study on representation of linear canonical transformation
TRANSCRIPT
Study on Representation of Linear Canonical Transformation in Quantum mechanics Raoelina Andriambololona1, Hanitriarivo Rakotoson2, Ravo Tokiniaina Ranaivoson3
[email protected]; ,[email protected] ; [email protected]
Theoretical Physics Department
Institut National des Sciences et Techniques NuclΓ©aires (INSTN- Madagascar)
BP 4279 101-Antananarivo βMadagascar, [email protected]
Abstract: Linear canonical transformations are particularly well known in the framework of
signal theory as integral transformations. Particular example of Linear canonical
transformations are the Fourier and Fractional Fourier Transforms. We show that Linear
canonical transformations can be well described in framework of quantum mechanics using
properties of momentum and coordinates operators, linear algebra and group theory.
Keywords: Linear Canonical Transformation, Fractional Fourier Transform, Quantum theory, Quantum operators, Lie algebra
1-Introduction
Let π be a function of the time variable π‘ and π the real matrix
π = π ππ π
Verifying πππ‘π = ππβ ππ = 1 i.e π β ππΏ(2,β). In framework of signal theory, a Linear Canonical Transformation (LCT) may be defined as the integral transformation [1], [2]
πΉ π’ =12ππ π
!" π(π‘)π!!π(!"!
π!!!π!!! )ππ‘
In which π is a real constant. For the case
π = πππ πΌπ = π ππ(πΌ) π = βπ ππ(πΌ) π = πππ (πΌ)
πππ = π!!(!!
!!)
we obtain the Fractional Fourier Transformations [1], [3]
πΉ! π’ =1β ππππ‘ πΌ
2π π(π‘)π![!"# ! (!!!!!! )! !"
!"#(!)]ππ‘
And for the case πΌ = !!, we have the Fourier transformation
πΉ!!π’ =
12π
π(π‘)π!!"#ππ‘
It can be shown that in the framework of quantum mechanics, a linear canonical
transformation can be defined as a linear transformation mixing coordinate and momentum operators π and π and leaving invariant the canonical commutation relation [1].
π = ππ+ πππ = ππ+ ππ
[π,π]! = [ π,π]! = i
In fact combining the three relation, we can deduce the condition which links the
parameters, π,π, π and π, of the linear canonical transformation
ππβ ππ = 1
And we may show that the above integral transformation, defining the LCT, corresponds to the law of transformation of wave functions [1]
π¦ π =12ππ π
!" π₯ π π!!π(!"!
π!!!π!!! )ππ₯
π¦ and π₯ being respectively the eigenstates of the operators π and π and π a state vecto
r.This analogy justify definition of LCT based on the use of the momentum and coordinates o
perators. This definition gives a natural way to study them and their multidimensional general
ization in framework of quantum theory.
2-Dispersion operator algebra
Let π,π,π,βπ be the quantum state of a particle admitting as wave functions, respectively in momentum and coordinates representations the functions [1],[4]
π! π₯,π,π,Ξπ =π»!(
π₯ β π2Ξπ₯
)
2!π! 2πΞπ₯π!(
!!!!"! )
!!!"#
π! π,π,π,Ξπ =12π
π! π₯,π,π,Ξπ π!!"#ππ₯ =(βπ)!π»!(
π β π2Ξπ
)
2!π! 2πΞππ!(
!!!!"!)
!!!"(!!!)
i.e π₯ π,π,π,Ξπ = π! π₯,π,π,Ξπ , and π π,π,π,Ξπ = π! π,π,π,Ξπ . π»! being a Hermite polynomial. The momentum operator and coordinates operator mean values and statistical dispersion (variance) corresponding to a state π,π,π,βπ are respectively π,π, 2π + 1 Ξπ₯ !, (2π + 1) Ξπ !. The parameters Ξπ and Ξπ₯ verifying the relation
Ξπ Ξπ₯ =12
Let β¬ = Ξπ ! , it can be shown that a state π,π,π,βπ is an eigenstate of the operator [1],[4],[5]
βΆ! =12 πβ π ! + 4 β¬ ! πβ π !
with the eigenvalues (2π + 1) Ξπ !. βΆ! is called momentum dispersion operator. If we introduce the two hermitian operators [5]
βΆ! =12 πβ π ! β 4 β¬ ! πβ π !
βΆΓ = β¬ πβ π πβ π + πβ π πβ π
It can be shown, using the commutation relation [π,π]! = π, that the three operators βΆ!, βΆ!
and βΆΓ verify the commutation relation
[βΆ!, βΆ!]! = 4πβ¬βΆΓ
[βΆ!, βΆΓ]! = β4πβ¬βΆ!
