The relations between d-dimensional isotropic oscillator and D-dimensional like-hydrogen atom

TL;DR

This paper establishes a mathematical relationship between d-dimensional isotropic oscillators and D-dimensional like-hydrogen atoms, enabling the transfer of solutions and properties across different quantum systems and dimensions.

Contribution

It introduces a special transformation that relates Schrödinger equations of different dimensions for these systems, generalizing solutions between isotropic oscillators and like-hydrogen atoms.

Findings

Derived a transformation linking quantum systems in different dimensions.

Generalized energy spectra and wave functions across dimensions.

Provided a method to solve hydrogen-like problems using oscillator solutions.

Abstract

Being comparable in quantum systems makes it possible for spaces with varying dimensions to attribute each other using special conversions can attribute schrodinger equation with like-hydrogen atom potential in defined dimensions to a schrodinger equation with other certified dimensions with isotropic oscillator potential. Applying special transformation provides a relationship between different dimensions of two quantum systems. The result of the quantized isotropic oscillator can be generalized to like-hydrogen atom problem in different dimensions. The connection between coordinate spaces in different dimensions can follow a specific relation that by using it and applying the parametric definition in two problems, energy spectrum and like-hydrogen atom potential wave functions problem will be solved.

Equations

$$u_1^2+u_2^2+...+u_d^2=x_1^2+x_2^2+...+x_D^2,$$

$$x_i=\sum _{j=1}^d{T}_{ij}{u}_j,i=1,2,3,...,D$$

$$\sum _{j=1}^d{T}_{ij}{T}_{kj}=u^2{\delta }_{ik},i,k=1,2,...,D,$$

$$\sum _{j=1}^d\frac{\partial T_{ij}}{\partial u_j}=0.$$

$$\frac{\partial x_i}{\partial u_j}=2T_{ij},i=1,...,D.$$

$$\frac{\partial \tau (u)}{\partial u_j}=2\sum _{k=1}^D{T}_{kj}\frac{\partial \tau (u)}{\partial x_k},j=1,2,...,d.$$

$$\frac{\partial \tau (u)}{\partial x_i}=\frac{1}{2u^2}\sum _{j=1}^d{T}_{ij}\frac{\partial \tau (u)}{\partial u_j}.$$

$$\frac{\partial }{\partial x_i}=\frac{1}{2u^2}\sum _{j=1}^d{T}_{ij}\frac{\partial }{\partial u_j},$$

$$\frac{\partial }{\partial u_j}=2\sum _{k=1}^D{T}_{kj}\frac{\partial }{\partial x_k},$$

$${\hat{\phi }}_i=i{u}^2\frac{\partial }{\partial q_i}=\frac{i}{2}\sum _{j=1}^d{T}_{ij}\frac{\partial }{\partial u_j},i=D,D+1,...,d-1,d.$$

$${\hat{\phi }}_i{\hat{x}}_k=i{u}^2{\delta }_{ik}=0,i=D,...,d,k=1,...,D.$$

$${\Delta }_x=\frac{1}{4u^2}{\Delta }_u+\frac{1}{u^4}{\phi }^2,$$

$${\hat{\phi }}^2={\hat{\phi }}_D^2+...+{\hat{\phi }}_d^2.$$

$$(-\frac{{\hslash }^2}{2\mu }{△}_x-\frac{e^2}{r})\psi (x)=E\psi (x),$$

$$(-\frac{{\hslash }^2}{2\mu }{△}_u-4E{u}^2)\psi (x)=4e^2\psi (x).$$

$$(-\frac{{\hslash }^2}{2\mu }{△}_u+\frac{1}{2}\mu {\omega }^2{u}^2)\phi (u)=\epsilon \phi (u).$$

$$\frac{1}{2}\mu {\omega }^2=-4E,4e^2=\epsilon .$$

$$\epsilon =\hslash \omega (N+2^{m-1}),N=0,1,2,...,m=1,2,3,....$$

$$E=\frac{-2\mu e^4}{{\hslash }^2(N+2^{m-1}{)}^2},$$

$${\psi }_n(x)=\sum C_{n_{\alpha }}{\phi }_{N_{\alpha }}(u),$$

$$T_{ij}=\left(\begin{array}[]{cccc}u_{1}&-u_{2}&-u_{3}&u_{4}\\ u_{2}&u_{1}&-u_{4}&u_{3}\\ u_{3}&-u_{4}&u_{1}&u_{2}\\ u_{4}&-u_{3}&-u_{2}&-u_{1}\\ \end{array}\right),$$

