Quantum Mechanical Propagators Related to Classical Orthogonal Polynomials

DEGREE PROJECT IN TECHNOLOGY, FIRST CYCLE, 15 CREDITS STOCKHOLM, SWEDEN 2021

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 förpropaga- torn, dvs. integrationskärnanförtidsutvecklingsoperatorn, härleds.I fallet med Hermitepolynom ärsystemet den harmoniska oscillatorn, medan det för Laguerre- och Gegenbauerpolynom ärekvivalent med tv˚apartikel-Calogero- Sutherland-system. Abstract

A few quantum systems on the line with weighted classical orthogonal polynomials 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 quantum 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 Schrödingerequation. 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 transformed 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 Schrödingerpicture, the solution to a system is described by a state |ψ(t)i in Hilbert space, which in the position basis looks like a classical field, ψ(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 |ψ(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 |ψ(t + τ)i = U(τ)|ψ(t)i, (1.1) that is, the operator U(τ) evolves a state for the time τ. For a Quantum mechanical 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 |ψi in t ∞ X τ n dn|ψ(t)i |ψ(t + τ)i = (1.3) n! dtn n=0 ∞ n X 1 d = τ |ψ(t)i (1.5) n! dt n=0 and applying the Schrödingerequation d i |ψi = H|ψi (1.6) ~dt we obtain ∞ n! X 1 −i |ψ(t + τ)i = τH |ψ(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 {|xi}x∈Rn , we can describe the state by a wave function ψ(x, t) = hx|ψ(t)i. By rewriting the identity operator as R |yihy|dy, Rn we can write Z Z |ψ(t)i = U(t)|ψ(0)i = U(t)|yihy|ψ(0)idy = U(t)|yiψ(y, 0)dy Rn Rn Z =⇒ ψ(x, t) = hx|ψ(t)i = hx|U(t)|yiψ(y, 0)dy. (1.8) Rn We deﬁne

K(x, y; τ) = hx|U(τ)|yi (1.9) to be the propagator and we thus have Z ψ(x, t) = K(x, y; t)ψ(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)|xihy|dxdy. (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 continuous 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 ~ |nihn| (1.13) n∈σ

3 where |ni 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 ~ ψn(x)ψn(y). (1.14) n∈σ where ψn(x) = hx|ni. 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 integral 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 ψk(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 |kihk|dk. (1.15) R Thus, using equation 1.9 and the proper normalization of momentum eigenstates |ki such that hk|k0i = 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. To get the propagator for a free particle in higher dimensions, one could just generalize the argument above. However, one can also use a trick which turns out to be very useful in other cases as well. Consider a time-independent Hamiltonian which is a sum of commuting P operators H = n Hn. Then the time-evolution operator is simply U(t) = Q n Un(t). In particular, if the full Hamiltonian consists of a sum of the same Hamiltonian acting on different variables xi, as in this case, the full Qn propagator Kn satisfies the relation that Kn(x, y; t) = i K(xi, yi; t), which can be easily seen from the definition of the propagator in terms of the time- evolution operator. The propagator for a free particle in dimension n is then just

n m 2 − m |x−y|2 K(x, y; t) = e 2i~t . (1.19) 2πi~t Figure 1.1 shows a color plot of the real part of the mathematical function 2 1 − (x−y) K(x, y; t) = √ e 2it corresponding to the propagator in appropriate 2πt units. The plot can be thought of as a matrix ﬂipped along the y-axis. The plot is at diﬀerent times t and in the region (−10, 10) on both x and y. At t ≈ 0, the propagator approaches the identity, as K(x, y; 0) = δ(x − y). For larger t, the propagator is slowly ”decaying” to lower and lower frequencies.

Figure 1.1: Image of the real part of the free particle propagator K(x, y; t). Yellow indicates positive and blue indicates negative values.

