ðrÞ; HΨ
ð1Þ
1 ∂2 ∂2 iγ ∂ ∂ 1 Z H^ ¼ − x −y þ γ 2 ðx2 þ y2 Þ− : þ − 2 ∂x2 ∂y2 2 ∂y ∂x 8 r
ð2Þ
Here, the effective Rydberg constant: Rn ¼ mn e4 =16π 2 ε20 ℏ2 is an energy unit; the coordinates are measured in units of the effective
Bohr radius: an ¼ 4πε0 ℏ2 =mn e2 and the dimensionless parameter γ is defined by the formula γ ¼ ℏωc =2Rn with the cyclotron frequency ωc ¼ eB=2πmn and the magnetic field intensity B; mn , ε are the electron effective mass and the static dielectric constant, respectively; Z is the charge of the hole, which equals 1 in this case to compare the obtained results with those in Refs. [13,14]. In this work, we consider a wide range of γ covering both weak and strong magnetic field regions. We will now consider the Eqs. (1) and (2) in another space which is more convenient for calculation through the Levi-Civita transformation [21]: ( x ¼ u2 −v2 ð3Þ y ¼ 2uv: The transformation (3) connects the two real two-dimensional spaces ðx; yÞ and (u,v). We can easily prove the following equalities: dx dy ¼ 4ðu2 þ v2 Þ du dv; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ x2 þ y2 ¼ u2 þ v2
ð4Þ
which will be used in the following calculation. From Eq. (4) we see that the Jacobian of the transformation (3) is not a constant but is instead 4ðu2 þ v2 Þ, so it will appear as a weight in the equation for calculating the scalar product of two state vectors when transforming from the ðx; yÞ space to the ðu; vÞ space. This means that if a certain operator K^ is hermitian in the ðx; yÞ space then the operator K~ ¼ 4ðu2 þ v2 ÞK^ is also hermitian in the ðu; vÞ space. Hence, in order to ensure that the Hamiltonian is hermitian through the transformation (3), we need to rewrite Eq. (1) as follows: ^ rðH−EÞΨ ðrÞ ¼ 0: In the ðu; vÞ space, this equation reads ~ ðu; vÞ ¼ ZΨ ðu; vÞ HΨ with the Hamiltonian: 1 ∂2 ∂2 γ γ2 H~ ¼ − þ 2 − E− L^ z ðu2 þ v2 Þ þ ðu2 þ v2 Þ3 : 2 8 ∂u 2 8 ∂v
ð5Þ
ð6Þ
We see in Eqs. (5) and (6) an interchange of Z and E in the role of the eigenvalue. The energy E is no longer the eigenvalue but is instead a parameter, while Z becomes the eigenvalue of Eq. (5). However, we can still solve the Schrödinger equation to find E when keeping Z¼1 as presented in the following section. Furthermore, in the Hamiltonian (6) there is also the operator L^ z which is the orbital angular momentum operator. Because the system under investigation is two-dimensional, this operator is also its projection operator on a direction perpendicular to the plane of motion. In the ðu; vÞ space, this operator reads i ∂ ∂ u −v : ð7Þ L^ z ¼ − 2 ∂v ∂u We can also prove that L^ z commutes with the Hamiltonian of Eq. (5). This means that angular momentum is conserved in the problem under consideration. We will use this conservation by constructing a basis set for solving Eq. (5) which contains the eigenfunctions of the orbital angular momentum operator. Then we replace L^ z by its eigenvalue in the Hamiltonian (5). Now we can see that the Eqs. (5) and (6) represent a two-dimensional anharmonic oscillator. In other words, we have transformed a twodimensional exciton in magnetic field into an anharmonic oscillator via the Levi-Civita transformation. This result allows the application of the FK operator method to find the exact numerical solutions of Eqs. (5) and (6).
N.-T. Hoang-Do et al. / Physica B 423 (2013) 31–37
2.2. The FK operator method for solving the Schrödinger equation The FK operator method is introduced in details in the work [15,16]. Here, we will only present the basic steps via solving the Schrödinger equations (5) and (6). The method includes four steps, as follows. Step 1 is writing the Hamiltonian in algebraic form ∂ ∂ ^ a^ þ ; b^ þ ; γÞ ~ a; ^ b; H~ ; ; u; v; γ -Hð ð8Þ ∂u ∂v using the creation and annihilation operators defined as follows: rffiffiffiffi rffiffiffiffi ω 1 ∂ ω n 1 ∂ þ ^ ξþ ξ − ; a^ ðωÞ ¼ aðωÞ ¼ n ; 2 ω ∂ξ 2 ω ∂ξ rffiffiffiffi rffiffiffiffi þ ω n 1 ∂ ω 1 ∂ ^ ξ þ ; b^ ðωÞ ¼ ξ− bðωÞ ¼ ; ð9Þ 2 ω ∂ξ 2 ω ∂ξn in which the complex coordinates are defined as ξ ¼ u þ iv, ξn ¼ u−iv. We can easily check that the operators (9) satisfy the well-known commutation relations: þ
^ a^ ¼ 1; ½a;
^ b^ þ ¼ 1: ½b;
ð10Þ
In definition (9), we use complex coordinates for convenience in writing only. The positive real number ω given in Eq. (9) is considered a free parameter whose role in the method will be discussed in the next steps. Plugging (9) into (6) we obtain the algebraic form of the Hamiltonian as follows: 2 ^ γÞ ¼ ω −2E þ mγ ða^ þ a^ þ b^ þ b^ þ 1Þ ~ a^ þ ; b^ þ ; a; ^ b; Hð 4ω ω2 þ 2E−mγ þ ^ þ ^ ða^ b þ a^ bÞ − 4ω þ γ2 þ þ þ ða^ b^ þ a^ b^ þ a^ a^ þ b^ b^ þ 1Þ3 : þ 64ω3
We also have an algebraic form of the orbital angular momentum operator: þ þ ^ ^ b^ bÞ; L^ z ¼ −12ða^ a−
ð12Þ
which is a neutral operator. Step 2 is splitting the Hamiltonian (12) into two components– the major component and the perturbation one: þ
þ
þ
^ γ; ωÞ þ V~ ða^ þ ; b^ ; a; ^ γÞ ¼ H~ ða^ þ a; ^ γ; ωÞ: ~ a^ þ ; b^ ; a; ^ b^ b; ^ b; ^ b; Hð 0
ð13Þ
The separation (13) is done based on a completely different principle from that of the perturbation method. Here, the major component contains only neutral operators which are products of equal number of creation and annihilation operators: 2 þ þ ^ γ; ωÞ ¼ ω −2E þ mγ ða^ þ a^ þ b^ þ b^ þ 1Þ ^ b^ b; H~ 0 ða^ a; 4ω þ þ γ2 þ þ þ ða^ a^ þ b^ b^ þ 1Þ½ða^ a^ þ b^ b^ þ 1Þ 64ω3 þ þ ða^ a^ þ b^ b^ þ 4Þ þ þ þ 6a^ a^ b^ b^ þ 2:
þ þ ^ γ; ωÞ commutes with the ^ b^ b; We see that the operator H~ 0 ða^ a; þ þ ^ hence its exact solutions are the waveoperators a^ a^ and b^ b, functions of the harmonic oscillator described by these operators. Furthermore, notice that although the Hamiltonian of the system does not depend on the free parameter ω, the split components þ þ ^ γ; ωÞ and V~ ða^ þ ; b^ þ ; a; ^ γ; ωÞ do. This means that we ^ b^ b; ^ b; H~ 0 ða^ a; can adjust the correlation between the major and the perturbation components by changing the value of ω. Step 3 is finding the zeroth-order energy and wavefunction þ þ ^ γ; ωÞ. This ^ b^ b; using the approximate Hamiltonian H~ 0 ða^ a; þ þ ^ so its Hamiltonian contains only neutral operators a^ a^ and b^ b, eigenfunctions have the following form: þ þ ða^ Þj ðb^ Þk j0ðωÞ〉
ð16Þ
in which j, k are non-negative integers and the vacuum state j0ðωÞ〉 is defined from the equations: ^ aðωÞj0ðωÞ〉 ¼ 0;
^ bðωÞj0ðωÞ〉 ¼ 0;
ð14Þ
The rest is the perturbation term: þ þ ^ γ; ωÞ ^ b; V~ ða^ ; b^ ; a; 2 2 ω þ 2E−mγ þ ^ þ ^ þ γ ½ða^ þ b^ þ Þ3 þ ða^ bÞ ^ 3 ða^ b þ a^ bÞ ¼− 4ω 64ω3 þ þ þ þ þ þ þ þ þ3a^ b^ ða^ a^ þ b^ b^ þ 1Þ2 þ 3ða^ b^ Þ2 ða^ a^ þ b^ b^ þ 1Þ þ þ ^ 2 þ 3ða^ þ b^ þ Þ2 ða^ bÞ ^ þ 3a^ þ b^ þ ða^ bÞ ^ 2 þ3ða^ a^ þ b^ b^ þ 1Þða^ bÞ þ þ þ þ þ þ þ3ða^ a^ þ b^ b^ þ 1Þ2 a^ b^ þ 9a^ b^ ða^ a^ þ b^ b^ þ 1Þ þ þ þ þ ^ 2 þ 6a^ þ b^ þ þ 6a^ b: ^ þ9ða^ a^ þ b^ b^ þ 1Þa^ b^ þ 6ða^ b^ Þ2 þ 6ða^ bÞ
ð15Þ
ð17Þ
and the normalization equation: 〈0ðωÞj0ðωÞ〉 ¼ 1:
ð18Þ
As said in Section 2.1, angular momentum is conserved in the problem under consideration. We will use this conservation by constructing a basis set which contains the eigenfunctions of the orbital angular momentum operator L^ z . This operator in the algebraic form (12) is also a neutral operator, hence its eigenfunctions are also in the form (16). Therefore, the wavefunction vectors (16) are rewritten in normalization form as follows: þ 1 þ jnðmÞ〉 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða^ Þn−m ðb^ Þnþm j0ðωÞ〉 ðn−mÞ!ðn þ mÞ!
