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Quantum Mechanical Propagators Related to Classical Orthogonal Polynomials

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Quantum Mechanical Propagators Related to Classical Orthogonal Polynomials DEGREE PROJECT IN TECHNOLOGY, FIRST CYCLE, 15 CREDITS STOCKHOLM, SWEDEN 2021 Quantum Mechanical Propagators Related to Classical Orthogonal Polynomials VALDEMAR MELIN KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES Quantum Mechanical Propagators Related to Classical Orthogonal Polynomials Valdemar Melin June 8, 2021 Sammanfattning N˚agraendimensionella kvantmekaniska system med viktade klassiska ortog- onala polynom som egentillst˚andstuderas. Explicita uttryck f¨orpropaga- torn, dvs. integrationsk¨arnanf¨ortidsutvecklingsoperatorn, h¨arleds.I fallet med Hermitepolynom ¨arsystemet den harmoniska oscillatorn, medan det f¨or Laguerre- och Gegenbauerpolynom ¨arekvivalent med tv˚apartikel-Calogero- Sutherland-system. Abstract A few quantum systems on the line with weighted classical orthogonal poly- nomials as eigenstates are studied. Explicit expressions of the propagators, i.e. the integral kernels of the time evolution operators, are derived. In the case of Hermite polynomials, the system is the Harmonic Oscillator, while for generalized Laguerre and Gegenbauer polynomials, the corresponding quan- tum system are equivalent to two-particle Calogero-Sutherland systems. Acknowledgements I would like to express my sincere gratitude to my supervisor Edwin Lang- mann for guiding and encouraging me throughout the making of this report. It has been a pleasure to follow the trails of the ideas that he presented to me. I would also like to thank my colleagues Atabak Jalali and Hugo Akesson˚ for our valuable discussions and for all their support. Contents 1 Introduction 1 1.1 The Propagator . .2 1.2 Free Particle Propagator . .4 1.3 Classical Orthogonal Polynomials . .6 2 Harmonic Oscillator - Mehler Kernel 7 2.1 Mehler's formula . .8 2.2 A proof of Mehler's formula . .8 2.3 Properties of the Harmonic Oscillator Propagator . 10 3 Rational Calogero-Sutherland Propagator - Laguerre Ker- nel 11 3.1 The Laguerre Kernel . 11 3.2 From a particle in a Two-dimensional Harmonic Oscillator . 13 3.2.1 Classical Mechanics . 13 3.2.2 Quantum Free Particle System . 14 3.2.3 Quantum Harmonic Oscillator . 17 3.3 Properties of the Propagator . 18 4 Trigonometric Calogero-Sutherland Propagator - Gegenbauer Kernel 19 4.1 From a Three Dimensional Free Particle . 20 5 Concluding Remarks 22 Bibliography 23 Chapter 1 Introduction The study of exactly solvable systems is a very active area of research, due to the wide range of applications. For example, exactly solvable systems are great models to test numerical algorithms on. As a rule of thumb, a physical system is said to be exactly solvable if the equations of motion can be solved analytically and in terms of a set of "known" functions. In quantum mechanics, the equations of motion of a system are fully encapsulated in the propagator which is essentially the Green function of the equations of motion given by the Schr¨odingerequation. If one can find an analytical expression for the propagator, everything about the system is known and can be calculated exactly by evaluating an integral. A simple example is the free particle, where the propagator is just the Wick trans- formed heat kernel. The most famous non-trivial exactly solvable system is the quantum harmonic oscillator. It is intimately connected to Hermite polynomials, as the energy eigenstates are Hermite polynomials weighted by a gaussian. The propagator of the harmonic oscillator can be calculated analytically and is known as the Mehler kernel. It turns out that there are systems related to other classical orthogonal polynomials, and like the harmonic oscillator, they are also exactly solvable. In fact, they correspond to well-known exactly solvable quantum mechanical models [1] which have many-variable generalizations known as Calogero- Moser-Sutherland systems [2]; see also [3]. The purpose of this report is to investigate different methods to derive the propagators of quantum mechanical one-dimensional systems related to the classical orthogonal polynomials. The polynomials which will be studied are Hermite, Laguerre and Gegenbauer polynomials. The methods used in this report give closed-form expressions for the propagators of the first two, 1 and an integral expression for the last. 1.1 The Propagator A physical system is described completely by the initial