5 1.3 Classical Orthogonal Polynomials

The classical orthogonal polynomials are the Hermite, Laguerre and Jacobi polynomials. Many other orthogonal polynomials turn out to be special cases of these. The Laguerre polynomials can be generalized to the generalized Laguerre polynomials which include Hermite polynomials as a special case. Jacobi polynomials are denoted P (α,β) and an important special case is α = β, when the polynomials are called Gegenbauer polynomials. The classical orthogonal polynomials have in common a wide range of applications and possess many interesting and useful properties. Each family of the classical orthogonal polynomials is a set of real polynomials {Pn}n∈N that satisfy

Z 2 Pn(x)Pm(x)w(x)dx = δnmCn (1.20) X for X ⊆ R, non-zero real constants Cn and a real function w(x), called the weight. The eigenstates in quantum mechanics are orthogonal to the standard L2-norm, and thus the eigenstates of the systems related to these polynomials have one-dimensional wave-functions on the form

1 p ψn(x) = w(x)Pn(x). (1.21) Cn Using the expression 1.14, the kernel for such a model obtains the form

it − En p X e ~ K(x, y; t) = w(x)w(y) 2 Pn(x)Pn(y). (1.22) Cn n∈N In the case of real Re(it) > 0 the sum behaves much nicer as it is exponen- tially decaying. Therefore, a Wick transformation is often useful, and the sum can be calculated using the limit that the real part vanishes. The Wick transformed version of the above expression is a famous expression for the heat kernel. For instance, in the case of Jacobi polynomials (α,β) Pn = Pn , it is sometimes called the ultra-spherical heat kernel or Jacobi kernel. The rest of this report will look at the expression for the propagator of the quantum systems related to the orthogonal polynomials, and (α) studying the Hermite (Hn), Laguerre and generalized Laguerre (Ln ) and Gegenbauer(Jacobi with α = β) kernels.

6 Chapter 2

Harmonic Oscillator - Mehler Kernel

The quantum system related to Hermite polynomials is the one-dimensional Harmonic Oscillator. The harmonic oscillator has the property that the higher dimensional Hamiltonians are just a sum of n one-dimensional Hamil- tonians. Thus, we look at the one-dimensional case

2 ∂2 1 H = − ~ + mω2x2. (2.1) 2m ∂x2 2 To make the notation cleaner, natural units for length and energy is chosen such that the Hamiltonian obtains the form 1 ∂2 1 H = − + x2, (2.2) 2 ∂x2 2 in terms of the now dimensionless quantity x. The eigenfunctions and spectrum can be found using the famous method of creation and annihilation operators. The solution is

1 − 1 x2 1 ψn(x) = √ e 2 Hn(x),En = n + (2.3) p π2nn! 2 where the ψn are normalized. The expression for the propagator is

e−(x2+y2)/2 Xe−(n+1/2)it K(x, y; t) = √ H (x)H (y), (2.4) π 2nn! n n n≥0 and computing this sum, one obtains the Mehler kernel.

7 2.1 Mehler’s formula

To evaluate the sum, we will use Mehler’s formula [4, Eq. 18.18.28]. Using a Wick transformation ρ = e−it the propagator becomes

−it/2 n e 2 2 X(ρ/2) √ e−(x +y )/2 H (x)H (y). (2.5) π n! n n n≥0 We use Mehler’s formula, n X (ρ/2) 1 ρ2(x2 + y2) − 2ρxy Hn(x)Hn(y) = exp − (2.6) n! p 2 1 − ρ2 n≥0 1 − ρ to obtain 1 r ρ (1 + ρ2)(x2 + y2) − 2ρxy K(x, y; t) = √ exp − . (2.7) π 1 − ρ2 2(1 − ρ2) Indentifying the expressions for trigonometric functions, e−it 1 1 csc t = = = (2.8) 1 − e−2it eit − e−it 2i sin t 2i 1 + e−2it eit + e−it i cos t = = = i cot t (2.9) 1 − e−2it eit − e−it sin t we get the ﬁnal expression for the propagator, rcsc t K(x, y; t) = exp −i (x2 + y2)/2 cot t + xy csc t. (2.10) 2iπ This propagator is called the Mehler kernel, which is the propagator of the harmonic oscillator in natural units.

2.2 A proof of Mehler’s formula

P (ρ/2)n Let us denote the expression n≥0 n! Hn(x)Hn(y) by E(x, y). We use the Rodrigues formula [4, Eq. 18.5.5] for Hermite polynomials, Hn(x) = x2 d n −x2 e dx e , on E, to obtain n n n X (ρ/2) 2 d 2 2 d 2 E(x, y) = ex e−x ey e−y n! dx dy n≥0 2 2 d d 2 2 = ex +y exp ρ/2 e−(x +y ) (2.11) dx dy

8 or equivalently, d d 2 2 2 2 exp ρ/2 e−(x +y ) = E(x, y)e−(x +y ). (2.12) dx dy