$$x_1=u_1^2-u_2^2-u_3^2+u_4^2$$

$$x_2=2(u_1{u}_2-u_3{u}_4)$$

$$x_3=2(u_1{u}_3+u_2{u}_4)$$

$${\hat{\phi }}_4=\frac{i}{2}(u_4\frac{\partial }{\partial u_1}-u_3\frac{\partial }{\partial u_2}+u_2\frac{\partial }{\partial u_3}-u_1\frac{\partial }{\partial u_4}),$$

$${\hat{\phi }}^2={\hat{\phi }}_4^2.$$

$$\epsilon =\hslash \omega (N+2).$$

$$E=\frac{-2\mu e^4}{{\hslash }^2(N+2{)}^2},$$

$$\tiny{T_{ij}=\left(\begin{array}[]{cccccccccccccccc}u_{9}&-u_{10}&-u_{11}&u_{12}&-u_{13}&u_{14}&-u_{15}&u_{16}&u_{1}&u_{2}&u_{3}&u_{4}&-u_{5}&u_{6}&-u_{7}&u_{8}\\ -u_{10}&-u_{9}&-u_{12}&u_{11}&-u_{14}&-u_{13}&-u_{16}&u_{15}&u_{2}&-u_{1}&u_{4}&u_{3}&-u_{6}&-u_{5}&-u_{8}&u_{7}\\ u_{11}&-u_{12}&u_{9}&u_{10}&u_{15}&-u_{16}&u_{13}&u_{14}&u_{3}&u_{4}&u_{1}&-u_{2}&-u_{7}&u_{8}&u_{5}&-u_{6}\\ -u_{12}&-u_{11}&-u_{10}&u_{9}&u_{16}&u_{15}&u_{14}&u_{13}&u_{4}&-u_{3}&-u_{2}&-u_{1}&u_{8}&u_{7}&u_{6}&u_{5}\\ u_{13}&u_{14}&-u_{15}&-u_{16}&u_{9}&-u_{10}&u_{11}&u_{12}&u_{5}&-u_{6}&u_{7}&u_{8}&u_{1}&u_{2}&-u_{3}&-u_{4}\\ u_{14}&u_{13}&u_{16}&-u_{15}&u_{10}&u_{9}&-u_{12}&u_{11}&u_{6}&u_{5}&u_{8}&-u_{7}&u_{2}&-u_{1}&-u_{4}&u_{3}\\ u_{5}&-u_{16}&u_{13}&-u_{14}&-u_{11}&u_{12}&u_{9}&u_{10}&u_{7}&u_{8}&-u_{5}&u_{6}&u_{3}&-u_{4}&u_{1}&-u_{2}\\ -u_{16}&-u_{15}&-u_{14}&-u_{13}&u_{12}&-u_{11}&-u_{10}&u_{9}&u_{8}&-u_{7}&-u_{6}&u_{5}&-u_{4}&-u_{3}&-u_{2}&-u_{1}\\ u_{1}&u_{2}&u_{3}&u_{4}&u_{5}&u_{6}&u_{7}&u_{8}&-u_{9}&-u_{10}&-u_{11}&-u_{12}&-u_{13}&-u_{14}&-u_{15}&-u_{16}\\ -u_{9}&u_{10}&-u_{11}&-u_{12}&u_{13}&-u_{14}&u_{15}&-u_{16}&u_{1}&-u_{2}&u_{3}&u_{4}&-u_{5}&u_{6}&-u_{7}&u_{8}\\ u_{10}&u_{11}&u_{12}&u_{13}&u_{14}&u_{15}&u_{16}&u_{9}&-u_{8}&-u_{1}&-u_{2}&-u_{3}&-u_{4}&-u_{5}&-u_{6}&-u_{7}\\ -u_{11}&u_{12}&u_{13}&u_{14}&u_{15}&-u_{16}&-u_{9}&u_{10}&u_{7}&-u_{8}&u_{1}&-u_{2}&-u_{3}&-u_{4}&-u_{5}&-u_{6}\\ u_{12}&u_{13}&u_{14}&u_{15}&u_{16}&u_{9}&u_{10}&u_{11}&-u_{6}&-u_{7}&-u_{8}&-u_{1}&-u_{2}&-u_{3}&-u_{4}&-u_{5}\\ u_{9}&-u_{10}&u_{11}&-u_{12}&-u_{13}&-u_{14}&-u_{15}&u_{16}&-u_{1}&u_{2}&-u_{3}&u_{4}&u_{5}&u_{6}&u_{7}&-u_{8}\\ -u_{10}&-u_{11}&-u_{12}&-u_{13}&-u_{14}&-u_{15}&-u_{16}&-u_{9}&u_{8}&u_{1}&u_{2}&u_{3}&u_{4}&u_{5}&u_{6}&u_{7}\\ u_{11}&u_{12}&-u_{13}&-u_{14}&u_{15}&u_{16}&-u_{9}&-u_{10}&u_{7}&u_{8}&-u_{1}&-u_{2}&u_{3}&u_{4}&-u_{5}&-u_{6}\\ \end{array}\right),}$$

