Path Integral for the Hydrogen Atom

Solutions in two and three dimensions

Vägintegral för Väteatomen Lösningar i två och tre dimensioner

Anders Svensson

Faculty of Health, Science and Technology Physics, Bachelor Degree Project 15 ECTS Credits Supervisor: Jürgen Fuchs Examiner: Marcus Berg June 2016

Abstract The path integral formulation of quantum mechanics generalizes the action principle of classical mechanics. The Feynman path integral is, roughly speaking, a sum over all possible paths that a particle can take between fixed endpoints, where each path contributes to the sum by a phase factor involving the action for the path. The resulting sum gives the probability amplitude of propagation between the two endpoints, a quantity called the propagator. Solutions of the Feynman path integral formula exist, however, only for a small number of simple systems, and modifications need to be made when dealing with more complicated systems involving singular potentials, including the Coulomb potential. We derive a generalized path integral formula, that can be used in these cases, for a quantity called the pseudo-propagator from which we obtain the fixed-energy amplitude, related to the propagator by a Fourier transform. The new path integral formula is then successfully solved for the Hydrogen atom in two and three dimensions, and we obtain integral representations for the fixed-energy amplitude.

Sammanfattning Vägintegral-formuleringen av kvantmekanik generaliserar minsta-verkanprincipen fr˚anklassisk meka- nik. Feynmans vägintegral kan ses som en summa över alla möjligavägaren partikel kan ta mellan tv˚a givna ändpunkterA och B, därvarje vägbidrar till summan med en fasfaktor inneh˚allandeden klas- siska verkan förvägen.Den resulterande summan ger propagatorn, sannolikhetsamplituden att partikeln g˚arfr˚anA till B. Feynmans vägintegral ärdock bara lösbarförett f˚atalsimpla system, och modifika- tioner behöver görasnärdet gällermer komplexa system vars potentialer inneh˚allersingulariteter, s˚asom Coulomb–potentialen. Vi härlederen generaliserad vägintegral-formel som kan användasi dessa fall, för en pseudo-propagator, fr˚anvilken vi erh˚aller fix-energi-amplituden som ärrelaterad till propagatorn via en Fourier-transform. Den nya vägintegral-formeln lösessedan med framg˚angförväteatomeni tv˚aoch tre dimensioner, och vi erh˚allerintegral-representationer förfix-energi-amplituden. Acknowledgements First of all I would like to thank my supervisor, Professor JürgenFuchs, for the interesting discussions and for helping me out with all of the hard questions. I would also like to thank my father, for teaching me elementary mathematics and physics in the beginning of my studies, as well as the rest of my family and friends for their great support over the years. Last, but not least, I must thank all of the great physicists including Leonard Susskind, Brian Greene, and Richard Feynman himself, for making me love physics and inspiring me to continue on to the next level. Contents

1 Introduction 1

2 Basic Concepts 3 2.1 Classical Mechanics ...... 3 2.2 Quantum Mechanics ...... 4

3 Propagators 7 3.1 The Propagator and its Properties ...... 7 3.2 The Retarded Propagator and Fixed-Energy Amplitude ...... 9

4 Path Integrals 13 4.1 The Short-time Propagator ...... 13 4.2 The Finite-time Propagator From the Short-time Propagator ...... 14 4.3 The Phase Space Path Integral ...... 14 4.4 The Conﬁguration Space Path Integral ...... 16

5 Finding a More Flexible Path Integral Formula 18 5.1 The Pseudo-propagator ...... 18 5.2 New Path Integral Formula: Phase Space ...... 22 5.3 New Path Integral Formula: Conﬁguration Space ...... 23

6 Exact Solution for the Hydrogen Atom 26 6.1 The Hydrogenic Path Integral in D Dimensions ...... 26 6.2 Solution for the Two-Dimensional H-atom ...... 27 6.3 Solution for the Three-Dimensional H-atom ...... 32

7 Conclusion 39

A Gaussian Integrals 40

B Exact Solutions for some Simple Path Integrals 43 B.1 The Free Particle ...... 43 B.2 The Harmonic Oscillator ...... 45

