alexa
Reach Us +44-1522-440391
The Normal Ordering Procedure and Coherent State of the Q-Deformed Generalized Heisenberg Algebra | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
Make the best use of Scientific Research and information from our 700+ peer reviewed, Open Access Journals that operates with the help of 50,000+ Editorial Board Members and esteemed reviewers and 1000+ Scientific associations in Medical, Clinical, Pharmaceutical, Engineering, Technology and Management Fields.
Meet Inspiring Speakers and Experts at our 3000+ Global Conferenceseries Events with over 600+ Conferences, 1200+ Symposiums and 1200+ Workshops on Medical, Pharma, Engineering, Science, Technology and Business
All submissions of the EM system will be redirected to Online Manuscript Submission System. Authors are requested to submit articles directly to Online Manuscript Submission System of respective journal.

The Normal Ordering Procedure and Coherent State of the Q-Deformed Generalized Heisenberg Algebra

Won Sang Chung*

Department of Physics and Research Institute of Natural Science, College of Natural Science, Gyeongsang National University, Jinju 660-701, Korea

*Corresponding Author:
Won Sang Chung
Department of Physics and Research Institute of Natural Science
College of Natural Science
Gyeongsang National University
Jinju 660-701, Korea
E-mail: [email protected]

Received date: December 05, 2013; Accepted date: October 24, 2014; Published date: October 31, 2014

Citation: Chung WS (2014) The Normal Ordering Procedure and Coherent State of the Q-Deformed Generalized Heisenberg Algebra. J Generalized Lie Theory Appl 8:213. doi:10.4172/1736-4337.1000213

Copyright: © 2014 Chung WS. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Visit for more related articles at Journal of Generalized Lie Theory and Applications

Abstract

In this paper we discuss the normal ordering procedure of the q-deformed generalized Heisenberg algebra. We also obtain the coherent state for some types of characteristic functions.

Keywords

Heisenberg algebra; q-deformed

Introduction

In the last few years quantum algebras and quantum groups have been the subject of intensive research in several physics and mathematics fields. Quantum groups or q-deformed Lie algebra implies some specific deformation of classical Lie algebra. From the mathematical point of view, it is a non-commutative associative Hopf algebra. The structure and representation theory of quantum groups have been developed extensively by Jimbo [1] and Drinfeld [2]. In the study of the basic hypergeometric function Jackson [3] invented the Jackson derivative and integral, which is now called q-derivative and q-integral. Jackson's pioneering research enabled theoretical physicists and mathematician to study the new physics or mathematics related to the q-calculus. Much was accomplished in this direction and work is under way to find the meaning of the deformed theory. By using the q- calculus, Arik and Coon [4] proposed the q-deformation of the Heisenberg algebra as follows;

image

image(1)

Where imageis called a number operator and imageFollowing the approach of the authors of ref [4], several deformed Heisenberg algebra has been proposed in the literature [5-10]. In most of deformed Heisenberg algebra, authors adopted the same commutation relations between the number operator and step operators and deformed the commutation relation between a and a†

In 2000, the new generalization of the Heisenberg algebra was introduced by Rego-Monteiro and Curado [11-13], which takes the form

image(2)

where H is a hamiltonian of the physical system under consideration and f(H) is an analytic function of H, called a characteristic function of the algebra. In this algebra, the commutation relations between the number operator and step operators were changed into the more general form which is characterized in terms of the function of the number operator. The authors of the ref [11,12] called this function a characteristic function and discussed the cases when the characteristic function is linear and quadratic in the number operator [12].

In this paper we change the algebra (2) by introducing the parameter q and discuss the normal ordering procedure of the q-deformed generalized Heisenberg algebra (GHA). We use some operator identities to construct the q-deformed generalized Stirling operator of the second kind and its generating function. We also present the q-deformed generalized Heisenberg algebra whose characteristic function is a MÄobius transformation. Finally we discuss some characteristic functions giving a Klauder's coherent state.