Fourier transforming the left hand side of the above equation from x and y to u and v... ρ d d 2 2 F exp e−(x +y ) (u, v) 2 dx dy   Z ∞ Z ∞ n X 1 ρ d d 2 2 = e−(x +y ) e−ixu−iyvdxdy  n! 2 dy dx  −∞ −∞ n≥0   Z ∞ Z ∞ 2 2 X 1 ρ n = e−(x +y ) − uv e−ixue−iyvdxdy  n! 2  −∞ −∞ n≥0 2 2 = πe−(u +v +2ρuv)/4 (2.13)

...and then back again...

h 2 2 i F −1 πe−(u +v +2ρuv)/4 (x, y) Z ∞ Z ∞ 1 −(u2+v2+2ρuv)/4+iux+ivy = 2 πe dudv 4π −∞ −∞ Z ∞ Z ∞ 1 −v2/4+ivy − 1 (u2+2(ρv−2ix)u) = e e 4 dudv 4π −∞ −∞ ∞ 1 Z 2 2 2 2 = √ e−v /4+ivy+ρ v /4−x −iρvxdv 2 π −∞ ∞ 2 1 Z − 1−ρ v2+2 2i(y−ρx) v −x2 = √ e 4 1−ρ2 dv 2 π −∞ 2 2 1 − y +x −2ρxy = e 1−ρ2 (2.14) p1 − ρ2

... and ﬁnally multiplying by ex2+y2 , we get

1 ρ2(x2 + y2) − 2ρxy E(x, y) = exp − . (2.15) pρ2 − 1 1 − ρ2

9 2.3 Properties of the Harmonic Oscillator Propa- gator

Figure 2.1 shows a color plot of the real part propagator, thought of as a matrix, at diﬀerent times t and in the region (−10, 10) on both x and y. The imaginary part looks very similar to the real part as the absolute value is independent of x and y. One can see that for t ≈ 0 the propagator appears to approximate the identity. Although the absolute value is constant over the image, the rapid oscillations away from the diagonal makes the inner product with respect to a continuous function to approach 0. For t > π, the motion reverses to complete a full cycle at t = 2π.

Figure 2.1: Image of the real part of the Mehler kernel K(x, y; t). Yel- low indicates positive and blue indicates negative value. All quantities are dimensionless.

10 Chapter 3

Rational Calogero-Sutherland Propagator - Laguerre Kernel

The first system of Calogero type that was found is the rational Calogero- Sutherland system. This system describes particles interacting with an inverse squared (centrifugal) potential, and a harmonic interaction can be added without effecting the exactly solvable property. The particle number needs to be at least 2 to get any interesting behaviour, but because the center of mass is not effected by the interaction potential one degree of freedom can be extracted effectively giving a one-particle system with an inverse-squared potential and optionally a harmonic potential. This system is related to the Laguerre polynomials and the propagator can be obtained by the use of the Laguerre kernel, but it can also be solved by using a transformation on a two-dimensional single-particle harmonic oscillator.

3.1 The Laguerre Kernel

The Hamiltonian diﬀerential operator is, in natural units, ! 1 ∂2 α2 − 1 H = − + x2 + 4 . (3.1) 2 ∂x2 x2

11 Because of the singularity at x = 0, it is convenient to restrict the space to x > 0. However, the problem and solutions can be extended to the negative half-line [5]. The solution consists of normalized eigenfunctions with eigenvalues s 2n! 2 ψ = e−x /2xα+1/2L(α)(x2),E = 2n + α + 1 (3.2) n Γ(n + α + 1) n n that are normalized on the positive half-line. The propagator is

−(2n+α+1)it X 2n!e − 1 x2 α+ 1 (α) 2 − 1 y2 α+ 1 (α) 2 K(x, y; t) = e 2 x 2 L (x )e 2 y 2 L (y ) Γ(n + α + 1) n n n≥0 −2nit − 1 (x2+y2) α+ 1 −(α+1)it X n!e (α) 2 (α) 2 = 2e 2 (xy) 2 e L (x )L (y ). Γ(n + α + 1) n n n≥0 (3.3)

