Taban operatörlerine açılım yöntemi ile enerji spektrumunun belirlenmesi: Kuvantum anharmonik salıncı
Başlık çevirisi mevcut değil.
- Tez No: 55486
- Danışmanlar: PROF.DR. METİN DEMİRALP
- Tez Türü: Yüksek Lisans
- Konular: Mühendislik Bilimleri, Engineering Sciences
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 1996
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 47
Özet
ÖZET Kuvaııtum fiziğinde yapılan çalışmalarda ele alman sistemlerin davranışları sistem relativistik bir durumda değilse Schrödiuger denkleminin çözümüyle be lirlenebilir. Bu denklem kullanılarak, incelemeler belli lineer operatörlerin öz- değer problemlerinin çözümüne dönüştürülür. Bu çalışmada ele alman sistem genellikle kristal ve katı hal fiziğindeki sistemleri modellemede kullanılmakta olan bir anlıarmonik şahmadır. Bu türden sistemler ayrık spektruma sahip olmaları nedeniyle sistemin davranışının belirlenmesi için çözülmesi gereken özdeğer problemlerine kolaylıklar getirirler. Burada gözönüne alman sistem dördüncü dereceden anlıarmonik şahmadır. Ele alınan sistemin enerji durumunun bulunması için özdeğerlerin belirlenmesi aşamasında Taban Operatörlerine Açılım yöntemi kullanılmıştır. Bu yöntemde taban op ar atollerinin sistemin Hamilton operatörü ile komütatör ve anti-komü- tatör ilişkileri kullanılarak elde edilen iki denklem takımı üzerinden kullanılır. Konum operatörlerinin beklenen değerleri ve sistemin enerji özdeğerleri biri lineer diferansiyel denklem diğeri de bir özdeğer denklemi yolu ile belirlenen iki sonsuz boyutlu matris kullanılarak hesaplanmaya çalışılmıştır. Bu amaçla yapılan hesaplar Ek-A, Ek-B ve Ek-C'te verilen program parçaları ile kontrol edilmişlerdir. Bu yöntem anlıarmonik salıma için uygun sonuçlar vermeyince sistem har- monik salınıcıya indirgenerek denenmiştir. Anlıarmonik salıma için uygun çözümler bulmak amacıyla özdeğer problemi ikinci dereceden bir fark denklemi olarak yeniden düzenlenip konum operatörlerinin beklenen değerleri ile sistemin taban özdeğeri belirlenmeye çalışılmıştır. Yöntemin anlıarmonik sahmcmın uyarılmış durumlarmdaki enerji özdeğerleriniıı belirlenmesinde kullanılıp kul lanılamayacağını belirlemek üzere harmonik salıma için bir sınama yapılmış ve sonuçların doğruluğu görülmüştür.
Özet (Çeviri)
SUMMARY DETERMINATION OF ENERGY SPECTRUM VIA THE BASIS OPERATOR EXPANSION: QUANTUM ANHARMONIC OSCILLATORS In the early years of this century quantum mechanics was a brand new subject for both theoritical and experimental researchers. After decades, today the best known results of solid-state and crystal physics are widely used in daily life such as radios, digital watches, calculators, etc. Semi-conductor and laser technology are based on quantum mechanics. For this reason advanced researches on these fields like super computer design and femtosecond lasers are related with quantum mechanical rules. Laser applications in scientific researches necessitate high technology. Es pecially in chemical reactions femtosecond lasers are used for effecting the reac- tants or product in such a way that result would be more productive or appro priate for the aim. The shorter wave length the laser has the more possibility is obtained to effect the molecules. Shapes and properties of the molecules can be changed during the reaction using external forces. Magnetic fields or laser interactions are used as external forces. Although nuclear magnetic resonance is a well known subject, controlling chemical reactions via laser interaction is under study. Development of laser devices gives a lot of chance to choose the right laser. Laser effects on molecular motion are not considered in this work. However it is an extension of that kind of studies. Molecules are not relativistic particles. So their motions obey Schrödinger's equation which is a partial differential equation. Even the solution of the Schrödinger's equation for a diatomic molecule's is hardly possible, analytical solution for the molecules containing many atoms is nearly impossible. Difficul ties in controlling molecular motion is due to nature of the quantum mechanical laws and the nonliuearities in these calculations. On this topic some numerical techniques were developed. One of these techniques is Basis Operator Ex pansion proposed by Metin Demiralp for the solution of the optimal control problems in molecular motions. VIBasis Operator Expansion method is based on the expactation values of the chosen basis operators. During determination process of the molecular motion the object is to determine the wave function by solving Schrödinger's equation. The wave function also appeal's in the definition of expectation val ues. The expectation values depend on only time and store information about the molecular system. On this aspect, basis operators and Hamilton operator of the system would be defined. When basis operator set is infinite dimen sional commutator relation between expectation values of the basis operators and the Hamiltoniau, constitutes a set of linear ordinary differential equation. Also anti-commutator relation is proportional to the expectation value of the operator itself. This new relation represents an eigenvalue problem. The scope of this work is to test the efficiency of the method on a non linear Hamiltouian system. For this purpose one dimensional quartic oscillator is considered. In this work basis operators are chosen as m 0“ Qm,n =.'