C Square-root Coordinates for the 3-D H-atom 52 1 Introduction

Developed by Richard Feynman in the 1940s, the path integral formulation of quantum mechanics generalizes the action principle of classical mechanics. In classical mechanics, extremizing the action functional S[x(t)] determines the unique path x(t) taken by a particle between two endpoints xa, xb. In quantum mechanics there is no such path describing the motion of the particle. Instead, the quantum particle has a probability amplitude for going from xa to xb. Feynman showed that this probability amplitude is obtained by summing i up phase factors exp S[x(t)] over each and every path connecting xa and xb. This sum is called the ~ Feynman path integral, written as

xb i D[x(t)] exp S[x(t)] . (1.1) ˆxa ~

This expression is to be viewed as a functional integral. While an ordinary integral xb dx f(x) sums up xa xb values of a function f(x) over all numbers x from xa to xb, a functional integral ´D[x(t)] F [x(t)] sums xa up values of a functional F [x(t)] over all functions x(t) with endpoints x(ta) = xa´ and x(tb) = xb. More explicitly, the Feynman path integral may be expressed in D dimensions as

xb DN/2 i m D D i D[x(t)] exp S[x(t)] = lim d xN−1 ··· d x1 exp S[x{x }(t)] (1.2) N→∞ i ˆxa ~ 2πi~δt ˆ ˆ ~

where m is the particle’s mass, δt = (tb − ta)/N and x{x1,...,xN−1}(t) is a piecewise linear path with values xk at the times tk = ta + kδt (k = 1,...,N − 1) as well as the endpoints xa and xb at the times ta and D tb, respectively. The integrals on the right hand side of (1.2) are understood to go over the whole of R .

It is important to understand that the resulting ”sum” is over all possible paths x{xi}(t) taking the values {xa, x1, x2,..., xb} at the times {ta, t1, t2, . . . , tb} – even those that are absurd from a classical viewpoint. In the limit N → ∞ we have tk − tk−1 → 0, but |xk − xk−1| will in general be large for an arbitrary such path, resulting in a highly discontinuous path. Only a small subset of paths will be continuous and differentiable. In general, it is hard to give the functional integral (1.1) a precise mathematical meaning. Accordingly, (1.1) should be viewed as a formal expression that needs to be supplemented by a proper prescription on how to evaluate it. In particular, it is possible to define (1.1) as a sum over the subset of continuous paths (see Glimm and Jaffe [5], chapter 3). For a standard form of the action, one can then show that the path integral resulting from this definition coincides with the right-hand side of (1.2) [5]. This means that the discontinuous paths do not contribute to the overall sum in the continuum limit. Consequently, when evaluating the path integral (1.2) one can make approximations such as |xk+1|/|xk| → 1 to first order in δt. In mathematics, the basic idea of the path integral can be traced back to the Wiener integral, introduced by Norbert Wiener for solving problems dealing with Brownian motion and diffusion. In physics, the idea was further developed by Paul Dirac in his 1933 paper [1], for the use of the Lagrangian in quantum mechanics. Inspired by Dirac’s idea, Feynman worked out the preliminaries in his 1942 doctoral thesis, before developing the complete formulation in 1948 [1]. The Feynman path integral has since become one of the most prominent tools in quantum mechanics and quantum field theory. Other areas of application include • quantum statistics, where the quantum mechanical partition function can be written as, or obtained from, a path integral in imaginary time; • polymer physics, where path integrals are useful for studying the statistical fluctuations of chains of molecules, modelled as random chains consisting of N links; and • financial markets, where the time dependence of prices of assets can be modelled by fluctuating paths. In physics, path integrals have found their main application in perturbative quantum field theory. In elementary quantum mechanics, however, the formulation has not had as much impact due to the difficulties in dealing with the resulting path integrals, with only a few standard problems having been solved analytically.