ð11Þ
33
ð19Þ
in which the principal quantum numbers n are non-negative integers: n ¼0, 1, 2,…, and the magnetic quantum numbers m are integers satisfying the condition −n ≤m ≤ n. Furthermore, we use the notation n ¼ nr þ jmj in which nr ¼ 0, 1, 2, … are radial quantum numbers. Thus we have the zeroth-order wavefunction corresponding to the state of principal quantum number n and the magnetic quantum numbers m: jψ nðmÞ 〉ð0Þ ¼ jnðmÞ〉;
ð20Þ
Now we let the Hamiltonian (14) act on the wavefunction (20) and consider the Eq. (5). As a result, we obtain the zeroth-order energy: Eð0Þ ¼ −
2ωZ 1 γ2 1 þ ω2 þ ð5n2 þ 5n−3m2 þ 3Þ þ mγ: 2n þ 1 2 2 16ω2
ð21Þ
Here the free parameter is determined from the condition ∂Eð0Þ =∂ω ¼ 0 [15], which leads to the following equation: −
2Z γ2 þ ω− 3 ð5n2 þ 5n−3m2 þ 3Þ ¼ 0: 2n þ 1 8ω
ð22Þ
A numerical analysis of the analytical solutions (21) will be given in the next section on results. Step 4 is calculating high-order corrections to obtain exact numerical solutions. In principle, we may use different schemes, e. g. the perturbation theory scheme for calculating high-order corrections in order to obtain the energy and the wavefunction with higher precision. If that scheme leads to a result that converges to a certain value with any given precision then we have the exact numerical solution. In this work, we will propose an iteration scheme to calculate the energy and wavefunction with a given precision. For convenience we rewrite the Eqs. (5) and (6) as follows: ~ ðH~ −ERÞjψ〉 ¼0 R
ð23Þ
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N.-T. Hoang-Do et al. / Physica B 423 (2013) 31–37
R in which the operator H~ and R~ takes the following forms: þ R ω2 þ mγ þ ω2 −mγ þ ^ þ ^ ða^ a^ þ b^ b^ þ 1Þ− ða^ b þ a^ bÞ H~ ¼ 4ω 4ω þ γ2 þ þ þ ða^ b^ þ a^ b^ þ a^ a^ þ b^ b^ þ 1Þ3 −1; þ 64ω3 þ 1 þ ^þ þ ða^ b þ a^ b^ þ a^ a^ þ b^ b^ þ 1Þ: R~ ¼ 2ω
in which the coefficients C ðsÞ ðk ¼ jmj; jmj þ 1; …; n−1; n þ 1; …; n þ k sÞ are determined by a system of n þ s−jmj linear equations: nþs
∑
k ¼ jmj;k≠n;k≠j
ð24Þ
We will use the basic set of wavefunctions (19) to construct the wavefunctions of the problem at hands. The matrix elements of the operators (24) corresponding to this basic set will be calculated through purely algebraic transformation. In fact, using the commutations (10) and the Eqs. (17) and (18), we easily obtain following formula: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ þ a^ b^ jnðmÞ〉 ¼ ðn þ 1Þ2 −m2 jn þ 1ðmÞ〉; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ a^ bjnðmÞ〉 ¼ n2 −m2 jn−1ðmÞ〉; þ þ ^ ¼ 2njnðmÞ〉 ð25Þ ða^ a^ þ b^ bÞjnðmÞ〉 from which we calculate the matrix elements as follows: H Rnn ¼ 〈nðmÞjH~ jnðmÞ〉 ω2 −mγ γ2 ð2n þ 1Þ þ ¼ ð2n þ 1Þð5n2 þ 5n þ 3−3m2 Þ−Z; 4ω 32ω3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R
R H Rn;nþ1 ¼ 〈nðmÞjH~ jn þ 1ðmÞ〉 ¼ ðn þ 1Þ2 −m2 ω2 þ mγ 3γ 2 þ − ð5n2 þ 10n þ 6−m2 Þ ; 3 4ω 64ω
H Rn;nþ2 ¼ 〈nðmÞjH~ jn þ 2ðmÞ〉 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3γ 2 ð2n þ 3Þ ðn þ 1Þ2 −m2 ðn þ 2Þ2 −m2 ; ¼ 64ω3 R H Rn;nþ3 ¼ 〈nðmÞjH~ jn þ 3ðmÞ〉 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ2 ðn þ 1Þ2 −m2 ðn þ 2Þ2 −m2 ðn þ 3Þ2 −m2 ; ¼ 3 64ω 2n þ 1 ~ Rnn ¼ 〈nðmÞjRjnðmÞ〉 ; ¼ 2ω qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ þ 1ðmÞ〉 ¼ − 1 Rn;nþ1 ¼ 〈nðmÞjRjn ðn þ 1Þ2 −m2 : 2ω R
ð26Þ
Besides (26), we can calculate other non-zero matrix elements using the symmetry properties H Rnk ¼ H Rkn , Rnk ¼ Rkn . We can write the exact wavefunction as a linear combination of the basic functions (19): jψ nðmÞ 〉 ¼ jnðmÞ〉 þ
þ∞
∑
j ¼ jmj;j≠n
C j jjðmÞ〉;
ð27Þ
and define the approximate wavefunction at the s-th order approximation (s-th iteration loop) as jψ nðmÞ 〉ðsÞ ¼ jnðmÞ〉 þ
nþs
∑
j ¼ jmj;j≠n
C ðsÞ j jjðmÞ〉
ð28Þ
corresponding to the approximate energy EðsÞ . If lims-∞ C ðsÞ ¼ Ck k then the approximate wavefunction (28) converges to the exact wavefunction (27). If then we have EðsÞ -ET while s- þ ∞, we also have the exact energy. We will use the notations for quantum − states as in the work [13,14], e.g.: 1s, 2s, 2p− , 2pþ , 3s, 3p− , 3pþ , 3d , þ 3d , etc. The first digit stands for the state level n þ 1, in which n is the principal quantum number defined in Eq. (19); the letters symbolize the orbital quantum numbers l ¼ jmj: s corresponding to l ¼ 0, p to l ¼1, d to l ¼2, f to l¼ 3, g to l¼4, etc; the 7 sign is of the magnetic quantum number m. Putting Eq. (28) into the Schrödinger equation (23), we obtain the expression for the approximate s-th order energy: EðsÞ ¼
H Rnn þ ∑nþs C ðsÞ H Rnk k ¼ jmj;k≠n k Rnn þ ∑nþs C ðsÞ Rnk k ¼ jmj;k≠n k
ð29Þ
ðH Rjk −Eðs−1Þ ÞC ðsÞ ¼ Rnj Eðs−1Þ −H Rnj ; k
ð30Þ
in which j ¼ jmj; jmj þ 1; …; n−1; n þ 1; …; n þ s. So, by plugging the solutions of Eq. (30) into Eq. (29) we obtain the energy of the system at the s-th iteration loop, which is called the s-th order approximation energy. Numerical results show that with appropriate choice of ω, for the considered problem we obtain a series of approximate energies Eð0Þ ; Eð1Þ ; Eð2Þ ; …; EðsÞ ; …
ð31Þ
which rapidly converges to a value ET. We call them the exact numerical solutions because their values can be obtained with any given precision. In this work, we calculate these solutions with precision of up to 20 decimal places due to the limitation of the default precision for real numbers in FORTRAN and the computing speed of computer. However, this is an important progress. As we know, the precision for real numbers in FORTRAN is limited to 15 decimal places. By appropriate programming, we can increase this precision up to 50 decimal places. In our final results for energies and wavefunctions, we require a precision of up to 20 decimal places in order to avoid the accumulation of errors in calculating. Besides energy, the coefficients C ðsÞ also converge rapidly so we k obtain not only exact numerical energy but also exact numerical wavefunctions. The parameter ω is chosen using the method described in Refs. [20–23]. In principle, this parameter does not affect the exact numerical results. However, investigation shows that the convergence rate to the exact solutions depends significantly on the choice of ω. In this work, the equation ∂Eð0Þ =∂ω ¼ 0 provides the first value ω0 which is not the optimal value. For high excited states nr, such a choice of ω even does not lead to convergence to exact values. We can scan to find the optimal value of parameter around the first value of ω0 . The results in this paper reconfirm the conclusions of Refs. [22,23] about the existence of the range of parameter such that the convergence rate of Eq. (28) is highest.