£ q~^ (1) and the set of differential equations is obtained via the following equations. d I m ön \ */ tf < in ro-2 d”n* m-1 9“+1 dt \ dx11 / h\ 2//. v ' dxn //. dxn+* -kMn - l)xm-£ =? - ktnxm+1 -£ r 2 K ' dxn~2 dxn~l 1-kMn-l)(n~2)(n-3)xm-^ (2) k2n(n - l)(n - 2).rm+1 - - Qxn-Z ^n(n-l),-+2#-¥-fc2n.^+3d dxn~2 ' dx.r.n-1 vnSimilarly, eigenvalue problem is given as A ( x «Z=: > = - 7T”l(«* - 1) ( *ro 2 7T- ) - ^-m ( a;“1 J 9arM / 4ft, K J\ dxn J 2ft, \ dxn+1 »a l.,n dn+2 \Mh / a» \ ^ a* / ro+4 a- *m7T-^r + if ( *”'^^7 + ^ * 2/i \ a.c»+2 / 2 \ ö;c» / 4 \' a.cM + -i ??.(n - 1) ( xm-- 7 > + -^ n ( zm+1 4 v ' \ öa;»-2 / 2 \ Ox»“1 +.£»(n-l)(n-2)(n-3)(xm n- 4 8 v /v /v ' \ cfo”-4 + ^n(n-l)(n-2)(,^£^). 3&2,,v/ m+2 dn~2 \ h I m+, o“”1 4 v ' \ dxn~* / 2 \ öre'1-1 (3) Initially basis operators vector is chosen from the null-space of the Schrö- dinger equation. This choice makes expectation values time-independent. In particular, differential equation set becomes a- homogeneous, linear equation set. This means, two infinite dimensional equation set is obtained and expec tation values would be determined by solving them simultaneously. For this reason basis operators are sorted by a convenient, index. In any case, aforementioned matrices are sparse matrices. Even after lin ear dependence between position and momentum operators are declared and reconstruction of the equations is done, matrices keep their form but in a more compact manner. Solving the eigenvalue problem in this new form gives com plex eigenvalues. When undesirable solutions were obtained, method tested on harmonic oscillator. Obtaining roots that are pure complex but acceptable ab solute values means finite truncation of matrices are insufficient. Presumably adequate dimension would be hundreds or thousands to get real eigenvalues. Because of the high demand of computational capabilities of this high dimen sions with not a guaranteed result, it is avoided here to go forth. viiiThe equation which represents eigenvalue problem would be treated as a second order finite difference equation. A (;c»>) = _|!m(m_ 1) /*«-') + *'/W +f) (*'»+») + y W +^ (*«+') (4) N ' 8/t v ;x ' 2(ro + l) x ' 4(7?i + l) x ' K ' Obviously the anharmonicity constant &2 will bring a considerable difficulty. Taylor series expansion of all unknowns around the point &2 = 0 gives another equation set. The first of these equations is a homogeneous one and eigenvalues of harmonic oscillator satisfy it. Other equations are non-homogeneous. In the.non-homogeneous case for the test of the convergence a test matrix was constructed. It is known that if the matrix is diagonal- dominant then the non-homogeneous finite difference equations would be convergent. Finite differences method is a tool for solving differential equations. So, when homogeneous solution is known, non-homogeneous solution would be ob tained via homogeneous solution as in the solutions of the differential equations. In order to get a solution the non-homogeneous solution is taken as /“2m\ _/,.2m\ ”k _ ^" F(m + V2) _fc (rs The homogeneous equation is also used for reconstruction of the non-homoge neous equations. (m + l)a2ra+2 - o2m - ma2m_2 = -n(m+i)(2m _ 1)r(m _ 1/2)a2m (6) Here «2m's are constants known through previously determined eigenvalues and expectation values. Then representation of the unknown function a-2m with the appropriate variables reduces the order of the finite difference equation. The time elapsing part of the problem is successive solutions of the first order finite difference equations. Solution of this new first-order finite differences scheme also brings another difficulty. During solution process of the second non-homogeneous equation, solution of the first non-homogeneous equation could not be added analytically. So there would be some restrictions to avoid this kind of solutions. When unknown function cr2m treated as a polynomial type function with respect to m, successive solutions of non-homogeneous equations become easier. This kind of approach does not have an undesirable result over solution. The goal of this work is the determination of the molecular motion of the quartic anharmonic oscillator via The Basis Operator Expansion and the ef ficiency of the method on that system. When results were evaluated for the ixOentire interval of the expansion variable hi numerically, same results with the values in the references were obtained. Lastly it can be shown that the series expansion makes the solution convergent in the unit ball centered at k% = 0. The last study is the determination of the expectation values and eigenval ues of the harmonic oscillator by using this method and same basis operators when harmonic oscillator's excited states are considered. When such kind of shifting between those cases is needed to put k% = 0 in equations would be enough. So eigenvalues and expectation values of basis operators were obtained easily.
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