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Kuantum matrisleri ve bessel fonksiyonları

Quantummatrices and bessel functions

  1. Tez No: 39253
  2. Yazar: ABDULKERİM SARACOĞLU
  3. Danışmanlar: PROF.DR. METİN ARIK
  4. Tez Türü: Doktora
  5. Konular: Matematik, Mathematics
  6. Anahtar Kelimeler: Belirtilmemiş.
  7. Yıl: 1993
  8. Dil: Türkçe
  9. Üniversite: İstanbul Teknik Üniversitesi
  10. Enstitü: Fen Bilimleri Enstitüsü
  11. Ana Bilim Dalı: Belirtilmemiş.
  12. Bilim Dalı: Belirtilmemiş.
  13. Sayfa Sayısı: 75

Özet

ÖZET a, b, c, d operatörler olmak üzere fa b\ kuantum matrisinin öz değer problemini inceledik. t2 ç = er = l+r + - +... olmak üzere A = erM =I + tM+ - kuantum matrisini seriye açtık. A mu elemanlarının sağladığı komutasyon bağıntılarım kullanarak, M matrisinin komutasyon bağıntılarım bul duk. Seriden birinci mertebe terimlerin alınması ile elde edilen M ye ait komutasyon bağıntılarının, yüksek mertebe terimler alınarak elde edilen komutasyon bağıntıları ile aynı olduğunu belirttik. M nin öz değeri a ve öz vektörü X olmak üzere, A nm öz değerinin qa ve öz vektörünün de aynı X vektörü olduğunu gösterdik. M = v K matrisinin M = az XqX -fio(x) - x - şeklinde temsil edilebileceğini bulduk. M ye ait öz vektör X = (f(x),g(x)) ise, a öz değer olmak üzere, f(x) fonksiyonunun sağlaması gerekli denklem olarak x2f" + 2xft0(x)f' + (u0X0x2 + [iQ2 + xfiQ - fJ,o - a? + a)f(x) = 0 değişken katsayılı ikinci mertebe lineer diferansiyel denklemini bulduk. Bunun çözümünün f(x) = y/xe~J * * Zk\f Vq\qX olduğunu gösterdik. Burada Zk ile mertebesi k olan Bessel denkleminin çözümü gösterilmiş olup, k2 = (a - -)2 dir.

Özet (Çeviri)