3. Results and discussions Analytical solutions: Fig. 1 shows the energies of the system corresponding to the ground state and some excited states 1s, 2s, − 2p− , 3d . The dotted line represents the results, which can be considered analytical solutions, obtained with zeroth-order approximation using the Eqs. (20) and (21). The solid line represents the exact numerical results for comparison purposes. These exact results are obtained from the iteration Eqs. (26) and (27). On this figure, we see that the analytical solutions are pretty precise for the ground and excited states in magnetic field region γ ≤ 1. In stronger magnetic field, this precision decreases while the correlation between the energy levels stays the same. In order to obtain analytical solutions which are highly precise and are uniformly suitable in the whole range of magnetic field, we need to consider the asymptotic behavior of the wavefunction in strong magnetic field region. We will discuss this case in another work. Exact numerical solutions: In this work, we focus on exact numerical solutions. Table 1 presents energies with precision of up to 20 decimal places for the ground state 1s (n¼ m ¼0) and − some excited states 2p− ðn ¼ 1; m ¼ −1Þ, 3d ðn ¼ 2; m ¼ −2Þ. These states were calculated with the precision of up to seven decimal places in Refs. [13,14]. Here, the results in Refs. [13,14] are not shown because all of these seven places are included in our results shown in Table 1. The energy of the 2pþ ðn ¼ 1; m ¼ þ 1Þ,
N.-T. Hoang-Do et al. / Physica B 423 (2013) 31–37 þ
3d ðn ¼ 2; m ¼ þ 2Þ states can be calculated from that of the 2p− , − 3d states based on the relation EnðmÞ ¼ Enð−mÞ þ mγ. The energies with precision of up to 20 decimal places presented in Table 1 are a new record presented for the first time in this paper. By choosing the optimal value of the free parameter, to obtain the ground state energy with seven decimal places we need s ¼6 iterations loops for γ′ ¼ 0:05 (weak magnetic field), s¼ 8 iteration loops for γ′ ¼ 0:55 (medium magnetic field), s ¼26 iteration loops for γ′ ¼ 0:95 (strong magnetic field); and to obtain the precision of 20 decimal places we need s ¼16, 35, 124 iterations loops for the three cases above, respectively. For excited states, more iteration loops are required, but not more than 300. Although the results presented in Table 1 are with 20 decimal places, we did attempt to run the program to get up to 50 decimal places for the ground state and some low excited states. Hence, we conclude that the method introduced in this paper provides numerical results with any given precision. In other words, they are exact numerical solutions. The wavefunction of the system in the form (28) is also obtained by calculating the coefficients C ðsÞ . The more iteration k
Fig. 1. Energies obtained from analytical equations are compared with exact energies of the system for the 1s, 2s, 2p−, 3d− states. For the ground state, the analytical solution is pretty precise, while its error increases for the excited states in the region of strong magnetic field.
35
loops are carried out, the more coefficients are obtained. These coefficients converge to a certain value. Hence, we obtain the exact numerical wavefunction. For example, for the ground state in weak magnetic field γ′ ¼ 0:05, when the energy is obtained with precision of 20 decimal places, the obtained wavefunction has 18 coefficients C ð15Þ , three of which are with precision of 20 decimal k places and the others with precision ranging from 1 to 19 decimal places. Similarly, for the case γ′ ¼ 0:55, there are 38 coefficients C ð35Þ with precision ranging from 1 to 19 decimal places. Furtherk more, we can increase the precision of these coefficients up to 20 decimal places by running some more iteration loops. We notice that the states presented in Table 1 and considered in the works [13,14] are special cases. Indeed, they are 1s (n ¼0, − m ¼0), 2p− ðn ¼ 1; m ¼ −1Þ, 3d ðn ¼ 2; m ¼ −2Þ which correspond to the principal quantum number n ¼ jmj. It means the radial quantum number nr ¼ 0 for all the states mentioned above. All of them are the lowest states when the magnetic quantum numbers are fixed, so they can be considered the ground states of a one-dimensional motion remaining after the motion related to angular momentum is taken off. For these states, the variational method works well but for states with nr ≠0, it does not. We have not found any follow-up works of [13,14] on these states. Therefore, our success in obtaining the exact solutions not only for the ground state but also for any excited states corresponding to nr ¼ 0 as well as nr ≠0 is a significant development. The computing programs are tested for excited states with principal quantum number of up to n ¼150. The exact energies for the states with the − − principal quantum number of up to n ¼4: 2s, 3s, 3p− , 4p− , 4d , 4f , − − − 5d , 5f , 5g are presented in Tables 2–4 for illustration. These data which are original and are presented for the first time in this paper would be interesting for further reference. We also found an interesting result while analyzing data in Table 2. For the case γ′ ¼ 0:8, corresponding to γ ¼ 4, the energy of the 2s state is exactly equal to 4. Similarly, for the 3s state and γ′ ¼ 0:4, corresponding to γ ¼ 2=3, the energy is exactly equal to 1. That means there are some values of magnetic field at which exact analytical solutions of the problem can be derived. This detection needs to be investigated in more details in another research which we will introduce in a follow-up work. The numerical results in the tables are also illustrated in Figs. 2 and 3. We see that for magnetic field γ 4 0:1, the levels of energies corresponding to the principal quantum numbers begin
Table 1 Exact energies for the ground state and some low-order excited states with different values of magnetic field. These results are presented with 20 decimal places and are included results of other authors with seven decimal places obtained from the variational method combined with 1=N expansion [13] and by the asymptotic perturbation method [14]. For comparison purpose, the magnetic field strength is represented by γ′ ¼ γ=ðγ þ 1Þ. −
γ′
1s
2p−
3d
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95
−1.99987017095990613170 −1.99942166507712501693 −1.99854256186645709957 −1.99707966222489033807 −1.99482091674920995739 −1.99146906712008245351 −1.98660128273873874036 −1.97960585091255737141 −1.96958003578721667516 −1.95515968324675916048 −1.93422334282651768145 −1.90335295328070946150 −1.85679038695312423867 −1.78426176250820398205 −1.66598114826115218564 −1.45958713448989797632 −1.05902943085968226100 −0.12110157606243169047 3.23173503617812205039
−0.24474134557798881450 −0.26197520208978872077 −0.27410756889059385484 −0.28145662051014027268 −0.28409801197305611464 −0.28179705884272357348 −0.27398063451910408445 −0.25967324608666750763 −0.23736745684644382973 −0.20479038588298687027 −0.15849200397641926789 −0.09310153718673498788 0.00008428907312278363 0.13597809871739040778 0.34214588246038558324 0.67521872604493890722 1.27112332610485384793 2.55062439464316783261 6.70030514510251978855
−0.11440510079734191774 −0.13025445178430606781 −0.13674173884597401955 −0.13659636693603053469 −0.13064401380817189943 −0.11888179385839621521 −0.10076026619266554000 −0.07521698217897997805 −0.04057485324928431054 0.00569412879010892499 0.06740516636374326600 0.15045496413924365797 0.26432778727006523882 0.42515253567577483784 0.66251797890286951771 1.03673987328882310035 1.69141167638720390304 3.06707775217066545341 7.42860734182398557718
36
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Table 2 Energies of some excited states corresponding to different values of the magnetic field. γ′
2s
3s
3p−
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95
−0.21728279396968785397 −0.20160591618074023247 −0.17457048111136615243 −0.13546551574551668372 −0.08298007150252920818 −0.01502952750059654076 0.07142846360607761967 0.18070180520537274524 0.31887814627104300055 0.49467963883695179668 0.72091579831145684540 1.01705668872877975106 1.41405521488835508770 1.96407032991584636290 2.76203075990704173477 4.000000000000000000 6.13177508892586718321 10.53825335361872298997 24.24854780554921085193
−0.05137115812285705179 0.01598790970267759145 0.10955543404170749556 0.22721834540118963879 0.37047446264365413831 0.54292733023387127727 0.75013268675298689141 1.000000000000000000 1.30366017149884873723 1.67697071075735877489 2.14309105414624009242 2.73703806534873385927 3.51419634203232951294 4.56741386886003820741 6.06476489650834377662 8.34434942653877459541 12.19997778752228732308 20.02986405533713661963 43.93175572792464946928
−0.08036721744503109760 −0.04916051442900800150 0.00108637565485044813 0.06738328574548576958 0.14986742555393444603 0.25028633092701018244 0.37170668580947103502 0.51867414992190953559 0.69769322374130095062 0.91810396914076738153 1.19360373208799551825 1.54495188613538814666 2.00503229789835964832 2.62903276273533536486 3.51694576840641184652 4.87008864382744499370 7.16153917283062994829 11.82192792999323245234 26.07692664470341686614
Table 3 Energies of some high excited states with n¼ 3 corresponding to different values of the magnetic field. −
γ′
4p−
4d
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95
0.00090050861670000681 0.09390796453557639448 0.21323373403753960833 0.35668048473475479870 0.52606940637774516311 0.72533207396702501971 0.96036057230843709310 1.23944041995229360305 1.57415466587235095938 1.98093584143111358970 2.48370977153057734079 3.11856112568506923453 3.94243430849379882861 5.05058514254628752285 6.61507130462100771829 8.98122289598444673098 12.95785147539592703023 20.98177530017893504943 45.30314463505568435467
−0.03603819112188519016 0.00747840526168139693 0.06842568965737196928 0.14447370643180047832 0.23628341974950389743 0.34594127952399409512 0.47676457059152532284 0.63351735049084262159 0.82292516365874272605 1.05458031287213984997 1.34249561828604384524 1.70785470976146371028 2.18414967307751588723 2.82749853910929518464 3.73942662403763270927 5.12403034105099778843 7.46015607265407629634 12.19329771792373821144 26.60790678762850209795
4f
−
−0.07763315014992853255 −0.08803520910560353969 −0.08888339654701696510 −0.08315384587850637174 −0.07160076440801957770 −0.05412267756001832601 −0.03006828179726522415 0.00173399981723935616 0.04308672404388300910 0.09667213220019815981 0.16650643696559374134 0.25876268139530569281 0.38332586831977878052 0.55694141965414866451 0.81021701467250590897 1.20531843995110106672 1.88968201934687586146 3.31375703786321820117 7.78159090037994763851
Table 4 Energies of some excited states with n¼ 4 corresponding to different values of the magnetic field. −
γ′
5d
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95
0.02905749316697708979 0.13341069872580500192 0.26190026516127302629 0.41356740099965750564 0.59077510317782206889 0.79776115680429092118 1.04063336998574952266 1.32786142397127676506 1.67121288127923035467 2.08732965304714088906 2.60039856986566405608 3.24685460813888191684 4.08414376941622929138 5.20828847495739630265 6.79260902069767290808 9.18472403676149577875 13.19819451070249609938 21.28207427023502943214 45.73489523603524444325
5f
−
−0.01420058007237175099 0.03570019722639813037 0.10204170420617281261 0.18308786847417280362 0.27975368917130985634 0.39427787158829185045 0.53009357416698937372 0.69207101474623163562 0.88704788903212652389 1.12474823987081387279 1.41935251349090562787 1.79227191808221737000 2.27732643588718193308 2.93113687655407534814 3.85605806748978018262 5.25769124636313341999 7.61800718398620959057 12.39055080521797139007 26.89161351897346612252
5g− −0.06007875519489516057 −0.06604657997486724854 −0.06292240231373530831 −0.05344145079515578704 −0.03821140930631437516 −0.01703067582967511760 0.01083372779566949526 0.04663124135459961939 0.09224854476563663044 0.15046790575342597724 0.22543362638168408645 0.32349322334645939986 0.45478350257101208093 0.63643631212786443537 0.89969714506691749633 1.30788909126295026871 2.01085156632785457493 3.46522653749514333096 7.99955272626774688799
N.-T. Hoang-Do et al. / Physica B 423 (2013) 31–37
Fig. 2. The dependence of energy on magnetic field strength for the states with principal quantum number n¼ 3: 1s, 2s, 2p 7 , 3s, 3p 7 , 3d 7 , 4p 7 , 4d 7 , 4f 7 . We see that in strong magnetic fields, the degenerated Landau levels split out due to Coulomb interaction although they are still close to each other and are magnified in (a), (b), (c), and (d).