To simplify this, we utilize the Hardy-Hille formula, a generalization of Mehler’s formula to Laguerre polynomials [4, Eq. 18.18.27]. The Hardy- Hille formula states exp − ρ (x + y) √ X n! 1−ρ 2 xyρ ρnL(α)(x)L(α)(y) = I Γ(n + α + 1) n n (xyρ)α/2(1 − ρ) α 1 − ρ n≥0 (3.4) where Iα is the modified Bessel function of the first kind. Using this formula, replacing x and y by x2 and y2 and setting ρ = e−2it, we can simplify the expression for the propagator. We get √ √ √ ρ (1 + ρ)(x2 + y2) ρ K (x, y; t) = 2 xy exp − I 2xy α 1 − ρ 2(1 − ρ) α 1 − ρ √ xy csc t i = exp − (x2 + y2) cot t I (−ixy csc t) (3.5) i 2 α when α ± 1/2, the x−2 term in the differential operator vanishes, and we expect the system to behave like a harmonic oscillator. We use the fact that

12 x −x x −x I (x) = e√−e ,I (x) = e√+e in the formula and we get 1/2 2πx −1/2 2πx

r 2 2 csc t − i (x2+y2) cot t−ixy csc t −i x +y cot t+ixy csc t K (x, y; t) = e 2 − e 2 , 1/2 2iπ (3.6)

r 2 2 csc t − i (x2+y2) cot t−ixy csc t −i x +y cot t+ixy csc t K (x, y; t) = e 2 + e 2 . −1/2 2iπ (3.7) Both of these expressions look like a superposition of two particles in a Har- monic oscillator, with center of mass at x = 0, a remnant from the fact that this one-particle system is a transformed version of two Calogero-Sutherland particles. However, only the α = −1/2 can be formally interpreted as a harmonic oscillator. The other case, α = 1/2, gives rise to a paradox, as explained in [6].

3.2 From a particle in a Two-dimensional Har- monic Oscillator

To understand the intuition behind the following derivation, let us start by considering the classical version of the model.

3.2.1 Classical Mechanics Consider the Harmonic Oscillator Lagrangian in two dimension polar coordinates mv2 kr2 m kr2 L = − = r˙2 + r2θ˙2 − . (3.8) 2 2 2 2 Notice that ∂L = 0 (3.9) ∂θ which by Noether’s theorem implies the quantity

∂L 2 ˙ pθ = = mr θ (3.10) ∂θ˙ is conserved. The radial momentum is ∂L p = = mr˙ (3.11) r ∂r˙

13 This is the angular momentum L. This fact is equivalent to the content of the Euler-Lagrange equations in the variable θ. Now, the Hamiltonian for the system 3.8 is

m kr2 H =p ¯ · r¯ − L = r˙2 + r2θ˙2 + (3.12) 2 2 and substituting θ˙ in the Hamiltonian, one obtains

mp2 L2 kr2 H = r + + . (3.13) 2m 2r2 2 which is the Hamiltonian for a one dimensional potential, and the equations of motion can be obtained from Hamilton’s equations. This is related to the classical Calogero-Moser system.

3.2.2 Quantum Free Particle System We ﬁrst consider a free particle in two dimensions before adding the harmonic potential. In natural units, the Hamiltonian is given by

P 2 p2 + p2 1 ∂2 ∂2 H = = x y = − + . (3.14) 2 2 2 ∂x2 ∂y2

We introduce the variables r and θ, such that x = r cos θ and y = r sin θ. In this basis, the Hamiltonian is

1 1 ∂ ∂ 1 ∂2 H = − r + , (3.15) 2 r ∂r ∂r r2 ∂θ2 and the propagator,

0 0 1 − 1 r2+r02−2rr0 cos(θ−θ0) K(r, θ − θ , t; r , 0) = e 2it [ ]. (3.16) 2iπt Notice that the angular dependency of the propagator only depends on θ−θ0. The operator ∂ L = −i = xp − yp (3.17) ~∂θ y x commutes with the Hamiltonian

[H,L] = 0 (3.18)

14 and is thus a symmetry of the system. This means that an eigenstate of L will remain an eigenstate after acted on by the time evolution operator. The eigenstates are ∂ Lφ(θ) = −i φ(θ) = αφ(θ) =⇒ φ(θ) = Ceiαθ (3.19) ∂θ and using φ(2nπ) = φ(0), we see that α ∈ Z. How will an eigenstate of the angular momentum operator L evolve with time? Using the propagator for the free particle, we obtain