conditions and the equations of motion. In a classical particle theory, solving a system means finding functions qi(t) and pi(t), the position and momentum, of all particles at any time. In a field theory, this changes to finding the field strength ui(x; t) and canonical momentum πi(x; t). In quantum mechanics, in the Schr¨odingerpicture, the solution to a system is described by a state j (t)i in Hilbert space, which in the position basis looks like a classical field, (x; t). However, because of the superposition principle, one can give a solution which is in a sense independent of the initial condition, such that the solution with a certain initial condition is given by the action of this general solution on the initial state. This general solution is the time-evolution operator. Let j (t)i : R !H denote a quantum state at time t. Then the Time- evolution operator is the unitary one-parameter group U(τ) that satisfies j (t + τ)i = U(τ)j (t)i; (1.1) that is, the operator U(τ) evolves a state for the time τ. For a Quantum me- chanical system with a time-independent Hamiltonian H, the time evolution operator is given by −i U(τ) = exp τH : (1.2) ~ An informal way to see this is to Maclauraint expand j i in t 1 X τ n dnj (t)i j (t + τ)i = (1.3) n! dtn n=0 1 n X 1 d = τ j (t)i (1.5) n! dt n=0 and applying the Schr¨odingerequation d i j i = Hj i (1.6) ~dt we obtain 1 n! X 1 −i j (t + τ)i = τH j (t)i: (1.7) n! n=0 ~ 2 Using definition of the exponential of an operator, we get back our expression for U. In the position basis fjxigx2Rn , we can describe the state by a wave func- tion (x; t) = hxj (t)i. By rewriting the identity operator as R jyihyjdy, Rn we can write Z Z j (t)i = U(t)j (0)i = U(t)jyihyj (0)idy = U(t)jyi (y; 0)dy Rn Rn Z =) (x; t) = hxj (t)i = hxjU(t)jyi (y; 0)dy: (1.8) Rn We define K(x; y; τ) = hxjU(τ)jyi (1.9) to be the propagator and we thus have Z (x; t) = K(x; y; t) (y; 0)dy: (1.10) Rn The propagator and the time-evolution operator both completely describe the behaviour of a system, but one is represented as an operator and one as a function. One can intuitively think of the propagator K(x; y; t) as the probability amplitude for a particle to travel from x to y in time t. One can recover the time-evolution operator from the propagator by Z U(t) = K(x; y; t)jxihyjdxdy: (1.11) Rn×Rn Formally, the Hamiltonian operator H can be spectrally decomposed together with the time-evolution operator using a projection-valued measure dP (E), such that Z − it E U(t) = e ~ dP (E); (1.12) σ(H) where σ(H) is the spectrum of H. This translates to integration over contin- uous parts of the spectrum and summing over discrete parts. In particular, if H has a pure-point spectrum σ (meaning no continuous part), the above expression of U(t) is it X − En e ~ jnihnj (1.13) n2σ 3 where jni is an energy eigenstate and En is the correspoding energy eigen- value. Thus the propagator obtains the simple form −it X En ∗ K(x; y; t) = e ~ n(x) n(y): (1.14) n2σ where n(x) = hxjni. The above form of the propagator is very useful when studying systems that are similar to the harmonic oscillator. If the spectrum has a continuous part equation 1.14 will include an in- tegral over the continuous set of eigenstates. This will be the case for the free particle propagator where the spectrum is R. It is worth looking at the propagator of the free particle. 1.2 Free Particle Propagator The Hamiltonian for a free particle in dimension n is nothing but a scaled 2 −~ Laplacian H = 2m ∆. To get the propagator, consider the one-dimensional 2 2 −~ @ ikx case where H = 2m @x2 . Eigenfunctions are given by k(x) = e with 2 2 ~ k eigenvalues Ek = 2m . This means that the Hamiltonian and time-evolution operator are diagonal in momentum space, such that Z 2 −i~k t U = e 2m jkihkjdk: (1.15) R Thus, using equation 1.9 and the proper normalization of momentum eigen- states jki such that hkjk0i = δ(k − k0), we get Z 1 −i~t k2+i(x−y)k K(x; y; t) = e 2m dk: (1.16) 2π R To compute the integral, we use a Wick transformation, setting ρ = it and rewriting 2 m(x−y)2 Z ρ im(x−y) 1 − − ~ k− K(x; y; t) = e 2ρ~ e 2m ρ~ dk: (1.17) 2π R The integral is now just a shifted Gaussian, which can be calculated for Re(ρ) > 0. Computing the integral and then taking the limit that the real part vanishes so that we get back t, the result is r m(x−y)2 r m m − m (x−y)2 K(x; y; t) = e 2ρ~ = e 2i~t (1.18) 2πρ~ 2πi~t 4 which is the analytical closed form expression for the propagator of the one- dimensional free particle.
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