37
(2) Successful development of the FK operator method for a twodimensional exciton in a magnetic field. This development is significant, universal and applicable to various problems related to two-dimensional exciton in electromagnetic field. (3) Obtaining exact numerical solutions for a two-dimensional exciton in a constant magnetic field with arbitrary strength. For the ground state and the excited states considered in previous works of other authors, we set a new record on precision of up to 20 decimal places. Furthermore, exact solutions for high excited states are presented for the first time in this work. The FORTRAN program for calculating energies and wavefunctions of any excited states with principal quantum number of up to hundreds can be provided to interested researchers. These results can be used for research related to the energy levels of a two-dimensional exciton in a magnetic field. (4) Detecting some concrete values of magnetic field strength at which the problem of a two-dimensional exciton in a magnetic field has exact analytical solutions. This detection is very interesting and needs more detailed research.
Acknowledgments We would like to thank Professor Feranchuk I. D. (Belarusian State University, Minsk – Belarus) for careful reading and helpful comments on this work. This work is sponsored by the National Foundation for Science and Technology Development (NAFOSTED) Grant no. 103.012011.08. The author Hoang-Do also would like to thank for the institutional grant provided by Ho Chi Minh City University of Pedagogy. References [1] [2] [3] [4] [5] Fig. 3. Energy levels in γ ≤1 region of magnetic field.
[6] −
to disarrange. For example, 2s and 2pþ levels are higher than 3d − − − and 4f levels; 3s and 3pþ are higher than 4d , 4f levels. That is because we use the principal quantum numbers of the Coulomb problem which is reasonable only in weak magnetic field. In strong magnetic field, Coulomb interactions are considered a perturbation which eliminates the degenerate Landau levels. Thus, in this case the principal quantum numbers must follow Landau levels in the problem of motion of electrons in a uniform magnetic field. Our finding of exact solutions for high excited states with principal quantum number of up to hundreds allows investigation of not only the degenerate separation of Landau levels but also quantum chaos effects. These are some suggestions for further research.
4. Conclusion and outlook
[7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
In this work we obtain the following original results: [23]
(1) Transforming the problem of a two-dimensional exciton in a magnetic field into that of a two-dimensional anharmonic oscillator through Levi-Civita transformation. This is an important result because the problem of an anharmonic oscillator is much simpler and is well-investigated using several methods.
[24] [25] [26] [27]
D. Paquet, T.M. Rice, K. Ueda, Phys. Rev. B 32 (1985) 5208. W. Edelstein, Phys. Rev. B 39 (1989) 7697. Jia-Lin Zhu, Y. Cheng, Jia-Jiong Xiong, Phys. Rev. B 41 (1990) 10792. A.B. Dzyubenko, Phys. Rev. B 64 (2001) 241101 (R). L.V. Butov, C.W. Lai, D.S. Chemla, Yu.E. Lozovik, K.L. Campman, A.C. Gossard, Phys. Rev. Lett. 87 (2001) 216804. G.V. Astakhov, D.R. Yakolev, V.V. Rudenkov, P.C.H. Christianen, T. Barrick, S. A. Gooker, A.B. Dzyubenko, W. Ossau, J.C. Maan, G. Karczewshi, T. Wojtowicz, Phys. Rev. B 71 (2005) 201312 (R). A. Bruno-Alfonso, L. Candido, G.Q. Hai, J. Phys.: Condens. Matter 22 (2010) 125801. A. Poszwa, Phys. Scripta 84 (2011) 055002. D. Nandi, A.D.K. Finck, J.P. Eisenstein, L.N. Pfeiffer, K.W. West, Nature 488 (2012) 481. Z. Zeng, C.S. Garoufalis, S. Baskoutas, J. Phys. D 45 (2012) 235102. Z. Zeng, E. Paspalakis, C.S. Garoufalis, Andreas F. Terzis, S. Baskoutas, J. Appl. Phys. 113 (2013) 054303. M. Rieth, W. Schommers, S. Baskoutas, Int. J. Mod. Phys. B 16 (2002) 4081. V.M. Villalba, R. Pino, Physica B 315 (2002) 289. A. Soylu, I. Boztosun, Physica E 40 (2008) 443. I.D. Feranchuk, L.I. Komarov, Phys. Lett. A 88 (1982) 212. I.D. Feranchuk, L.I. Komarov, I.V. Nichipor, A.P. Ulyanenkov, Ann. Phys. 238 (1995) 370. I.D. Feranchuk, A. Ivanov, in: Etude on Theoretical Physics, World Scientific, Singapore, 2004, pp. 171–188. I.D. Feranchuk, A.V. Leonov, Phys. Lett. A 373 (2009) 517. A.V. Leonov, I.D. Feranchuk, J. Appl. Spectrosc. 77 (2011) 832. F.M. Fernandez, A.M. Meson, E.A. Castro, Phys. Lett. A 18 (1984) 401. F.M. Fernandez, A.M. Meson, E.A. Castro, Phys. Lett. A 18 (1985) 104. Chan Za An, I.D. Feranchuk, L.I. Komarov, L.S. Nakhamchik, J. Phys. A 19 (1986) 1583. Quoc-Khanh Hoang, Van-Hoang Le, L.I. Komarov, Proceed. Acad. Sci. Belarus (Phys. Math. Ser.) 3 (1997) 71. T Levi-Civita, Opere Matematiche. Memorie e note. Vol. II. 1901-1907, (Nicola Zanichelli Editore, Bologna, 1956). Van-Hoang Le, Thu-Giang Nguyen, J. Phys. A 26 (1993) 1409. J. Zhu, S.L. Ban, S.H. Ha, Phys. Status Solidi B 248 (2011) 384. F. Milota, J. Sperling, A. Nemeth, T. Mancal, H.F. Kauffmann, Acc. Chem. Res. 42 (2009) 1364.
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Physica B journal homepage: www.elsevier.com/locate/physb
Exact numerical solutions of the Schrödinger equation for a two-dimensional exciton in a constant magnetic field of arbitrary strength Ngoc-Tram Hoang-Do a, Dang-Lan Pham b, Van-Hoang Le a,n a b
Department of Physics, Ho Chi Minh City University of Pedagogy 280, An Duong Vuong Street, District 5, Ho Chi Minh City, Vietnam Institute for Computational Science and Technology, Quang Trung Software Town, District 12, Ho Chi Minh City, Vietnam
art ic l e i nf o
a b s t r a c t
Article history: Received 22 March 2013 Received in revised form 10 April 2013 Accepted 11 April 2013 Available online 30 April 2013
Exact numerical solutions of the Schrödinger equation for a two-dimensional exciton in a constant magnetic field of arbitrary strength are obtained for not only the ground state but also high excited states. Toward this goal, the operator method is developed by combining with the Levi-Civita transformation which transforms the problem under investigation into that of a two-dimensional anharmonic oscillator. This development of the non-perturbation method is significant because it can be applied to other problems of two-dimensional atomic systems. The obtained energies and wave functions set a new record for their precision of up to 20 decimal places. Analyzing the obtained data we also find an interesting result that exact analytical solutions exist at some values of magnetic field intensity. & 2013 Elsevier B.V. All rights reserved.