∞ 2π Z Z 0 eiαθψ(r, t) = K(r, θ − θ0, t; r0, 0)eiαθ ψ(r, 0)r0dr0dθ0 0 0 ∞ 2π Z Z 0 =⇒ ψ(r, t) = r0 K(r, θ − θ0, t; r0, 0)eiα(θ −θ)dθ0 ψ(r, 0)dr0 0 0 Z ∞ 0 0 = Kα(r, t; r ; 0)ψ(r, 0)dr (3.20) 0 which, substituting θ − θ0 = ϕ and simplifying, becomes

Z 2π 0 0 0 −iαϕ Kα(r, t; r ; 0) = r K(r, ϕ, t; r , 0)e dϕ (3.21) 0 0 Z 2π 0 r − 1 r2+r02 1 rr cos ϕ−iαϕ = e 2it [ ] e it dϕ (3.22) it 2π 0 0 0 r − 1 r2+r02 α rr = e 2it [ ]i J − (3.23) it α t 0 0 r − 1 r2+r02 −irr = e 2it [ ]I (3.24) it α t where we in the last step have used Bessel’s integral. The function ψ(r, t) satisﬁes the equation

∂ψ(r, t) 1 1 ∂ ∂ψ(r, t) α2 i = − r − ψ(r, t) ∂t 2 r ∂r ∂r r2 1 ∂2ψ(r, t) 1 ∂ψ(r, t) 1 α2 = − + + ψ(r, t) (3.25) 2 ∂r2 r ∂r 2 r2 which can not be interpreted as an ordinary Schrödingerequation in one dimension. To get a Schrödingerequation corresponding to the propagator in eq 3.24, we need find a transformation to get rid of the first order derivative.

15 Using the ansatz ψ(r, t) = w(r)ξ(r, t) (3.26) equation 3.25 becomes ∂ξ(r, t) 1 ∂2ξ(r, t) iw(r) = − w(r) ∂t 2 ∂r2 1 1 ∂ξ(r, t) − 2w0(r) + w(r) 2 r ∂r 1 1 α2 − w00(r) + w0(r) − w(r) ξ(r, t) (3.27) 2 r r2 and now setting 1 C 2w0(r) + w(r) = 0 =⇒ w(r) = √ (3.28) r r the ﬁrst order derivatives vanish. The third term becomes V (r)w(r)ξ(r, t) where 1 3 1 1 1 α2 − 1 V (r) = − w(r) − − α2 = 4 . (3.29) 2 4 2 r2 2 r2 Finally, multiplying equation 3.27 the result is ∂ξ(r, t) 1 ∂2ξ(r, t) i = − + V (r)ξ(r, t) (3.30) ~ ∂t 2 ∂r2 which is the Schr¨odingerequation for a one dimensional particle. √1 We notice that the weight w(r) = r is precisely the function which transforms the induced inner product with weight W (r) = r, Z ∞ ∗ hψ|φiW = ψ (r)φ(r)W (r)dr, (3.31) 0 to the ordinary inner product Z ∞ ∗ hw(r)ψ|w(r)φiW = ψ (r)φ(r)dr = hψ|φi, (3.32) 0 − 1 because w(r) = W (r) 2 . It remains to transform the propagator Kα. Using equation 3.20, we obtain √ r 0 0 0 r 0 rr − 1 r2+r02 rr K(r, t; r , 0) = K (r, t; r , 0) = e 2it [ ]I (3.33) r0 α it α it which we now can interpret as the propagator for a one-dimensional particle 1 in a r2 potential.

16 3.2.3 Quantum Harmonic Oscillator We will repeat the argument above for the 2D harmonic oscillator. The Hamiltonian is 1 ∂2 ∂2 1 H = − + + (x2 + y2) (3.34) 2 ∂x2 ∂y2 2 which can be separated in to two independent Hamiltonians implying the propagator is a product of the one-dimensional case. The two-dimensional harmonic oscillator propagator in polar coordinates can be obtained from equation 2.10 and is

0 0 csc t −i (r2+r02) cot t+2rr0 cos (θ−θ0) csc t K(r, θ − θ , t; r , 0) = e 2 ( ) (3.35) 2iπ The Hamiltonian in polar coordinates is

1 1 ∂ ∂ 1 ∂2 1 H = − r + + r2. (3.36) 2 r ∂r ∂r r2 ∂θ2 2

We notice that [H,L] = 0, and that the inner product has the weight W (r) = √1 iαθ r. Applying the Schr¨odingerequation to r e ψ(r, t), we obtain