$$x_1=2(u_1{u}_9+u_2{u}_{10}-u_3{u}_{11}+u_4{u}_{12}-u_5{u}_{13}+u_6{u}_{14}-u_7{u}_{15}+u_8{u}_{16}),$$

$$x_2=2(-u_1{u}_{10}+u_2{u}_9+u_3{u}_{12}+u_4{u}_{11}-u_5{u}_{14}-u_6{u}_{13}+u_7{u}_{16}+u_8{u}_{15}),$$

$$x_3=2(u_1{u}_{11}-u_2{u}_{12}+u_3{u}_{10}+u_4{u}_{10}+u_5{u}_{15}-u_6{u}_{16}-u_7{u}_{13}+u_8{u}_{14}),$$

Full text

PACS numbers: 03.65. w, 03.65.Fd 03.65.Ge,11.30.Pb

1 Introduction

The hidden dynamical symmetry in some quantum systems and the beauty of solving such issues causes imputation of other quantum systems solvable models by using some special transformations. The solvability of such systems provides enough motivations for simulation of other physical systems with them. Such solvable quantum systems are included in quantum isotropic oscillators with dimensions 2, 4, 8, 16, 32 that can be attributed to like-hydrogen atom quantum systems with dimensions 2, 3, 5, 9. The type of transformation is so that special dimensions of two quantum systems can be related to each other. Therefore, these transformations can’t be useful for any dimension of considered quantum system. These transformations create a duality between discrete quantum systems named oscillators with different dimensions and continues quantum systems that is called like-hydrogen atom quantum systems. These transformations help attributing solvability of a quantum system to the solvability of desired quantum system, so different algebra methods that are applied in solvable models can be expanded in the desired problems. This expansion causes parametrical relations between two quantum systems.

Algebra method such as dynamic symmetry groups corresponded with the like-hydrogen atom can be related to the dynamic symmetry groups of isotropic oscillators [1]. Also, the explicit form of the transformations binding like-hydrogen atom wave function and isotropic oscillator wave function can be available by using the definitions of parameters in two considered systems [2-4].

Since it is known that the quantized oscillator is a principle concept model in the quantum field theory and the problem of D-dimensional oscillator can be related to the problem of d-dimensional like-hydrogen atom, by using special transformations, so many well developed methods applied in quantum field theory and nuclear physics can be also effectively used in the investigations of the behavior of like-hydrogen atom in the external electric and magnetic fields [5-8]. There are some special transformations that can convert d-dimensional like-hydrogen atom to D-dimensional isotropic oscillator. These bilinear transformations are called Levi-Civita [9], kustaan- Steif [10],Hurwitz transformations [11] that can depend the dimensions of d=2, 3, 5, 9, 17 to the dimensions of D=2, 4, 8, 16, 32, correspondingly. Since, the above mentioned transformations connected two different coordinate spaces, dimensionally, so there isn’t an one by one relation between two spaces. The transformations that is applied to changing two spaces with different dimensions are called non-bijective quadratic transformations [12]. It should stressed that non-bijectivity of the $R^{d}\rightarrow R^{D}$ transformation means that for each element of $R^{d}$, there is a whole set of elements in $R^{D}$ .

In this paper, by special transformations, a general model is presented that shows how physical quantities can be converted to each other in two different quantum systems with two different dimensions. In this method, D-dimensional isotropic oscillator energy can be expanded to d-dimensional like-hydrogen atom energy by considering the parametrical relations between two systems. Also, it is shown that wave function of isotropic oscillator can be written based on wave function of like- hydrogen atom. The presented general method is applied in 5 and 16 dimensional isotropic oscillators in the sections (3) and (4), respectively.

