An algebraic method for Schrödinger equations in quaternionic quantum mechanics

Computer Physics Communications 178 (2008) 795–799 www.elsevier.com/locate/cpc

An algebraic method for Schrödinger equations in quaternionic quantum mechanics ✩ Tongsong Jiang a,b,∗ , Li Chen c a Department of Mathematics, Linyi Normal University, Shandong 276005, China b Department of Computer Science and Technology, Shandong University, Jinan 250100, China c Department of Physics, Linyi Normal University, Shandong 276005, China

Received 2 December 2007; received in revised form 8 January 2008; accepted 16 January 2008 Available online 31 January 2008

Abstract In the study of theory and numerical computations of quaternionic quantum mechanics and quantum chemistry, one of the most important ∂ |f  = −A|f  with A an anti-self-adjoint real quaternion matrix, and |f  an eigenstate to A. The tasks is to solve the Schrödinger equation ∂t quaternionic Schrödinger equation plays an important role in quaternionic quantum mechanics, and it is known that the study of the quaternionic Schrödinger equation is reduced to the study of quaternionic eigen-equation Aα = αλ with A an anti-self-adjoint real quaternion matrix (timeindependent). This paper, by means of complex representation of quaternion matrices, introduces concepts of norms of quaternion matrices, studies the problems of quaternionic Least Squares eigenproblem, and give a practical algebraic technique of computing approximate eigenvalues and eigenvectors of a quaternion matrix in quaternionic quantum mechanics. © 2008 Elsevier B.V. All rights reserved. Keywords: Schrödinger equation; Quaternion matrix; Least Squares eigenproblem; Quaternionic quantum mechanics

1. Introduction In recent year, there has been a wide interest in formulating quantum theories by using the non-commutative field of quaternions [1–13]. In the study of theory and numerical computations of quaternionic quantum mechanics and field theory, one of the most important tasks is to solve the Schrödinger equation ∂ (1.1) |f  = −A|f , ∂t with A an anti-self-adjoint real quaternion matrix, and |f  an eigenstate to A. The quaternionic Schrödinger equation (1.1) plays an important role in quaternionic quantum mechanics, and it is known that the study of the quaternionic Schrödinger ✩ This paper is partly supported by the National Natural Science Foundation of China (10671086) and Shandong Natural Science Foundation of China (Y2005A12). * Corresponding author at: Department of Mathematics, Linyi Normal University, Shandong 276005, China. E-mail address: [email protected] (T.S. Jiang).

0010-4655/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cpc.2008.01.038

equation (1.1) is reduced to the study of quaternionic eigenequation Aα = αλ,

(1.2)

with A an anti-self-adjoint real quaternion matrix (time-independent). In the study of theory and numerical computations of quaternionic quantum theory, in order to well understand the perturbation theory, experimental proposals and theoretical discussions underlying the quaternionic formulations of the Schrödinger equation and so on, one often meets problems of approximate solutions of quaternion problems, such as approximate solution of quaternion linear equations Aα ≈ αλ that is appropriate when there are errors in the vector α and λ, i.e. quaternionic Least Squares eigenproblem (QLSE) in quaternionic quantum mechanics. The main difficulty in obtaining the quaternionic approximate solutions of a physical problem is due to the fact of the non-commutation of quaternion in general, and the standard mathematical methods of resolution break down. It is known that the complex Least Squares eigenproblem (LSE) has been

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T. Jiang, L. Chen / Computer Physics Communications 178 (2008) 795–799

developed as a global fitting technique especially in physics for solving approximate solutions of complex linear equations Aα ≈ αλ if errors occur in the vector α and λ. But the QLSE problem has not been settled now. In this paper, by means of complex representation, we study the quaternionic Least Squares eigenproblem (QLSE), and give a practical algebraic technique of computing approximate eigenvalues and eigenvectors of a quaternion matrix in quaternionic quantum mechanics. Let R denote the real number field, C the complex number field, Q the quaternion number field. For any quaternion x = x0 + x1 i + x2 j + x3 k,