Keywords: Two-dimensional exciton Exact solution Schrödinger equation Magnetic field
1. Introduction A two-dimensional exciton in a magnetic field has been of great interest to both theoretical and experimental researchers for many years [1–5] and continues to be after several new and interesting physical effects were discovered in recent years [6–10]. The energy spectrum and wavefunction of an exciton in magnetic field, therefore, need to be calculated with increasing precision. Since the 1990s, the perturbation method, the variational method and some other numerical methods have been employed to calculate the energy of this system in weak and strong magnetic field [2–4]. The solution of the problem for a medium magnetic field was calculated using extrapolation (see [3] and references therein). Note that there are some powerful methods which were developed for finding the numerical solutions of the problems with interaction potentials of general shape, see for example Refs. [11,12] for the potential morphing method. In the last decade, however, solving the problem with higher precision was preferred. In the work [13], the problem was solved using the mixed-basis variational method in combination with the shifted 1=N method, while in the work [14], the asymptotic iteration method was employed. Both of these methods provided solutions with a precision of up to seven decimal places for the − ground state 1s and the excited states 2p− and 3d only. Numerical results for higher excited states have not been obtained up till
n
Corresponding author. Tel.: +84 8 38352020-109; fax: +84 8 38398946. E-mail address: [email protected] (V.-H. Le).
0921-4526/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physb.2013.04.040
now. As per our understanding, increasing the precision and applying these methods to higher excited states are not easy and are inefficient in terms of computing resources. Therefore, in our point of view, developing a new method to calculate energy and wavefunctions for not only the ground state but also other high excited states with any given precision is of interest. In this work, we will use the operator method originated in Refs. [15,16] to obtain exact numerical solutions of the Schrödinger equation for a two-dimensional exciton in a constant magnetic field of arbitrary strength. The operator method was constructed by Komarov and Feranchuk in the 1980s and was employed to solve the Schrödinger equation for several different physical systems (for an overview, see [17] and references therein; see also [18,19] for more recent works). To distinguish this method from many other operator methods, in this paper we will refer to it as the FK operator method. This method is similar to the perturbation theory method in the splitting of the Hamiltonian into two components: (i) the major component whose exact analytical solutions can be derived and (ii) the perturbation component which is taken into account through corrections to energies and wavefunctions. However, the splitting of the Hamiltonian in the operator method simply bases on the representation of the creation and annihilation operators. In addition, by changing the value of a free parameter we can regularize the comparison between the major component and the perturbation. That makes the FK operator method applicable to non-perturbation problems. Furthermore, schemes of calculating high-order terms of corrections allow finding exact numerical solutions with a given precision with a high convergence rate [20–23].
32
N.-T. Hoang-Do et al. / Physica B 423 (2013) 31–37
Unfortunately, the FK operator method can not be applied directly to atomic systems because the expression of Coulomb interaction which contains coordinates in the denominator can not be calculated using algebraic transformations of the creation and annihilation operators. We will overcome this difficulty of the FK operator method by using the Levi-Civita transformation [24]. According to the work [25], a two-dimensional exciton problem through the Levi-Civita transformation is equivalent to a harmonic oscillator problem. This connection makes it possible to obtain the wavefunctions of a two-dimensional exciton in an algebraic form which is very convenient for calculation. This leads us to using the Levi-Civita transformation to transform the problem of a twodimensional exciton in magnetic field into that of an anharmonic oscillator in order to apply the FK operator method. The obtained results are interesting in the meaning of developing a nonperturbation method for two-dimensional atomic systems, which can be applied to other specific problems of interest in recent years, such as those mentioned in Refs. [26,27]. In this work, new results are obtained while applying the developed FK operator method to the problem of a twodimensional exciton in a magnetic field. In principle, we can obtain both energies and wave functions with any given precision (exact numerical solutions) for any given state. Particularly, we develop an algorithm to get solutions with precision of up to 20 decimal places due to the high convergence rate of this method. − For the ground state 1s and the low excited states 2p− and 3d , whose results were presented with a precision of seven decimal places in Refs. [13,14], our results include these seven decimal places. Moreover, the computing program can be used to calculate energies and wavefunctions for any high excited states with the principal quantum number of up to 150. This is a new record. While analyzing the obtained data, we also find an interesting result: the values of magnetic field intensity at which the exact analytical solutions of the problem exist. The structure of this paper is as follows. After this introduction comes the section on theoretical and computational methods. We first introduce the Schrödinger equation of the system through the Levi-Civita transformation. Then we present the basic steps and the general principles of the FK operator method and derive the necessary formula for applying the FK operator method to the problem. In Section 3, we present the numerical results for the ground state as well as high excited states. Some discussions are given in this section too. Finally, the conclusion section summarizes the new results of this work and proposes some ideas for follow-up research.
2. Theoretical and computational methods 2.1. The Schrödinger equation through the Levi-Civita transformation In this section, the Levi-Civita transformation [24] is used to transform the Schrödinger equation of a two-dimensional exciton in a magnetic field into that of a two-dimensional anharmonic oscillator. The idea for this relation comes from our previous work [25] and the relation is originated in this paper. For a two-dimensional exciton in a magnetic field we consider the following Schrödinger equation written in atomic unit: ^ ðrÞ ¼ EΨ ðrÞ; HΨ
ð1Þ
1 ∂2 ∂2 iγ ∂ ∂ 1 Z H^ ¼ − x −y þ γ 2 ðx2 þ y2 Þ− : þ − 2 ∂x2 ∂y2 2 ∂y ∂x 8 r
ð2Þ
Here, the effective Rydberg constant: Rn ¼ mn e4 =16π 2 ε20 ℏ2 is an energy unit; the coordinates are measured in units of the effective
Bohr radius: an ¼ 4πε0 ℏ2 =mn e2 and the dimensionless parameter γ is defined by the formula γ ¼ ℏωc =2Rn with the cyclotron frequency ωc ¼ eB=2πmn and the magnetic field intensity B; mn , ε are the electron effective mass and the static dielectric constant, respectively; Z is the charge of the hole, which equals 1 in this case to compare the obtained results with those in Refs. [13,14]. In this work, we consider a wide range of γ covering both weak and strong magnetic field regions. We will now consider the Eqs. (1) and (2) in another space which is more convenient for calculation through the Levi-Civita transformation [21]: ( x ¼ u2 −v2 ð3Þ y ¼ 2uv: The transformation (3) connects the two real two-dimensional spaces ðx; yÞ and (u,v). We can easily prove the following equalities: dx dy ¼ 4ðu2 þ v2 Þ du dv; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ x2 þ y2 ¼ u2 þ v2
ð4Þ
which will be used in the following calculation. From Eq. (4) we see that the Jacobian of the transformation (3) is not a constant but is instead 4ðu2 þ v2 Þ, so it will appear as a weight in the equation for calculating the scalar product of two state vectors when transforming from the ðx; yÞ space to the ðu; vÞ space. This means that if a certain operator K^ is hermitian in the ðx; yÞ space then the operator K~ ¼ 4ðu2 þ v2 ÞK^ is also hermitian in the ðu; vÞ space. Hence, in order to ensure that the Hamiltonian is hermitian through the transformation (3), we need to rewrite Eq. (1) as follows: ^ rðH−EÞΨ ðrÞ ¼ 0: In the ðu; vÞ space, this equation reads ~ ðu; vÞ ¼ ZΨ ðu; vÞ HΨ with the Hamiltonian: 1 ∂2 ∂2 γ γ2 H~ ¼ − þ 2 − E− L^ z ðu2 þ v2 Þ þ ðu2 þ v2 Þ3 : 2 8 ∂u 2 8 ∂v
ð5Þ
ð6Þ
We see in Eqs. (5) and (6) an interchange of Z and E in the role of the eigenvalue. The energy E is no longer the eigenvalue but is instead a parameter, while Z becomes the eigenvalue of Eq. (5). However, we can still solve the Schrödinger equation to find E when keeping Z¼1 as presented in the following section. Furthermore, in the Hamiltonian (6) there is also the operator L^ z which is the orbital angular momentum operator. Because the system under investigation is two-dimensional, this operator is also its projection operator on a direction perpendicular to the plane of motion. In the ðu; vÞ space, this operator reads i ∂ ∂ u −v : ð7Þ L^ z ¼ − 2 ∂v ∂u We can also prove that L^ z commutes with the Hamiltonian of Eq. (5). This means that angular momentum is conserved in the problem under consideration. We will use this conservation by constructing a basis set for solving Eq. (5) which contains the eigenfunctions of the orbital angular momentum operator. Then we replace L^ z by its eigenvalue in the Hamiltonian (5). Now we can see that the Eqs. (5) and (6) represent a two-dimensional anharmonic oscillator. In other words, we have transformed a twodimensional exciton in magnetic field into an anharmonic oscillator via the Levi-Civita transformation. This result allows the application of the FK operator method to find the exact numerical solutions of Eqs. (5) and (6).