QUANTUM MATRICES and BESSEL FUNCTIONS SUMMARY The matrix elements of a quantum matrix, (a b\ \c dj satisfy the following commutation relations: ab = qba or [a, b] = (q - l)ba ac = qca or [a, c] - (q - l)ca bd = qdb or [b, d] - (q - l)db cd - qdc or [c, d] = (q - l)dc be = cb or [6, c] = 0 ad - da = (g - q~x)bc or [a, d\ = (q - q~x)bc where a,b,c,d are operators and q is a real or complex parameter. To compute the determinant of A, the Borel decomposition method is used 'a b \ /l W-J\ (a-bd-xc 0\ / 1 0' c d J \0 1 / \ 0 dj \d~l 1 From this, the determinant of A is det A - (a - bd~1c)d = ad - qbc = da - q~lbc. The inverse of A is found as A~l - ıh adiA where det A = ad - qbc is used. As a result d -q~H i-i det A. -qc is found. VIIt can be shown that when the matrix A is real, the complex parameter q is on the unit circle and when matrix A is unitary, the q parameter is required to be real. The parameter of An is qn where n is an integer [1,2]. In a Lie qroup, any element of the group can be expanded into a series around the unit element. First degree term of this series is a member of the Lie Algebra of the group. The exponential of a member of Lie Algebra gives an element of Lie group. We investigate whether this property is also valid for quantum groups. As a first step, 2 3 q = eT = 1 + T + L- + L- +... (2) where r is a parameter. For matrix A, written as [3]. M is a new matrix whose elements are also operators. M can be written as fj, v M=[ j. (4) A K To lowest order in t, the first term of (3) (A = I + tM) yields a - 1 + Tfj,, b = rv, c = rA, d=l + tk. Similarly, (2) becomes q = 1 + r. By neglecting higher order terms the elements of M satisfy the fol lowing commutation relations [iv - vfj, - v or [fj,, v\ = v fj,X - X/j, = A or [fx, A] = A uk - ku = v or [u, k] = u Xk - k\ = A or [A, k] = A vX = Xv or [v, A] = 0 [J,K = Kfl or [fi, k] = 0. viiThese relations axe obtained from the commutation relations expressed in (1). Note that these relations do not include the parameter r. When second and higher degree terms of (2) and (3) are considered, again (5) is obtained. Therefore there are no higher order corrections to (5). On the other hand, the relations (5) are not approximate ones, they are exact [1]. For p, and k of matrix M we can write H = a + ~p, k = a - ~p. a and ~p are new operators defined by fj, + « _ ft - K a= -IT- » A*= - ^-. Commutation relations (5) show that a is commutative with all ele ments of the matrix M. Therefore qa is also commutative with everything. Thus the nontrivial structure of the matrix M is given by (J, v M=( |. (6) A -pt According to this, commutation relations are decreased into three [fj,, u] = v b,A] = A [i/,A] = 0. Using above relations, elements of M can be expressed in terms of the derivative operator. As a result, the matrix M is shown to be given by 'fj,0(x) + x-^ PQX M=( |. (7) Aqs ~(io(x) - x--fc If a is eigenvalue and X is eigenvector of the matrix M MX = aX=> MnX = anX viiion the other hand, AX = erMX =>AX = qaX and AnX = qnaX are obtained, in other words, X is eigenvector of An matrix with eigen value qna. Here n is an integer. Eigenvalue of M and Bessel Functions : (J. v MX = aX or I J I I = a I I (8) A - (j, clearly, these can be written as Hf + vg = otf \f - pg = ag. From these, the equations for eigenfunctions are obtained. (fi2 - ft + v\ - a2 + a) f(x) = 0 (fj? - fj. + uX - a2 - a) g(x) = 0 (9) It can be seen that the second equation of (9) is very similar to the first equation, a, has opposite sign in the second one. Considering (7) x2f“ + 2xpo(x)f + (uqXox2 + /i02 + W -Ho-a2 + a) f(x) = 0 (10) is obtained. This is a second degree linear differential equation with variable coefficients, and it has two independent solutions. Therefore, it is sufficient to use only one of the equations in (9). In case jj,q{x) is constant it is shown that the solution of equation (10) is 1 - 2/*o f{x) = X-1~ Zk{yfcx) (C = P0X0) (11) where Zk(y/cx) is the solution of Bessel equation and k2 = (a - |). In general, when fiQ is constant or variable we prove that the solution is the following f(x) = v£e”/ ^dxZk(^x) (12) Solution of (11) is a special case of the solution of (12), fJ,o(x) = const. Eigenfuntions in some special cases : ixWhen v0 and Ao have different signs, the eigenfunctions are f(x) = x 2 cilk( y/-u0Xox) + c2İİTfc( V-^oAoaO ((J,o = const) where Ik is the first kind and Kk is the second kind modified Bessel Function. Again assuming y,Q is constant, in the case M is antisymetric, the eigenfunction is 1 - 2/tp f(x) = x 2 [ci Ifc(|A0|») + c2 J_fc(|A0x)|ar)] In the case M is symmetric, the eigenfunction is found to be l - 2^o f(x) = x 2 Zk(\Xo\x) (fx0 = const) Let us say that the argument of Ux is \ and the argument of Ao is 4>2- In the special case of \ + 2 = - f the eigenfunction with fio = const, is 1 - 2/in f = x 2 ci[Berk(c0x) + iBeik(c0x) + c2[Kerk(c0x) + iKeik(cQx)] where cq = ji'oAol. In case 6 = 0 the eigenfunction is / = x-kZk(y/^x), fc2 = (a-i)2. The special cases when the eigenvalue is half odd integer or an integer are also investigated. In addition, an example is given. For larger x values, calculating the invariant function of equation (10) different solution are also searched. In these, the Sonin-Polya theorem is used. It is shown that in the normalized case of the self-adjoint form of equation (10) with /^o = const the eigenvalue is in the interval 0 < a < 1 as x - > 0. It has to be emphasized that x = 0 is a regular singular point of equation (10).One other interesting result is : the coefficient A in the Helmholtz equation VV + A^ = 0, and i/oAo in our eigenfunctions can be expressed in terms of the same physical quantities. Lastly, some of our second degree linear differential equations with variable coefficients are in the forms x2y" + 2xjj,0(x)y' + [b(x, po,k) + cx2]y = 0, or X X1 where b(x,fj,o,k) - p,Q2 + Xfio' - fio - k2 + \. The solutions of these of types of equations are given by y = y/xe J * Zk{s/cx), where Z\. is the solution of the Bessel Equation. XI

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