(1.3)

i = j = k = −1,

(1.4)

2

2

2

ij = −j i = k,

in which xs ∈ R. For any quaternion x = x1 + x2 i + x3 j + x4 k where xs ∈ R, the conjugate of quaternion x is x = x1 − x2 i − x3 j − x4 k, and xx = x12 + x22 + x32 + x42 . For any quaternion matrix A, AT , A and AH denote the transpose, conjugate and conjugate transpose of A over quaternion field, respectively. Fm×n denotes the set of m × n matrices on a field F. For any quaternion matrix A = B1 + B2 i + B3 j + B4 k ∈ Qm×n , Bl ∈ Rm×n , l = 1, 2, 3, 4, A can be uniquely written as A = (B1 + B2 i) + (B3 + B4 i)j = A1 + A2 j , A1 , A2 ∈ Cm×n . It is easy to verify that for any A ∈ Cm×n , we have Aj = j A, and j Aj = −A. For any A ∈ Qn×n , A is unitary if AH A = In ; and A is anti-self-adjoint if AH = −A. Let A ∈ Qn×n , a quaternion λ is said to be a right (left) eigenvalue provided that Aα = αλ (Aα = λα), and α is said to be an eigenvector to corresponding eigenvalue λ. Two quaternions x and y are said to be similar if there exists a nonzero quaternion p such that p −1 xp = y, and this is written as x ∼ y. It is routine to check that ∼ is an equivalence relation on the quaternions. We denote by [x] the equivalence class containing x. By [14, Proposition 2.1] we easily get following result. Lemma 1.1. Let x = x0 + x1 i + x2 j + x3 k be a real quaternion. Then there exists a unit quaternion p (pp = 1) such that  h = x12 + x22 + x32  0, p −1 λp = x0 + hi, (1.5)  namely x ∈ [x0 + x12 + x22 + x32 i]. The complex number ν = x0 + hi in (1.5) is called principal number of the class [x]. 2. Norms of quaternion matrices In this section, we introduce concepts of norms of quaternion matrices by means of complex representation of quaternion matrices. For any quaternion matrix A = A1 + A2 j ∈ Qm×n , in paper [14], the author defined a complex representation   A1 A2 f A = (2.1) ∈ C2m×2n , −A2 A1 the complex matrix Af was called complex representation of A. Let A, B ∈ Qm×n , C ∈ Qn×s , a ∈ R. From [14] we have following results. (A + B)f = Af + B f ,

(aA)f = aAf ,

(2.2)

(AC)f = Af C f , Af

f = Q−1 m A Qn ,

0

(AH )f = (Af )H

(2.3) (2.4)

−It 

in which Qt = It 0 is a unitary matrix, It is t × t identity matrix. For A ∈ Qm×m , by (2.3) we easily know that the quaternion matrix A is nonsingular if and only if Af is nonsingular, and the quaternion matrix A is unitary if and only if Af is unitary. Proposition 2.1. (See [15].) Let A ∈ Qn×n . Then the real eigenvalues of complex representation Af appear in pairs, and the complex eigenvalues of complex representation Af appear in conjugate pairs. From the Frobenius norm (or Euclid norm)  · F and 2 norm (or spectral norm)  · 2 of complex matrices, we derived [16] following norms of quaternion matrices. That is for A ∈ Qm×n and x ∈ Qn×1 , define AF ≡ Af F , and A(F ) ≡



tr(AH A),

A2 ≡ Af 2 ,

A(2) ≡ max Ax2 , x2 =1

(2.5)