∂ψ(r, t) 1 ∂2ψ(r, t) 1 α2 − 1 1 i = − + 4 + r2. (3.37) ∂t 2 ∂r2 2 r2 2 And for the propagator, √ 0 0 rr csc t −i (r2+r02) cot t K(r, t; r , 0) = e 2 i 2π 1 Z 0 · e−irr csc t cos ϕ−iαϕdϕ (3.38) 2π √ 0 0 rr csc t −i (r2+r02) cot t α 0 = e 2 i Jα −rr csc t (3.39) √ i 0 rr csc t − i (r2+r02) cot t 0 = e 2 I −irr csc t (3.40) i α which is what we obtained in the previous section. This result also proves the Hardy-Hille formula for the special case where α is an integer.

17 3.3 Properties of the Propagator

Color plots of the real part of the propagator for the Laguerre system with α = 2 is shown in figure 3.1. Close to nπ, the propagator approaches δ(x−y). Comparing to the plot 2.1, we can see that this plot looks just like what the first quadrant of a superposition of two harmonic oscillators with a phase difference of π would look like, with one addition. Except for the familiar pattern of the harmonic oscillator, there are also quarter circles where the real part obtains the value zero(green), which are present except at π/2

Figure 3.1: Image of the real part of the propagator Kα(x, y; t) for α = 2. Yellow indicates positive and blue indicates negative values, while green indicates 0.

18 Chapter 4

Trigonometric Calogero-Sutherland Propagator - Gegenbauer Kernel

The two-particle trigonometric Calogero-Sutherland can be written in a one- particle form using the trigonometric P¨oschl-Teller potential

1 ∂2 1 α2 − 1 H = − + 4 . (4.1) 2 ∂x2 2 sin2 x The Schr¨odingerequation corresponding to this Hamiltonian has (not normalized) solutions given by

α+ 1 (α,α) ψn = (sin x) 2 Pn (cos x) (4.2)

1 2 (α,β) with energy eigenvalues (n+α+ 2 ) , where Pn (x) are Jacobi polynomials. This report is limited to the case of α = β. Then, the polynomials are called Gegenbauer polynomials. The system can however be generalized to α 6= β by adding a 1/ cos2-potential. The ﬁrst factor again comes from the weight of the inner product with which the polynomials are orthogonal, as the eigenstates are indeed orthogonal with respect to the standard inner product. Now, trying to compute the propagator kernel immediately from the eigenstate expansion leads to

19 ∞ α+ 1 X 1 −it(n+α+ 1 )2 (α,α) (α,α) K(x, y; t) = (sin x) 2 e 2 P (cos x)P (cos y) (4.3) C2 n n n=0 n where Cn is the normalization factor, which can be computed using the inner α+ 1 product. This is ((sin x) 2 times) the Gegenbauer kernel, or the bilinear generating function for Gegenbauer polynomials, a special case of the Jacobi kernel. In general, a closed form of this formula is not known, however for some special cases it can be calculated [7]. But there is no known formula corresponding to the Hardy-Hille formula or Mehler’s formula. Thus, we get no further, keeping the generality of α using this method. Let us now turn to the geometric method developed in the previous chapter.

4.1 From a Three Dimensional Free Particle

The analog of the method in the previous chapter would be to consider a free particle moving on the surface of a sphere. However, the propagator of a free particle on a sphere, or equivalently the spherical heat kernel. But we will attempt to let the particle move freely in ﬂat 3D space, and only restrict the movement to r = 1 after the azimuthal angle has been removed. Consider a free particle moving in three dimensional spherical coordinates. The Hamiltonian is simply the Laplacian in spherical coordinates which is given by

1 ∂2 2 ∂ 1 ∂2 cot θ ∂ 1 ∂2 H = − + + + + . (4.4) 2 ∂r2 r ∂r r2 ∂θ2 r2 ∂θ r2 sin2 θ ∂φ2 Now, consider a wave-function on the form

eiαφ Ψ(r, θ, φ, t) = √ ψ(θ, t) (4.5) r sin θ where α must be an integer because of periodicity of the variable φ. This precise expression can be derived using an ansatz w(r, θ) similar to the previous chapter. The Schr¨odingerequation for Ψ with the free particle Hamiltonian then reduces to the Schr¨odingerequation of ψ with Hamiltonian