[βΆΓ, βΆ!]! = 4πβ¬βΆ!
i.e they may be considered as the vector basis of a Lie algebra which can be called dispersion operator algebra [5]. Let be
βπ₯ = πΆ, βπ = π·, π = βπ₯ ! = (πΆ)! β¬ = (βπ)! = (π·)!
We can introduce the following reduced operator
π =πβ π2 Ξπ₯
=πβ π2πΆ
=πβ π2π
= 2 Ξπ πβ π = 2π· πβ π = 2β¬ πβ π
π =πβ π2 Ξπ
=πβ π2π·
=πβ π2β¬
= 2 Ξπ₯ πβ π = 2πΆ πβ π = 2π πβ π
and
βΆ! =βΆ!
4β¬ =14 ((π)
! + (π)π)
βΆ! =βΆ!
4β¬=14((π)! + (π)π)
βΆΓ =βΆΓ
4β¬ =14 ππ+ ππ
We have the commutation relations
[βΆ!, βΆ!]! = πβΆΓ
[βΆ!, βΆΓ]! = βπβΆ!
[βΆΓ, βΆ!]! = πβΆ!
[βΆ!,π]! =!!ππ
[βΆ!,π]! = β !!ππ
[βΆΓ,π]! =!!ππ
[βΆ!,π]! = β !!ππ
[βΆ!,π]! = β !!ππ
[βΆΓ,π]! = β !!ππ
The set βΆ!, βΆ!, βΆΓ is also a basis of the dispersion operator algebra. 3-Unitary representation
As we have already seen above, a linear canonical transformation can be defined as a linear transformation mixing the coordinate operator π and the momentum operator π and leaving invariant the commutator π,π = π [1], [5]. The operators π and π are linearly linked with the operators π and π we may also take a definition of linear canonical transformation as linear transformation mixing π and π
π! = Ξ π+ Ξππ! = Ξπ+ Ξπ
πβ²,πβ² = π,π = πβΊ π! π! = π π Ξ Ξ
Ξ Ξ , Ξ ΞΞ Ξ = Ξ Ξβ ΞΞ = 1
The matrix Ξ Ξ
Ξ Ξ is an element of the special linear group ππΏ(2,β). We may write it in the form
Ξ ΞΞ Ξ = πβ³ = π
β³! β³!β³! β³!
with β³ an element of the Lie algebra ππ(2,β) of the Lie group ππΏ(2,β), we have
Ξ Ξβ ΞΞ = 1βΊβ³! = ββ³! βΊβ³ = β³! β³!β³! ββ³!
Using the commutation relation between the operators βΆ!, βΆ!, βΆΓ,π and π, we may also establish an unitary representation of the linear canonical transformation [5]
π! = Ξ π+ Ξπ = πΌππΌ!
π! = Ξπ+ Ξπ = πΌππΌ!
with
Ξ ΞΞ Ξ = πβ³ = π
β³! β³!β³! β³! = π
!!
!!Γ !!!!!!!!!! !Γ πΌ = π!(!!βΆ!!!!βΆ!!!ΓβΆΓ)
The unitarity of πΌ results from the hermiticity of the operators βΆ!, βΆ!and βΆΓ.
4- Conclusion
The properties of momentums and coordinates operators permit to perform a well description of Linear Canonical Transformation in framework of quantum mechanics. The introduction of dispersion operator algebra permits to establish unitary representation.
References
1. RaoelinaAndriambololona, Ravo Tokiniaina Ranaivoson, Rakotoson Hanitriarivo, Wilfrid Chrysante Solofoarisina: Study on Linear Canonical Transformation in a Framework of a Phase Space Representation of Quantum Mechanics, arXiv:1503.02449[quant-ph], International Journal of Applied Mathematics and Theoretical Physics. Vol. 1, No. 1, 2015, pp. 1-8, 2015
2. Tian-Zhou Xu, Bing-Zhao Li: Linear Canonical Transform and Its Applications, Science Press, Beijing, China, 2013.
3. V. Ashok Narayanan, K.M.M. Prabhu, βThe fractional Fourier transform: theory, implementation and error analysisβ, Microprocessors and Microsystems 27 (2003) 511β521, Elsevier, 2003.
4. Ravo Tokiniaina Ranaivoson: Raoelina Andriambololona, Rakotoson Hanitriarivo, Roland Raboanary: Study on a Phase Space Representation of Quantum Theory, arXiv:1304.1034[quant-ph],International Journal of Latest Research in Science and Technology, ISSN(Online):2278-5299, Volume 2,Issue 2 :Page No.26-35,March-April, 2013
5. Raoelina Andriambololona, Ravo Tokiniaina Ranaivoson, Randriamisy Hasimbola Damo Emile, Rakotoson Hanitriarivo: Dispersion Operators Algebra and Linear Canonical Transformations, arXiv:1608.02268 [quant-ph], 2016