(2.6)

then by [16] we get following result. Proposition 2.2.  · F ,  · (F ) ,  · 2 and  · (2) are unitarily invariant norms of quaternion matrices, and √ AF = 2A(F ) , (2.7) A2 ≡ A(2) . We say  · F and  · 2 , respectively, to be Frobenius norm (or Euclid norm) and 2 norm (or spectral norm) of quaternion matrices. 3. Quaternionic Least Squares eigenproblem In this section, by means of complex representation, we study the quaternionic Least Squares eigenproblem (QLSE), and give a practical algebraic technique of computing approximate eigenvalues and eigenvectors of a quaternion matrix in quaternionic quantum mechanics. In the study of theory and practical numerical computations in quaternionic quantum mechanics, the eigenproblem of finding eigenvalues and eigenvectors of a quaternion matrix is a very important and difficult problem. In practical, the eigenequation Aα = αλ do not always has precision solutions for an approximate eigenvalue λ of a quaternion matrix, and one have to find approximate solutions α and λ. That is one meets following problems of approximate solution of quaternion equations Aα ≈ αλ that is appropriate when there are errors in the vector α and λ, i.e. quaternionic Least Squares eigenproblem Aα − αλF = min,

(3.1)

in which  · F the Frobenius norm of quaternion matrices. A ∈ Qn×n , α ∈ Qn×1 and λ ∈ Q. We say (λ, α) a solution of (3.1)

T. Jiang, L. Chen / Computer Physics Communications 178 (2008) 795–799

if there exist a quaternion λ and a quaternion vector α such that (3.1). If (λ, α) is a solution of (3.1), and let ν = p−1 λp be the principal number of [λ] for a unit quaternion p with Im ν  0, then the (3.1) is equivalent to 

 A(αp) − (αp) p−1 λp  = Aγ − γ νF = min, (3.2) F in which γ = αp and ν = p −1 λp is the principal number of [λ] for a unit quaternion p with Im ν  0. Therefore, (λ, α) is a solution to (3.1) if and only if (p−1 λp, αp) is a solution to (3.1) for any unit quaternions p. By the statement above, we study the quaternionic Least Squares eigenproblem (3.1) with complex eigenvalues λ and Im λ  0 from now on. Consider following complex Least Squares eigenproblem of corresponding complex representation matrices of (3.1) Af β − βλf F = min.

(3.3)

From (3.5)–(3.8) we obtain following result. Proposition 3.2. If (μ, ζ ) is a solution of complex Least Squares eigenproblem (3.5), then (μ, (ζ, Qn ζ )) is a solution of the complex Least Squares eigenproblem (3.3) and vice versa. Next we study the relationship between the solutions of the quaternionic Least Squares eigenproblem (3.1) and that of the complex Least Squares eigenproblem (3.3). In the first place, if (λ, α) is a solution to (3.1), then by Proposition 3.1 (λ, α f ) is a solution to (3.3). Conversely if (λ, β) is a solution of (3.3), then Af β − βλf F = min .

(3.9)

By (2.4), we know that Af = Qn Af QTn , λf = Q1 λf QT1 , and 



 Af β − βλf F = Af QTn βQ1 − QTn βQ1 λf F 



 = Af QT βQ1 − QT βQ1 λf  . (3.10) n

By the definition of the Frobenius norm and (2.2)–(2.3), we have   Aα − αλF = (Aα − αλ)f F = Af α f − α f λf F . (3.4) By the statement (3.1)–(3.4) above, we get following result. Proposition 3.1. The quaternionic Least Squares eigenproblem (3.1) has a solution (λ, α) if and only if the complex Least Squares eigenproblem (3.3) has a solution (λ, α f ). Let A ∈ Qn×n , and Af be the complex representation of quaternion matrix A. It is known by Proposition 2.1 that the complex eigenvalues of complex representation Af appear in conjugate pairs. Therefore if μ an approximate eigenvalue of the complex representation Af , then so is μ and vice versa. Let μ be an approximate eigenvalue of the complex representation Af . Then we can find a complex vector ζ ∈ C2n×1 such that Af ζ − ζ μF = min

(3.5)

and Af ζ − ζ μF = min.