1 In particular, the path integral for the Hydrogen atom remained unsolved until Duru and Kleinert published their solution in 1979 [2]. The goal of this thesis is to provide an exact solution of the path integral for the Hydrogen atom, following the steps of Duru–Kleinert. Before doing so, we shall develop the necessary preliminaries, including a derivation of the path integral formalism. Moreover, due to the singular nature of the Coulomb potential, the corresponding path integral can be shown to diverge when written down in the original form [4], and hence a new, modified, path integral must be constructed. In the following Section we begin by reviewing some basic concepts from classical mechanics and quantum mechanics. In Section 3 we continue by studying the propagator and its related quantities, including the fixed-energy amplitude, which is related to the propagator by a Fourier transform. We then derive the basic path integral formulas in phase space and configuration space in Section 4, before deriving more flexible versions of these in Section 5 that can be applied to problems involving singular potentials. These new path integral formulas yield an auxiliary quantity known as the pseudo-propagator, from which the fixed-energy amplitude can be obtained. This modified formalism is then finally applied in Section 6 to the two- and three-dimensional Hydrogen atoms, for which we solve the corresponding modified path integral formulas in configuration space, thus obtaining integral representations for the fixed-energy amplitude.

2 2 Basic Concepts

This Section will serve as a review of the key ingredients from classical mechanics and quantum mechanics that are relevant to the subsequent sections.

2.1 Classical Mechanics

Throughout this thesis we will restrict our attention to a physical system consisting of a single spinless particle of mass m subjected to a time-independent potential V (x) in D dimensions. In the Lagrangian formulation of classical mechanics, the Lagrangian for this system is deﬁned by 1 L x, x˙ := mx˙ 2 − V (x) (2.1) 2 and the action functional by

t b S x(t); ta, tb := dt L x(t), x˙ (t) , (2.2) ˆta with x(t) an arbitrary differentiable path in configuration space, the D-dimensional space of points x. Let xcl(t) be the true classical path taken by the particle from the point xa at time ta, to the point xb at time tb. The principle of stationary action then states that the action functional for this path has a stationary value with respect to all infinitesimally neighbouring paths having the same endpoints. By extremizing the action with respect to all such neighbouring paths, we obtain the Euler-Lagrange equations of motion,

d ∂L ∂L − = 0. (2.3) dt ∂x˙ i ∂xi For the Lagrangian (2.1), these are nothing but Newton’s equation of motion

mx¨ = −∇V (x). (2.4)

The canonical momentum conjugate to the coordinate xi is generally deﬁned by

∂L p := , (2.5) i ∂x˙ i which for the Lagrangian (2.1) is nothing but the ordinary classical momentum p = mx˙ . In the Hamiltonian formulation of classical mechanics, the Hamiltonian is generally deﬁned by

X i H x, p := pix˙ − L x, x˙ (2.6) i and for the single particle,

p2 Hx, p = p · x˙ − L x, x˙ = + V (x), (2.7) 2m i.e. the total energy of the particle. The motion of the particle is in the Hamiltonian formulation described by a path (x(t), p(t)) in phase space, the 2D-dimensional space of points (x, p). We can write the action (2.2) in terms of the Hamiltonian (2.7) as

t b h i S x(t); ta, tb = dt p(t) · x˙ (t) − H x(t), p(t) (2.8) ˆta

3 where we have to remember that p(t) = mx˙ (t). We can also deﬁne a canonical action functional by

t b h i S x(t), p(t); ta, tb := dt p(t) · x˙ (t) − H x(t), p(t) (2.9) ˆta deﬁned for arbitrary paths in phase space. Thus in this expression we let x(t) and p(t) be completely independent, with no relation between p and x˙ . The Lagrangian action (2.2) and the canonical action (2.9) are related by

t 2 b p(t) − mx˙ (t) S x(t), p(t); ta, tb = S x(t); ta, tb − dt . (2.10) ˆta 2m The principle of stationary action also holds for the canonical action (2.9), except that there is no restriction on the endpoints of p(t). This leads to the Hamilton’s equations motion

∂H i ∂H p˙i = − i , x˙ = , (2.11) ∂x˙ ∂pi which are equivalent with the Euler-Lagrange equations (2.3) via (2.5) and (2.6).