N.-T. Hoang-Do et al. / Physica B 423 (2013) 31–37
2.2. The FK operator method for solving the Schrödinger equation The FK operator method is introduced in details in the work [15,16]. Here, we will only present the basic steps via solving the Schrödinger equations (5) and (6). The method includes four steps, as follows. Step 1 is writing the Hamiltonian in algebraic form ∂ ∂ ^ a^ þ ; b^ þ ; γÞ ~ a; ^ b; H~ ; ; u; v; γ -Hð ð8Þ ∂u ∂v using the creation and annihilation operators defined as follows: rffiffiffiffi rffiffiffiffi ω 1 ∂ ω n 1 ∂ þ ^ ξþ ξ − ; a^ ðωÞ ¼ aðωÞ ¼ n ; 2 ω ∂ξ 2 ω ∂ξ rffiffiffiffi rffiffiffiffi þ ω n 1 ∂ ω 1 ∂ ^ ξ þ ; b^ ðωÞ ¼ ξ− bðωÞ ¼ ; ð9Þ 2 ω ∂ξ 2 ω ∂ξn in which the complex coordinates are defined as ξ ¼ u þ iv, ξn ¼ u−iv. We can easily check that the operators (9) satisfy the well-known commutation relations: þ
^ a^ ¼ 1; ½a;
^ b^ þ ¼ 1: ½b;
ð10Þ
In definition (9), we use complex coordinates for convenience in writing only. The positive real number ω given in Eq. (9) is considered a free parameter whose role in the method will be discussed in the next steps. Plugging (9) into (6) we obtain the algebraic form of the Hamiltonian as follows: 2 ^ γÞ ¼ ω −2E þ mγ ða^ þ a^ þ b^ þ b^ þ 1Þ ~ a^ þ ; b^ þ ; a; ^ b; Hð 4ω ω2 þ 2E−mγ þ ^ þ ^ ða^ b þ a^ bÞ − 4ω þ γ2 þ þ þ ða^ b^ þ a^ b^ þ a^ a^ þ b^ b^ þ 1Þ3 : þ 64ω3
We also have an algebraic form of the orbital angular momentum operator: þ þ ^ ^ b^ bÞ; L^ z ¼ −12ða^ a−
ð12Þ
which is a neutral operator. Step 2 is splitting the Hamiltonian (12) into two components– the major component and the perturbation one: þ
þ
þ
^ γ; ωÞ þ V~ ða^ þ ; b^ ; a; ^ γÞ ¼ H~ ða^ þ a; ^ γ; ωÞ: ~ a^ þ ; b^ ; a; ^ b^ b; ^ b; ^ b; Hð 0
ð13Þ
The separation (13) is done based on a completely different principle from that of the perturbation method. Here, the major component contains only neutral operators which are products of equal number of creation and annihilation operators: 2 þ þ ^ γ; ωÞ ¼ ω −2E þ mγ ða^ þ a^ þ b^ þ b^ þ 1Þ ^ b^ b; H~ 0 ða^ a; 4ω þ þ γ2 þ þ þ ða^ a^ þ b^ b^ þ 1Þ½ða^ a^ þ b^ b^ þ 1Þ 64ω3 þ þ ða^ a^ þ b^ b^ þ 4Þ þ þ þ 6a^ a^ b^ b^ þ 2:
þ þ ^ γ; ωÞ commutes with the ^ b^ b; We see that the operator H~ 0 ða^ a; þ þ ^ hence its exact solutions are the waveoperators a^ a^ and b^ b, functions of the harmonic oscillator described by these operators. Furthermore, notice that although the Hamiltonian of the system does not depend on the free parameter ω, the split components þ þ ^ γ; ωÞ and V~ ða^ þ ; b^ þ ; a; ^ γ; ωÞ do. This means that we ^ b^ b; ^ b; H~ 0 ða^ a; can adjust the correlation between the major and the perturbation components by changing the value of ω. Step 3 is finding the zeroth-order energy and wavefunction þ þ ^ γ; ωÞ. This ^ b^ b; using the approximate Hamiltonian H~ 0 ða^ a; þ þ ^ so its Hamiltonian contains only neutral operators a^ a^ and b^ b, eigenfunctions have the following form: þ þ ða^ Þj ðb^ Þk j0ðωÞ〉
ð16Þ
in which j, k are non-negative integers and the vacuum state j0ðωÞ〉 is defined from the equations: ^ aðωÞj0ðωÞ〉 ¼ 0;
^ bðωÞj0ðωÞ〉 ¼ 0;
ð14Þ
The rest is the perturbation term: þ þ ^ γ; ωÞ ^ b; V~ ða^ ; b^ ; a; 2 2 ω þ 2E−mγ þ ^ þ ^ þ γ ½ða^ þ b^ þ Þ3 þ ða^ bÞ ^ 3 ða^ b þ a^ bÞ ¼− 4ω 64ω3 þ þ þ þ þ þ þ þ þ3a^ b^ ða^ a^ þ b^ b^ þ 1Þ2 þ 3ða^ b^ Þ2 ða^ a^ þ b^ b^ þ 1Þ þ þ ^ 2 þ 3ða^ þ b^ þ Þ2 ða^ bÞ ^ þ 3a^ þ b^ þ ða^ bÞ ^ 2 þ3ða^ a^ þ b^ b^ þ 1Þða^ bÞ þ þ þ þ þ þ þ3ða^ a^ þ b^ b^ þ 1Þ2 a^ b^ þ 9a^ b^ ða^ a^ þ b^ b^ þ 1Þ þ þ þ þ ^ 2 þ 6a^ þ b^ þ þ 6a^ b: ^ þ9ða^ a^ þ b^ b^ þ 1Þa^ b^ þ 6ða^ b^ Þ2 þ 6ða^ bÞ
ð15Þ
ð17Þ
and the normalization equation: 〈0ðωÞj0ðωÞ〉 ¼ 1:
ð18Þ
As said in Section 2.1, angular momentum is conserved in the problem under consideration. We will use this conservation by constructing a basis set which contains the eigenfunctions of the orbital angular momentum operator L^ z . This operator in the algebraic form (12) is also a neutral operator, hence its eigenfunctions are also in the form (16). Therefore, the wavefunction vectors (16) are rewritten in normalization form as follows: þ 1 þ jnðmÞ〉 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða^ Þn−m ðb^ Þnþm j0ðωÞ〉 ðn−mÞ!ðn þ mÞ!
ð11Þ
33
ð19Þ
in which the principal quantum numbers n are non-negative integers: n ¼0, 1, 2,…, and the magnetic quantum numbers m are integers satisfying the condition −n ≤m ≤ n. Furthermore, we use the notation n ¼ nr þ jmj in which nr ¼ 0, 1, 2, … are radial quantum numbers. Thus we have the zeroth-order wavefunction corresponding to the state of principal quantum number n and the magnetic quantum numbers m: jψ nðmÞ 〉ð0Þ ¼ jnðmÞ〉;
ð20Þ
Now we let the Hamiltonian (14) act on the wavefunction (20) and consider the Eq. (5). As a result, we obtain the zeroth-order energy: Eð0Þ ¼ −
2ωZ 1 γ2 1 þ ω2 þ ð5n2 þ 5n−3m2 þ 3Þ þ mγ: 2n þ 1 2 2 16ω2
ð21Þ
Here the free parameter is determined from the condition ∂Eð0Þ =∂ω ¼ 0 [15], which leads to the following equation: −
2Z γ2 þ ω− 3 ð5n2 þ 5n−3m2 þ 3Þ ¼ 0: 2n þ 1 8ω
ð22Þ
A numerical analysis of the analytical solutions (21) will be given in the next section on results. Step 4 is calculating high-order corrections to obtain exact numerical solutions. In principle, we may use different schemes, e. g. the perturbation theory scheme for calculating high-order corrections in order to obtain the energy and the wavefunction with higher precision. If that scheme leads to a result that converges to a certain value with any given precision then we have the exact numerical solution. In this work, we will propose an iteration scheme to calculate the energy and wavefunction with a given precision. For convenience we rewrite the Eqs. (5) and (6) as follows: ~ ðH~ −ERÞjψ〉 ¼0 R
ð23Þ
34
N.-T. Hoang-Do et al. / Physica B 423 (2013) 31–37
R in which the operator H~ and R~ takes the following forms: þ R ω2 þ mγ þ ω2 −mγ þ ^ þ ^ ða^ a^ þ b^ b^ þ 1Þ− ða^ b þ a^ bÞ H~ ¼ 4ω 4ω þ γ2 þ þ þ ða^ b^ þ a^ b^ þ a^ a^ þ b^ b^ þ 1Þ3 −1; þ 64ω3 þ 1 þ ^þ þ ða^ b þ a^ b^ þ a^ a^ þ b^ b^ þ 1Þ: R~ ¼ 2ω
in which the coefficients C ðsÞ ðk ¼ jmj; jmj þ 1; …; n−1; n þ 1; …; n þ k sÞ are determined by a system of n þ s−jmj linear equations: nþs
∑
k ¼ jmj;k≠n;k≠j
ð24Þ
We will use the basic set of wavefunctions (19) to construct the wavefunctions of the problem at hands. The matrix elements of the operators (24) corresponding to this basic set will be calculated through purely algebraic transformation. In fact, using the commutations (10) and the Eqs. (17) and (18), we easily obtain following formula: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ þ a^ b^ jnðmÞ〉 ¼ ðn þ 1Þ2 −m2 jn þ 1ðmÞ〉; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ a^ bjnðmÞ〉 ¼ n2 −m2 jn−1ðmÞ〉; þ þ ^ ¼ 2njnðmÞ〉 ð25Þ ða^ a^ þ b^ bÞjnðmÞ〉 from which we calculate the matrix elements as follows: H Rnn ¼ 〈nðmÞjH~ jnðmÞ〉 ω2 −mγ γ2 ð2n þ 1Þ þ ¼ ð2n þ 1Þð5n2 þ 5n þ 3−3m2 Þ−Z; 4ω 32ω3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R