2 Generalized transformations in two different spaces of quantum systems

As mentioned before, transformations of dimensions between two systems as $d\rightarrow D$ are included only in some cases of dimensions such as d=2, 3, 5, 9, 17 to the dimensions of D=2, 4, 8, 16, 32, correspondingly. In general form, there is a like-hydrogen atom with a certain dimension that can be related to isotropic oscillator with another specific dimension. Dimensional relations between two spaces $R^{d}$ and $R^{D}$ is considered by special transformation that only convert $d=2^{m}$ dimension of the isotropic oscillator to $D=2^{m-1}+1$ dimension of like-hydrogen, where $m$ is an integer number and $m=1,2,3,4,5$ based on restrict of $d$ and $D$.

The most important feature of non-bijective quadratic transformations is the validity of the Euler identity [13]:

[TABLE]

Where $x_{i}$ and $u_{k}$ are the D-dimensional like-hydrogen atom and d-dimensional isotropic oscillator coordinates, respectively.

Introducing non-bijective quadratic transformation, coordinates $(x_{1},...,x_{D})$ of space $R^{D}$ can be related to coordinates $(u_{1},...,u_{d})$ of the space $R^{d}$, by [12]:

[TABLE]

where $x_{i}$ is a generalized coordinate in space $R^{D}$ and $T_{ij}$ is a transformation matrix that shows the type of transformations applied to the problem.

Transformation matrix has basic properties. First, the elements are linear homogeneous functions of $u_{j}$. Second, matrix is orthogonal in the following sense:

(a) The scalar produces two different rows vanishes.

(b) Each row has the norm $u_{1}^{2}+u_{2}^{2}+...+u_{d}^{2}$.

Multiplying two transformation matrixes and summing over $j$ based on the Euler identify (2.1), it can be shown that [12]:

[TABLE]

and

[TABLE]

The transformation matrixes $T_{ij}$ should be defined so that:

[TABLE]

Considering an arbitrary function of $u$ as $\tau(u)$ and using equation (2.5):

[TABLE]

Multiplying (2.6) by $T_{ij}$ and using relation (2.3), it can be easily shown that:

[TABLE]

Therefore, relations (2.6) and (2.7) can be generalized by:

[TABLE]

[TABLE]

If $\varphi_{i}$ operators are introduced as:

[TABLE]

Considering relation (2.5), $\varphi_{i}$ will operate as follows:

[TABLE]

Also, it is clear that relation (2.11) is valid for an arbitrary functions of $x$, so that $\hat{\varphi}_{i}F(x)=0$. Considering differential relations between two different coordinates $x_{i}$ and $u_{j}$, the relation between like-hydrogen atom and isotropic oscillator Laplacians is as follows [12]:

[TABLE]

where operator $\hat{\varphi}^{2}$ is defined as:

[TABLE]

$\hat{\varphi}^{2}$ is the operator that causes laplacian of like-hydrogen atom, written based on the laplacian of isotropic oscillator. Therefore, D-dimensional Schrödinger equation for like-hydrogen atom as:

[TABLE]

can be transformed into the form:

[TABLE]

Since equation (2.15) is comparable with solvable Schrödinger equation for d-dimensional isotropic oscillator as:

[TABLE]

Thus, equation (2.15) that is related to D-dimensional like-hydrogen atom, can be solved considering the solution of d-dimensional isotropic oscillator. Applying d-dimensional isotropic oscillator as a solvable model will be perfect, if the following parametrical relations is satisfied between two equations (2.15) and (2.16):

[TABLE]

The energy spectrum in d-dimensional isotropic oscillator can be obtained as:

[TABLE]

where $m$ is an integer number and $N$ is a main quantum number that represents the number of energy discrete bound states levels. This energy discrete spectrum is corresponded with isotropic oscillator in the $2^{m}$ dimensions of space $R^{d}$. It should be mentioned that in equation (2.16), $\omega$ is considered as a potential constant parameter, so, $\varepsilon$ is quantized according to relation (2.18). Although, in equation (2.15), $\varepsilon=4e^{2}$ is assumed as a potential constant parameter and $\omega$ is quantized by $\frac{\varepsilon}{\hbar(N+2^{m-1})}$ where $N$ is a natural number. Substituting parametrical relations (2.17) in relation (2.18), energy discrete spectrum of D-dimensional like-hydrogen atom can be calculated by:

[TABLE]

where $N$ as a number of energy bound states level can be introduced by parametrical definitions in the like-hydrogen atom systems with dimensions of $2^{m-1}+1$.