Q Deformed Generalized Heisenberg Algebra

In this section we discuss the representation theory of the q-deformed generalized Heisenberg algebra. The q-deformed generalized Heisenberg algebra takes the following form;

image(3)

Where H is a hamiltonian of the physical system under consideration and f(H) is an analytic function of H, called a characteristic function of the algebra. The deformation parameter is related to the concrete form of f(H) and a large class of type Heisenberg algebra can be obtained by choosing the function f(H). From now on we restrict our concern to the case of

0 < q < 1

For example, if we take f(H) = 1 + qH, the algebra (3) reduces to the q-deformed Heisenberg algebra where the hamiltonian is related to the number operator N as follows

image(4)

The choice of f(H) gives a lot of deformed algebra, which is the reason why f(H) is called a characteristic function of the algebra. Now let us introduce the q-Casimir operator as follows;

image(5)

We demand that the q-Casimir operator obeys

image(6)

When q goes to a unity, the q-Casimir operator reduces to an ordinary Casimir operator. If We let C=CqCq-1 , we have

image(7)

Now let us construct the irreducible representation of the algebra (3) by introducing the ground state image with the lowest eigenvalue of H obeying

image(8)

Let imagebe a normalized eigenstate of H

image(9)

Applying H on image yields

image(10)

Which means that imageis an eigen state of H with eigen value imageApplying a† on the ground state successively, we have

image(11)

Where imagedenotes the n-th iterate of f defined as imageIf we assume that

imageis proportional to image, we have

image(12)

So all eigen values of H are determined fromimage through f.

Acting a on imagewe get

image(13)

We get

image(14)

Which shows that imageis also an eigen state of H with eigen value imageand imageis proportional to image

The representation of the q-deformed generalized Heisenberg algebra is then given by

image

image(15)

Where

image(16)

The relation between step operators and hamiltonian is given by

image(17)

image(18)

Where we have

image(19)

Here Pq is a q-projection operator satisfying

image(20)

Then we have

image(21)

Where f-number is defined as

image(22)

And

image(23)

The (n+1)-th eigenvalue en+1 of the Hamiltonian depends on the previous eigenvalue en

image(24)

So this algebra is sometimes called a one step algebra.

The representation can be rewritten in terms of the f-number as follows;

image(25)

Normal Ordering Process and F-stirling Operator

Now we discuss the normal ordering process for the q-deformed generalized Heisenberg algebra. From the second relation of the eq.(3), we have

image(26)

For an arbitrary function image. For imagewe get

image(27)

Replacing H → f −1(H) in the first relation of the eq.(3), we have

image(28)

or generally

image(29)

And

image(30)

Where

image(31)

Then we have the following formulas;

image(32)

image(33)

Where f0(H) = f(H).

The f-Stirling operator of the second kind is defined as

image(34)

Where S(n; k;H) is the f-Stirling operator of the second kind. Using

image

We can obtain the recurrence relation

image

image(35)

Where S(I, j,H) = 0 for i < j and we used the following formulas

image(36,37)

The first few Stirling operator of the second kind are

image(38)

We define the generating function of the f-Stirling operator of the second kind in the form

image(39)

The recurrence relations are then given by

image(40)

image(41)

If we set image, we get

image(42)

The eq.(42) can be written in terms of a sum of partial fractions

image(43)

Where

image(44)

Therefore the f-Stirling operator of the second kind takes the following form;

image(45)

The Deformed Heisenberg Algebra Related to the Mäobius Transformation

In this section we are going to find the representation for the algebra defined by the relation given in the eq.(3) considering

image(46)

Where α ,β ,γ ,δ are real. The ordinary Heisenberg algebra and q-deformed Heisenberg algebra are obtained from the suitable choice of α ,β ,γ ,δ inverse of the MÄobius transformation is given by

image(47)

Whereβγ −αδ ≠ 0 .