R H Rn;nþ1 ¼ 〈nðmÞjH~ jn þ 1ðmÞ〉 ¼ ðn þ 1Þ2 −m2 ω2 þ mγ 3γ 2 þ − ð5n2 þ 10n þ 6−m2 Þ ; 3 4ω 64ω
H Rn;nþ2 ¼ 〈nðmÞjH~ jn þ 2ðmÞ〉 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3γ 2 ð2n þ 3Þ ðn þ 1Þ2 −m2 ðn þ 2Þ2 −m2 ; ¼ 64ω3 R H Rn;nþ3 ¼ 〈nðmÞjH~ jn þ 3ðmÞ〉 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ2 ðn þ 1Þ2 −m2 ðn þ 2Þ2 −m2 ðn þ 3Þ2 −m2 ; ¼ 3 64ω 2n þ 1 ~ Rnn ¼ 〈nðmÞjRjnðmÞ〉 ; ¼ 2ω qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ þ 1ðmÞ〉 ¼ − 1 Rn;nþ1 ¼ 〈nðmÞjRjn ðn þ 1Þ2 −m2 : 2ω R
ð26Þ
Besides (26), we can calculate other non-zero matrix elements using the symmetry properties H Rnk ¼ H Rkn , Rnk ¼ Rkn . We can write the exact wavefunction as a linear combination of the basic functions (19): jψ nðmÞ 〉 ¼ jnðmÞ〉 þ
þ∞
∑
j ¼ jmj;j≠n
C j jjðmÞ〉;
ð27Þ
and define the approximate wavefunction at the s-th order approximation (s-th iteration loop) as jψ nðmÞ 〉ðsÞ ¼ jnðmÞ〉 þ
nþs
∑
j ¼ jmj;j≠n
C ðsÞ j jjðmÞ〉
ð28Þ
corresponding to the approximate energy EðsÞ . If lims-∞ C ðsÞ ¼ Ck k then the approximate wavefunction (28) converges to the exact wavefunction (27). If then we have EðsÞ -ET while s- þ ∞, we also have the exact energy. We will use the notations for quantum − states as in the work [13,14], e.g.: 1s, 2s, 2p− , 2pþ , 3s, 3p− , 3pþ , 3d , þ 3d , etc. The first digit stands for the state level n þ 1, in which n is the principal quantum number defined in Eq. (19); the letters symbolize the orbital quantum numbers l ¼ jmj: s corresponding to l ¼ 0, p to l ¼1, d to l ¼2, f to l¼ 3, g to l¼4, etc; the 7 sign is of the magnetic quantum number m. Putting Eq. (28) into the Schrödinger equation (23), we obtain the expression for the approximate s-th order energy: EðsÞ ¼
H Rnn þ ∑nþs C ðsÞ H Rnk k ¼ jmj;k≠n k Rnn þ ∑nþs C ðsÞ Rnk k ¼ jmj;k≠n k
ð29Þ
ðH Rjk −Eðs−1Þ ÞC ðsÞ ¼ Rnj Eðs−1Þ −H Rnj ; k
ð30Þ
in which j ¼ jmj; jmj þ 1; …; n−1; n þ 1; …; n þ s. So, by plugging the solutions of Eq. (30) into Eq. (29) we obtain the energy of the system at the s-th iteration loop, which is called the s-th order approximation energy. Numerical results show that with appropriate choice of ω, for the considered problem we obtain a series of approximate energies Eð0Þ ; Eð1Þ ; Eð2Þ ; …; EðsÞ ; …
ð31Þ
which rapidly converges to a value ET. We call them the exact numerical solutions because their values can be obtained with any given precision. In this work, we calculate these solutions with precision of up to 20 decimal places due to the limitation of the default precision for real numbers in FORTRAN and the computing speed of computer. However, this is an important progress. As we know, the precision for real numbers in FORTRAN is limited to 15 decimal places. By appropriate programming, we can increase this precision up to 50 decimal places. In our final results for energies and wavefunctions, we require a precision of up to 20 decimal places in order to avoid the accumulation of errors in calculating. Besides energy, the coefficients C ðsÞ also converge rapidly so we k obtain not only exact numerical energy but also exact numerical wavefunctions. The parameter ω is chosen using the method described in Refs. [20–23]. In principle, this parameter does not affect the exact numerical results. However, investigation shows that the convergence rate to the exact solutions depends significantly on the choice of ω. In this work, the equation ∂Eð0Þ =∂ω ¼ 0 provides the first value ω0 which is not the optimal value. For high excited states nr, such a choice of ω even does not lead to convergence to exact values. We can scan to find the optimal value of parameter around the first value of ω0 . The results in this paper reconfirm the conclusions of Refs. [22,23] about the existence of the range of parameter such that the convergence rate of Eq. (28) is highest.
3. Results and discussions Analytical solutions: Fig. 1 shows the energies of the system corresponding to the ground state and some excited states 1s, 2s, − 2p− , 3d . The dotted line represents the results, which can be considered analytical solutions, obtained with zeroth-order approximation using the Eqs. (20) and (21). The solid line represents the exact numerical results for comparison purposes. These exact results are obtained from the iteration Eqs. (26) and (27). On this figure, we see that the analytical solutions are pretty precise for the ground and excited states in magnetic field region γ ≤ 1. In stronger magnetic field, this precision decreases while the correlation between the energy levels stays the same. In order to obtain analytical solutions which are highly precise and are uniformly suitable in the whole range of magnetic field, we need to consider the asymptotic behavior of the wavefunction in strong magnetic field region. We will discuss this case in another work. Exact numerical solutions: In this work, we focus on exact numerical solutions. Table 1 presents energies with precision of up to 20 decimal places for the ground state 1s (n¼ m ¼0) and − some excited states 2p− ðn ¼ 1; m ¼ −1Þ, 3d ðn ¼ 2; m ¼ −2Þ. These states were calculated with the precision of up to seven decimal places in Refs. [13,14]. Here, the results in Refs. [13,14] are not shown because all of these seven places are included in our results shown in Table 1. The energy of the 2pþ ðn ¼ 1; m ¼ þ 1Þ,
N.-T. Hoang-Do et al. / Physica B 423 (2013) 31–37 þ
3d ðn ¼ 2; m ¼ þ 2Þ states can be calculated from that of the 2p− , − 3d states based on the relation EnðmÞ ¼ Enð−mÞ þ mγ. The energies with precision of up to 20 decimal places presented in Table 1 are a new record presented for the first time in this paper. By choosing the optimal value of the free parameter, to obtain the ground state energy with seven decimal places we need s ¼6 iterations loops for γ′ ¼ 0:05 (weak magnetic field), s¼ 8 iteration loops for γ′ ¼ 0:55 (medium magnetic field), s ¼26 iteration loops for γ′ ¼ 0:95 (strong magnetic field); and to obtain the precision of 20 decimal places we need s ¼16, 35, 124 iterations loops for the three cases above, respectively. For excited states, more iteration loops are required, but not more than 300. Although the results presented in Table 1 are with 20 decimal places, we did attempt to run the program to get up to 50 decimal places for the ground state and some low excited states. Hence, we conclude that the method introduced in this paper provides numerical results with any given precision. In other words, they are exact numerical solutions. The wavefunction of the system in the form (28) is also obtained by calculating the coefficients C ðsÞ . The more iteration k
Fig. 1. Energies obtained from analytical equations are compared with exact energies of the system for the 1s, 2s, 2p−, 3d− states. For the ground state, the analytical solution is pretty precise, while its error increases for the excited states in the region of strong magnetic field.