(3.6)

By (2.4) and the definition of the Frobenius norm, we have Af = Qn Af QTn and Qt is a unitary matrix, and  f  A (Qn ζ ) − (Qn ζ )μ = Af ζ − ζ μF = min . F Combine (3.5) and (3.7) we have  f  A (ζ, Qn ζ ) − (ζ, Qn ζ )μf  F      μ 0   f = A (ζ, Qn ζ ) − (ζ, Qn ζ )  0 μ   F  f  f  = A ζ − ζ μF + A (Qn ζ ) − (Qn ζ )μF = min.

(3.7)

(3.8)

797

n

F

This means that if (λ, β) is a solution to (3.3), then (λ, QTn βQ1 ) is also a solution to (3.3). Let

1 β + QTn βQ1 . 2 Since

βˆ =

(3.11)

Af β − βλf F ˆ f F  Af βˆ − βλ



 1 = (Af β − βλf ) + Af QTn βQ1 − QTn βQ1 λf F 2



 1 f 1  A β − βλf F + Af QTn βQ1 − QTn βQ1 λf F 2 2 f f = A β − βλ F , and hence ˆ f F . Af β − βλf F = Af βˆ − βλ

(3.12)

ˆ is also a Therefore, if (λ, β) is a solution to (3.3), then (λ, β) solution to (3.3). Let  β11 β12 β= (3.13) , βkl ∈ Cn×1 , k, l = 1, 2. β21 β22 By direct calculation we can easily get that  ˆ β1 βˆ2 ˆ , β= −βˆ 2 βˆ 1 where 1 1 βˆ2 = (β12 − β 21 ). βˆ1 = (β11 + β 22 ), 2 2 From (3.14) we construct a quaternion matrix  1 1 . α = βˆ1 + βˆ2 j = (In , −j In )βˆ j 2

(3.14)

(3.15)

(3.16)

Clearly α f = βˆ and (λ, α f ) is a solution to (3.3). So by Proposition 3.1 we have that (λ, α) is a solution to (3.1).

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The statement above implies following result Proposition 3.3. Let A ∈ Qn×n . Then the quaternionic Least Squares eigenproblem (3.1) has a solution (λ, α) if and only if the complex Least Squares eigenproblem (3.3) has a solution (λ, β). In which case, if (λ, β) is a solution to (3.3), then (λ, α) is a solution to (3.1), in which 

1 1 . α = (In , −j In ) β + QTn βQ1 (3.17) j 4 Now combine Propositions 3.1–3.3, we derive following theorem. Theorem 3.4. Let A ∈ Qn×n , λ is a complex number with Im λ  0. Then (1) The quaternionic Least Squares eigenproblem (3.1) has a solution (λ, α) if and only if the complex Least Squares eigenproblem (3.5) has a solution (λ, ζ ). In which case, if (λ, ζ ) is a solution to (3.5), then (λ, α) is a solution to (3.1), in which 

1 1 α = (In , −j In ) β + QTn βQ1 , β = (ζ, Qn ζ ). j 4 (3.18) (2) If (λ, α) is a solution of quaternionic Least Squares eigenproblem (3.1), then (p −1 λp, αp) is also a solution of quaternionic Least Squares eigenproblem (3.1) for any unit quaternions p. As an especial case, we have a corollary as follows. Corollary 3.5. Let A ∈ Qn×n , λ is a complex number with Im λ  0. Then (1) The eigenequation Aα = αλ has a solution (λ, α) if and only if the equation Af ζ = ζ μ has a solution (λ, ζ ). In which case, if (λ, ζ ) is a solution to Af ζ = ζ μ, then (λ, α) is a solution to Aα = αλ, in which 

1 1 α = (In , −j In ) β + QTn βQ1 , β = (ζ, Qn ζ ); j 4 (3.19) (2) If (λ, α) is a solution of Aα = αλ, then (p −1 λp, αp) is also a solution of Aα = αλ for any nonzero quaternions p. 4. Algorithm In last section, Theorem 3.4 sets up a bridge between the solutions of the quaternionic Least Squares eigenproblem (3.1) and that of the complex Least Squares eigenproblem (3.5), and suggests an algebraic technique of computing a solution of quaternionic Least Squares eigenproblem (3.1) by that of complex Least Squares eigenproblem (3.5). In this section, we list an algorithm for computing a solution of quaternionic Least Squares eigenproblem (3.1) by means of complex representation.