In the choice of the characteristic function given in the eq.(46), the algebra (3) takes the following form;

image(48)

Now let us introduce the following characteristic function;

image(49)

Where we assume that α > 0; 0 < γ < 1. Then we have the following algebra

image(50)

The inverse of the characteristic function is given by

image(51)

The n-the iterate of f is then given by

image(52)

Where image

Representation

For the characteristic function given in the eq.(49), we have

image(53)

image(54)

From imagefor all n, we have

image(55)

Where we assumed that

image

The representation takes the following form;

image(56)

Coherent state

We define a coherent state as a eigenvector of the annihilation operator as follows:

image(57)

Where z is a complex number. The coherent state can be represented by using the number state as follows :

image(58)

Inserting the eq.(58) into the eq.(57), we obtain the following relations :

image(59)

Solving the eq.(59), the coherent state is given by

image(60)

Where

image(61)

And

image(62)

Some Characteristic Functions giving Klauder's Coherent State

In this section we discuss the some characteristic functions giving q-deformed Klauder's coherent state (KCS). The KCS should satisfy the normalizability

image(63)

and the completeness

image(64)

Where we use the q-integral instead of the ordinary integral and

assume image

From the definition of the coherent state

image(65)

We have

image(66)

The case of image

In this choice we have

image(67)

Where we used

image(68)

If we choose imagewe have

image(69)

The coherent state is then given by

image(70)

Where

image(71)

And

image(72)

Now we have to find the weighting function w(x) so that the coherent sate may satisfy the Completeness. If we set image

and assume imagewe have

image(73)

We can easily find the weighting function

image(74)

Where we used

image(75)

The case of image

In this choice we have

image(76)

If we choose imagewe have

image(77)

The coherent state is then given by

image(78)

Where

image(79)

Now we have to find the weighting function w(x) so that the coherent sate may satisfy the completeness, which implies

image(80)

We can easily find the weighting function

image(81)

Where we used

image(82)

The case of imagep = 1; 2; 3;

In this choice we have

image(83)

If we chooseimage we have

image(84)

The coherent state is then given by

image(85)

Where

image(86)

and the q-polylogarithm function is defined by

image(87)

Now we have to find the weighting function w(x) so that the coherent sate may satisfy the completeness, which implies

image(88)

The eq.(70) is rewritten as

image(89)

If we set imagewe get image

Differentiating both sides of the eq.(89) p times with respect to n, we have

image(90)

Where

image(91)

If we set

image(92)

We have the following recurrence relation

image(93)

And

image(94)

For k = 1, the eq.(93) is as follows :

image(95)

Solving the eq.(95), we have

image(96)

For general k, we have

image(97)

And

image(98)

Where

image(99)

Using the formula (92), we have

image(100)

If we expand the summation of the eq.(100) in terms of a ¡ 1, we get

image(101)

Where

image(102)

Therefore we have the following formula;

image(103)

If we define the new function

image(104)

We have

image(105)

The first few Lp(x) are as follows;

image(106)

Therefore we obtain the weighting function as follows;

image(107)

Conclusion

In this paper we discussed the normal ordering procedure of the q-deformed generalized Heisenberg algebra, where we introduced the q-deformed generalized Stirling operator of the second kind instead of the Stirling number of the second kind and constructed its generating function. We also discussed the q-deformed generalized Heisenberg algebra whose characteristic function is a MÄobius transformation. Finally we discussed some types of characteristic functions giving a Klauder's coherent state. In fact, it is possible to construct more general algebra as

image(108)

It is tempting to investigate, as we did in this paper, the above algebra for some interesting characteristic function. We hope that this topic and its related one will become clear in the near future.

References

Select your language of interest to view the total content in your interested language
Post your comment

Share This Article

Relevant Topics

Article Usage

  • Total views: 12168
  • [From(publication date):
    December-2014 - Aug 22, 2019]
  • Breakdown by view type
  • HTML page views : 8332
  • PDF downloads : 3836
Top