35
loops are carried out, the more coefficients are obtained. These coefficients converge to a certain value. Hence, we obtain the exact numerical wavefunction. For example, for the ground state in weak magnetic field γ′ ¼ 0:05, when the energy is obtained with precision of 20 decimal places, the obtained wavefunction has 18 coefficients C ð15Þ , three of which are with precision of 20 decimal k places and the others with precision ranging from 1 to 19 decimal places. Similarly, for the case γ′ ¼ 0:55, there are 38 coefficients C ð35Þ with precision ranging from 1 to 19 decimal places. Furtherk more, we can increase the precision of these coefficients up to 20 decimal places by running some more iteration loops. We notice that the states presented in Table 1 and considered in the works [13,14] are special cases. Indeed, they are 1s (n ¼0, − m ¼0), 2p− ðn ¼ 1; m ¼ −1Þ, 3d ðn ¼ 2; m ¼ −2Þ which correspond to the principal quantum number n ¼ jmj. It means the radial quantum number nr ¼ 0 for all the states mentioned above. All of them are the lowest states when the magnetic quantum numbers are fixed, so they can be considered the ground states of a one-dimensional motion remaining after the motion related to angular momentum is taken off. For these states, the variational method works well but for states with nr ≠0, it does not. We have not found any follow-up works of [13,14] on these states. Therefore, our success in obtaining the exact solutions not only for the ground state but also for any excited states corresponding to nr ¼ 0 as well as nr ≠0 is a significant development. The computing programs are tested for excited states with principal quantum number of up to n ¼150. The exact energies for the states with the − − principal quantum number of up to n ¼4: 2s, 3s, 3p− , 4p− , 4d , 4f , − − − 5d , 5f , 5g are presented in Tables 2–4 for illustration. These data which are original and are presented for the first time in this paper would be interesting for further reference. We also found an interesting result while analyzing data in Table 2. For the case γ′ ¼ 0:8, corresponding to γ ¼ 4, the energy of the 2s state is exactly equal to 4. Similarly, for the 3s state and γ′ ¼ 0:4, corresponding to γ ¼ 2=3, the energy is exactly equal to 1. That means there are some values of magnetic field at which exact analytical solutions of the problem can be derived. This detection needs to be investigated in more details in another research which we will introduce in a follow-up work. The numerical results in the tables are also illustrated in Figs. 2 and 3. We see that for magnetic field γ 4 0:1, the levels of energies corresponding to the principal quantum numbers begin
Table 1 Exact energies for the ground state and some low-order excited states with different values of magnetic field. These results are presented with 20 decimal places and are included results of other authors with seven decimal places obtained from the variational method combined with 1=N expansion [13] and by the asymptotic perturbation method [14]. For comparison purpose, the magnetic field strength is represented by γ′ ¼ γ=ðγ þ 1Þ. −
γ′
1s
2p−
3d
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95
−1.99987017095990613170 −1.99942166507712501693 −1.99854256186645709957 −1.99707966222489033807 −1.99482091674920995739 −1.99146906712008245351 −1.98660128273873874036 −1.97960585091255737141 −1.96958003578721667516 −1.95515968324675916048 −1.93422334282651768145 −1.90335295328070946150 −1.85679038695312423867 −1.78426176250820398205 −1.66598114826115218564 −1.45958713448989797632 −1.05902943085968226100 −0.12110157606243169047 3.23173503617812205039
−0.24474134557798881450 −0.26197520208978872077 −0.27410756889059385484 −0.28145662051014027268 −0.28409801197305611464 −0.28179705884272357348 −0.27398063451910408445 −0.25967324608666750763 −0.23736745684644382973 −0.20479038588298687027 −0.15849200397641926789 −0.09310153718673498788 0.00008428907312278363 0.13597809871739040778 0.34214588246038558324 0.67521872604493890722 1.27112332610485384793 2.55062439464316783261 6.70030514510251978855
−0.11440510079734191774 −0.13025445178430606781 −0.13674173884597401955 −0.13659636693603053469 −0.13064401380817189943 −0.11888179385839621521 −0.10076026619266554000 −0.07521698217897997805 −0.04057485324928431054 0.00569412879010892499 0.06740516636374326600 0.15045496413924365797 0.26432778727006523882 0.42515253567577483784 0.66251797890286951771 1.03673987328882310035 1.69141167638720390304 3.06707775217066545341 7.42860734182398557718
36
N.-T. Hoang-Do et al. / Physica B 423 (2013) 31–37
Table 2 Energies of some excited states corresponding to different values of the magnetic field. γ′
2s
3s
3p−
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95
−0.21728279396968785397 −0.20160591618074023247 −0.17457048111136615243 −0.13546551574551668372 −0.08298007150252920818 −0.01502952750059654076 0.07142846360607761967 0.18070180520537274524 0.31887814627104300055 0.49467963883695179668 0.72091579831145684540 1.01705668872877975106 1.41405521488835508770 1.96407032991584636290 2.76203075990704173477 4.000000000000000000 6.13177508892586718321 10.53825335361872298997 24.24854780554921085193
−0.05137115812285705179 0.01598790970267759145 0.10955543404170749556 0.22721834540118963879 0.37047446264365413831 0.54292733023387127727 0.75013268675298689141 1.000000000000000000 1.30366017149884873723 1.67697071075735877489 2.14309105414624009242 2.73703806534873385927 3.51419634203232951294 4.56741386886003820741 6.06476489650834377662 8.34434942653877459541 12.19997778752228732308 20.02986405533713661963 43.93175572792464946928
−0.08036721744503109760 −0.04916051442900800150 0.00108637565485044813 0.06738328574548576958 0.14986742555393444603 0.25028633092701018244 0.37170668580947103502 0.51867414992190953559 0.69769322374130095062 0.91810396914076738153 1.19360373208799551825 1.54495188613538814666 2.00503229789835964832 2.62903276273533536486 3.51694576840641184652 4.87008864382744499370 7.16153917283062994829 11.82192792999323245234 26.07692664470341686614
Table 3 Energies of some high excited states with n¼ 3 corresponding to different values of the magnetic field. −
γ′
4p−
4d
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95
0.00090050861670000681 0.09390796453557639448 0.21323373403753960833 0.35668048473475479870 0.52606940637774516311 0.72533207396702501971 0.96036057230843709310 1.23944041995229360305 1.57415466587235095938 1.98093584143111358970 2.48370977153057734079 3.11856112568506923453 3.94243430849379882861 5.05058514254628752285 6.61507130462100771829 8.98122289598444673098 12.95785147539592703023 20.98177530017893504943 45.30314463505568435467
−0.03603819112188519016 0.00747840526168139693 0.06842568965737196928 0.14447370643180047832 0.23628341974950389743 0.34594127952399409512 0.47676457059152532284 0.63351735049084262159 0.82292516365874272605 1.05458031287213984997 1.34249561828604384524 1.70785470976146371028 2.18414967307751588723 2.82749853910929518464 3.73942662403763270927 5.12403034105099778843 7.46015607265407629634 12.19329771792373821144 26.60790678762850209795
4f
−
−0.07763315014992853255 −0.08803520910560353969 −0.08888339654701696510 −0.08315384587850637174 −0.07160076440801957770 −0.05412267756001832601 −0.03006828179726522415 0.00173399981723935616 0.04308672404388300910 0.09667213220019815981 0.16650643696559374134 0.25876268139530569281 0.38332586831977878052 0.55694141965414866451 0.81021701467250590897 1.20531843995110106672 1.88968201934687586146 3.31375703786321820117 7.78159090037994763851
Table 4 Energies of some excited states with n¼ 4 corresponding to different values of the magnetic field. −
γ′
5d
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95
0.02905749316697708979 0.13341069872580500192 0.26190026516127302629 0.41356740099965750564 0.59077510317782206889 0.79776115680429092118 1.04063336998574952266 1.32786142397127676506 1.67121288127923035467 2.08732965304714088906 2.60039856986566405608 3.24685460813888191684 4.08414376941622929138 5.20828847495739630265 6.79260902069767290808 9.18472403676149577875 13.19819451070249609938 21.28207427023502943214 45.73489523603524444325
5f
−
−0.01420058007237175099 0.03570019722639813037 0.10204170420617281261 0.18308786847417280362 0.27975368917130985634 0.39427787158829185045 0.53009357416698937372 0.69207101474623163562 0.88704788903212652389 1.12474823987081387279 1.41935251349090562787 1.79227191808221737000 2.27732643588718193308 2.93113687655407534814 3.85605806748978018262 5.25769124636313341999 7.61800718398620959057 12.39055080521797139007 26.89161351897346612252
5g− −0.06007875519489516057 −0.06604657997486724854 −0.06292240231373530831 −0.05344145079515578704 −0.03821140930631437516 −0.01703067582967511760 0.01083372779566949526 0.04663124135459961939 0.09224854476563663044 0.15046790575342597724 0.22543362638168408645 0.32349322334645939986 0.45478350257101208093 0.63643631212786443537 0.89969714506691749633 1.30788909126295026871 2.01085156632785457493 3.46522653749514333096 7.99955272626774688799
N.-T. Hoang-Do et al. / Physica B 423 (2013) 31–37
Fig. 2. The dependence of energy on magnetic field strength for the states with principal quantum number n¼ 3: 1s, 2s, 2p 7 , 3s, 3p 7 , 3d 7 , 4p 7 , 4d 7 , 4f 7 . We see that in strong magnetic fields, the degenerated Landau levels split out due to Coulomb interaction although they are still close to each other and are magnified in (a), (b), (c), and (d).
37
(2) Successful development of the FK operator method for a twodimensional exciton in a magnetic field. This development is significant, universal and applicable to various problems related to two-dimensional exciton in electromagnetic field. (3) Obtaining exact numerical solutions for a two-dimensional exciton in a constant magnetic field with arbitrary strength. For the ground state and the excited states considered in previous works of other authors, we set a new record on precision of up to 20 decimal places. Furthermore, exact solutions for high excited states are presented for the first time in this work. The FORTRAN program for calculating energies and wavefunctions of any excited states with principal quantum number of up to hundreds can be provided to interested researchers. These results can be used for research related to the energy levels of a two-dimensional exciton in a magnetic field. (4) Detecting some concrete values of magnetic field strength at which the problem of a two-dimensional exciton in a magnetic field has exact analytical solutions. This detection is very interesting and needs more detailed research.
Acknowledgments We would like to thank Professor Feranchuk I. D. (Belarusian State University, Minsk – Belarus) for careful reading and helpful comments on this work. This work is sponsored by the National Foundation for Science and Technology Development (NAFOSTED) Grant no. 103.012011.08. The author Hoang-Do also would like to thank for the institutional grant provided by Ho Chi Minh City University of Pedagogy. References [1] [2] [3] [4] [5] Fig. 3. Energy levels in γ ≤1 region of magnetic field.
[6] −
to disarrange. For example, 2s and 2pþ levels are higher than 3d − − − and 4f levels; 3s and 3pþ are higher than 4d , 4f levels. That is because we use the principal quantum numbers of the Coulomb problem which is reasonable only in weak magnetic field. In strong magnetic field, Coulomb interactions are considered a perturbation which eliminates the degenerate Landau levels. Thus, in this case the principal quantum numbers must follow Landau levels in the problem of motion of electrons in a uniform magnetic field. Our finding of exact solutions for high excited states with principal quantum number of up to hundreds allows investigation of not only the degenerate separation of Landau levels but also quantum chaos effects. These are some suggestions for further research.
4. Conclusion and outlook
[7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
In this work we obtain the following original results: [23]
(1) Transforming the problem of a two-dimensional exciton in a magnetic field into that of a two-dimensional anharmonic oscillator through Levi-Civita transformation. This is an important result because the problem of an anharmonic oscillator is much simpler and is well-investigated using several methods.
[24] [25] [26] [27]
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