Algorithm. Let A ∈ Qn×n . Step 1. Find the complex representation matrix Af ; Step 2. Find a solution (λ, ζ ) of the complex Least Squares eigenproblem (3.5); Step 3. Find a solution (λ, α) of the quaternionic Least Squares eigenproblem (3.1): If (λ, ζ ) is a solution of (3.5), let β = (ζ, Qn ζ ), and 

1 1 α = (In , −j In ) β + QTn βQ1 , j 4 then (λ, α) is a solution of (3.1). Moreover, if (λ, α) is a solution of quaternionic Least Squares eigenproblem (3.1), then (p −1 λp, αp) is also a solution of quaternionic Least Squares eigenproblem (3.1) for any unit quaternions p. Example. Let  2i − k 1+j A= . 1 − j −i − 2k Find a solution (λ, α) of quaternionic Least Squares eigenproblem (3.1). Solution. It is easy to find the complex representation Af of quaternion matrix A by (2.1) ⎛ ⎞ 2i 1 −i 1 −i −1 −2i ⎟ ⎜ 1 Af = ⎝ ⎠, −i −1 −2i 1 1 −2i 1 i of and |λI4 − Af | = (λ2 + 3)2 . Therefore the eigenvalues √ the√ complex representation matrix Af are λ1 = 3i, λ2 = − 3i = λ1 , and by direct calculation we easily derive that (λ1 , ζs ), s = 1, 2, are two solutions of (3.5), in which √ √ ζ1 = (2i, 3 − 3, 0, 1 − 3 )T , √ √ ζ2 = (0, −1 − 3, −i, 3 + 3 )T are two eigenvectors of Af corresponding to eigenvalues λ1 , namely Af ζ1 = ζ1 λ1 , Af ζ2 = ζ2 λ1 . Let β1 = (ζ1 , Q2 ζ 1 ), β2 = (ζ2 , Q2 ζ 2 ), and  

1 1 T α1 = (I2 , −j I2 ) β1 + Q2 β 1 Q1 j 4   2i√ = √ , ( 3 − 1)( 3 + j )  

1 1 T α2 = (I2 , −j I2 ) β2 + Q2 β 2 Q1 j 4   −2k √ √ = . −( 3 + 1)(1 + 3j ) Then (λ1 , αs ), s = 1, 2, are two solutions of (3.1). Therefore, for any nonzero quaternion p, (p −1 λ1 p, αs p), s = 1, 2, are also solutions of quaternionic Least Squares eigenproblem (3.1).

T. Jiang, L. Chen / Computer Physics Communications 178 (2008) 795–799

5. Conclusions In this paper, by means of complex representation of a quaternion matrix, we first introduce the norms of quaternion matrices, study the problems of quaternionic Least Squares eigenproblem, and give a practical algebraic technique of computing approximate eigenvalues and eigenvectors of a quaternion matrix. This paper sets up a bridge between the solutions of the quaternionic Least Squares eigenproblem and that of the complex Least Squares eigenproblem, turns the problems of solutions of quaternionic Least Squares eigenproblem to that of the complex Least Squares eigenproblem by means of complex representations of quaternion matrices. This paper also provides a practical formula technique for finding the solutions of the quaternionic Least Squares eigenproblem, and the algebraic technique will have potential applications in the study of theory and numerical computations in modern quaternionic quantum mechanics. References [1] D. Finkelstein, J.M. Jauch, S. Schiminovich, D. Speiser, Foundations of quaternion quantum mechanics, J. Math. Phys. 3 (1962) 207. [2] S.L. Adler, Quaternionic quantum field theory, Phys. Rev. Lett. 55 (1985) 783. [3] S.L. Adler, Quaternionic quantum field theory, Comm. Math. Phys. 104 (1986) 611.

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