According to the type of transformations in the problem $\psi(x)$ is an even function of variables $u$, so that $\psi(x(u))=\psi(x(-u))$. Therefore, $\psi(x)$ as the solution of equation (2.14) can be expanded in a full system of even solutions $\varphi_{N_{\alpha}}$ of equation (2.16) [12]:

[TABLE]

where $\alpha$ denotes all other quantum numbers.

3 $ks$ transformations in the four-dimensional quantum isotropic oscillator

Presented general model can be used for expansion of four-dimensional like-hydrogen atom by $Ks$ transformations. The $Ks$ transformations is presented by following matrix [10]:

[TABLE]

where mentioned before, basic properties. According to the relation (2.2), connection between $x_{i}$ coordinates of three-dimensional like-hydrogen atom and $u_{j}$ coordinates of four-dimensional isotropic oscillator is arranged as follows:

[TABLE]

The $\hat{\varphi}_{i}$ operator has only one form as $\hat{\varphi}_{4}$ that is defined based on relation (2.10) as:

[TABLE]

The $\hat{\varphi}^{2}$ operator can be written based on $\hat{\varphi}_{4}^{2}$ operator by considering relation (2.14):

[TABLE]

Therefore, schrödinger equation of like-hydrogen atom in three dimensions can be comparable with schrodinger equation of isotropic oscillator in four dimensions. Considering parametrical relations (2.17), $\varepsilon$ energy spectrum in four-dimensional isotropic oscillator as:

[TABLE]

where $m=2$ and $N$ is a level number of energy spectrum. This energy spectrum can be applied to calculation of energy discrete spectrum of three-dimensional like-hydrogen atom. According to relation (2.20), if the integer number $m$ is considered as $m=2$, energy spectrum of like-hydrogen atom in three-dimensions as follows:

[TABLE]

where $N$ parameter can be written by parametrical definitions for three-dimensional like-hydrogen atom.

4 $Hurwitz$ transformations in the sixteen-dimensional quantum isotropic oscillator

The transformation of nine-dimensional like-hydrogen atom to sixteen-dimensional isotropic oscillator will be followed by mentioned general model, if Hurwitz transformation is applied as [11]:

[TABLE]

Hurwitz transformation matrix is arranged so that the basic properties of transformations are satisfied in it. $x_{i}$ coordinates of nine-dimensional like-hydrogen atom are written based on the $u_{j}$ Coordinates of sixteen-dimensional isotropic oscillator by considering relation (2.2):

[TABLE]

$\hat{\varphi}_{i}$ differential operators are written by using relations (2.10) and (4.1):

[TABLE]

According to relation (2.13), the $\hat{\varphi}^{2}$ operator is defined as follows:

[TABLE]

$\hat{\varphi}^{2}$ is the operator that connects Laplacian of nine-dimensional like-hydrogen atom to sixteen-dimensional isotropic oscillator. Therefore, $\varepsilon$ energy spectrum of sixteen-dimensional isotropic oscillator as:

[TABLE]

where is obtained from relations (2.18), is connected to energy discrete spectrum of nine-dimensional like-hydrogen atom. $E$ energy spectrum of like-hydrogen atom in nine-dimensions, for $m=4$, can be obtained with relation(2.19):

[TABLE]

where $N$ parameter is defined based on like-hydrogen atom in nine-dimensions. Since there exists a hidden non-abelian monopole in the sixteen- dimensional isotropic oscillator, so, this system looks like a nine-dimensional like- hydrogen atom in a field of monopole desired by septet of potential vectors [8].

5 Conclusion

Using appropriate transformations and mathematical computations creates a relationship between d-dimensional isotropic oscillator and D-dimensional like-hydrogen atom coordinations. It is considered that mathematical calculations leads relationship between D-dimensional laplacian of like-hydrogen atom and d-dimensional laplacian of isotropic oscillator. Energy spectrum and wave function are calculated through replacing obtained relations in Schrödinger equation of D-dimensional like-hydrogen atom and comparing of this equation to d-dimensional isotropic oscillator problems.