" # 1 1 ∂2 1 α2 − 1 H0 = − + 4 . (4.6) r2 2 ∂θ2 2 sin2 θ

20 Thus, restricting the space to r = 1, we get our desired Hamiltonian. Next, we look at the propagator for Ψ. It is given by 3/2 0 1 − 1 r2+r2−2r r (cos(φ −φ ) sin θ sin θ +cos θ cos θ ) K(¯x, t;x ¯ , 0) = e 2it [ 1 2 1 2 1 2 1 2 1 2 ]. 2πit (4.7) The kernel equation for our Ψ becomes after some calculations

1 3/2 Z π Z 2π Z ∞ rr p 1 iα(φ2−φ1) ψ(θ1, t) = sin θ1 sin θ2 e 2πit 0 0 0 r2 1 2 2 − (r1+r2−2[cos(φ1−φ2) sin θ1 sin θ2+cos θ1 cos θ2]r1r2) 2 e 2it ψ(θ2, 0)r2dr2dφ2dθ2. (4.8) We can perform the φ integral. Using a version of the Bessel integral formula for (analytically continued to half-integer) spherical Bessel functions, Z 2π √ i(x cos φ+nφ) n n e dφ = 2πi Jn(x) = i 8πxjn+ 1 (x) (4.9) 0 2 we obtain iα+1/2 Z π ψ(θ1, t) = − 2 r1 sin θ1 sin θ2ψ(θ2, 0) πt 0 Z ∞ 1 2 2 r1 − 2it (r2−2r1 cos θ1 cos θ2r2+r1) 2 e jα+ 1 sin θ1 sin θ2r2 r2dr2 dθ2. 0 2 t (4.10) Thus, we have an integral of the form Z ∞ 2 iax(x−b) I(µ, a, b, c) = x jµ(cx)e dx, (4.11) 0 and our kernel can be written in terms of this integral according to

iα+1/2 K(θ , t; θ , 0) = − sin θ sin θ 1 2 πt2 1 2 − 1 r2 1 r1 r e 2it 1 I(α + 1/2, , 2r cos θ cos θ , − sin θ sin θ ). 1 2t 1 1 2 t 1 2 (4.12) A simple numerical study was performed comparing the result of the found expression with the truncated series of eigenstates, giving inconclusive result. A potential problem with this method is that Ψ does not have a ﬁnite 3 norm over R .

21 Chapter 5

Concluding Remarks

The propagator for Hamiltonians related to Hermite and Laguerre polynomials can be solved exactly using known expression for the kernel of these polynomials. There is no general closed form expression for the Gegenbauer or Jacobi kernel and the associated propagator with the inverse sine squared potential. However, an integral expression for the case of P (α,α) was found. This report presented a method to compute the two-particle Calogero- Sutherland propagator from a two-dimensional harmonic oscillator. During the making of this report an attempt was made to generalize this method to the n-particle rational Calogero-Sutherland system. In this general case, the two-dimensional harmonic oscillator is replaced by a Hermitian matrix satisfying the Harmonic Oscillator equations of motion, while the radial and angular part are replaced by the spectral decomposition into eigenvalues and a U(n)-matrix [3]. The method gave promising results. An expression for the propagator when α = 0 was found, but the general case was not completed.

22 Bibliography

[1] L. Infeld and T. E. Hull. The Factorization Method, Rev. Mod. Phys. 23, (1951)

[2] M. A. Olshanetsky and A. M. Perelomov. Quantum integrable systems related to lie algebras Phys. Rep. 94, (1983)

[3] A. P. Polychronakos. The physics and mathematics of Calogero particles J. Phys. A: Math. Gen. 39, 12793 (2006).

[4] NIST Digital Library of Mathematical Functions. http://dlmf.nist. gov/, Release 1.1.1 of 2021-03-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds.

[5] H. Miyazaki and I. Tsutsui. Quantum Tunneling and Caustics under Inverse Square Potential Ann. Phys. 299, 78 (2002)

[6] E. Langmann, A. Laptev and C. Pauﬂer. Singular factorizations, self- adjoint extensions, and applications to quantum many-body physics J. Phys. A: Math. Gen. 39, 1057 (2018)

[7] D. Andersson. Estimates of the Spherical and Ultraspherical Heat Ker- nel https://publications.lib.chalmers.se/records/fulltext/ 182086/182086.pdf (2013)

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