2.2 Quantum Mechanics

Classical mechanics is deterministic, meaning that by knowing the position xa and momentum pa of the particle at some initial time ta, we can with certainty predict the position xb and momentum pb at any later time tb. We now turn to quantum mechanics. The motion of a quantum particle cannot be described by some classical path x(t). If by a measurement the particle is determined to be at a point xa at time ta, we can only know the probability of the particle to found at xb at time tb. Moreover, the position and momentum cannot be known simultaneously due to the Heisenberg uncertainty principle, which states that the product of the uncertainties in position and momentum is always greater than, or the order of, Planck’s constant ~.

Any general state of the particle is represented by a ket vector ψ in a Hilbert space over the complex numbers. Conversely, each non-zero vector of the Hilbert space corresponds to some state of the particle. Two nonzero vectors that are proportional to each other represent the same physical state, and thus we always assume state vectors to be of unit norm. For each ket-vector ψ , there exists a bra-vector ψ in 0 0 the dual vector space, such that ψ acting on a ket ψ gives the inner product ψ ψ of the kets ψ and 0 ψ .

The state vector corresponding to the particle being at position x is denoted by x . A general state ψ is a superposition of such position-states:

D ψ = d x ψ(x) x . (2.12) ˆ

Here

ψ(x) ≡ x ψ (2.13)

is called the wave function of the system, or the probability amplitude for ﬁnding the particle at x. The probability of ﬁnding the particle in a volume element d3x about x is given by |ψ(x)|2 d3x = ψ∗(x)ψ(x) d3x. Observables such as position, momentum and energy are in quantum mechanics represented by Hermitian operators on the Hilbert space. It is postulated that every such operator possesses a complete set of eigenvectors (or eigenkets), complete in the sense that any general state may be expressed as a superposition

4 of these. The eigenvalues constitute all possible outcomes for a measurement of the observable. For example,

the position-kets x are eigenkets of the position operator xˆ with eigenvalues x.

Similarly, the momentum operator pˆ has eigenkets p with eigenvalues p. The state p describes the particle having a well-deﬁned momentum given by the corresponding eigenvalue p. The momentum operator may be deﬁned in the position representation by

x pˆ ψ = −i~∇ x ψ (2.14) where ∇ is the gradient diﬀerential operator acting on the wave function. By writing down the eigenvalue equation

pˆ p = p p (2.15)

and acting from the left with x , we get the diﬀerential equation

−i~∇ x p = p x p (2.16) for which the solutions are the momentum eigenfunctions

i exp p · x x p = ~ , (2.17) (2π~)D/2 up to normalisation. The position- and momentum eigenkets x and p are not strictly members of the Hilbert space, and cannot be normalized to unity. Instead, they satisfy the normalisations

0 D 0 x x = δ (x − x ) (2.18) and

0 D 0 p p = δ (p − p ) (2.19)

D QD i i where δ (x − x0) ≡ i=1 δ(x − x0) and δ(x − x0) is the Dirac delta function. The Hamilton operator Hˆ is obtained by replacing x and p in (2.7) by the corresponding operators:

Hˆ := H(xˆ, pˆ). (2.20)

To ﬁnd the energy eigenkets and the energy eigenvalues, we write down the eigenvalue equation

Hˆ E = E E . (2.21) where E denotes an eigenket of Hˆ with eigenvalue E. This is known as the time-independent Schr¨odinger equation. In the position representation, it becomes

H(x, −i~∇) x E = E x E (2.22) or, using (2.7) and writing ψE(x) ≡ x E , we obtain

2 − ~ ∇2 + V (x) ψ (x) = E ψ (x), (2.23) 2m E E for which the solutions are the energy eigenfunctions with energy eigenvalues E. In general the space of eigenkets corresponding to a particular eigenvalue has a dimensionality greater than one, in which case the eigenvalue is said to be degenerate. The number α(E) of linearly independent eigenkets having eigenvalue E is called the degeneracy of the eigenvalue. An eigenvalue E is said to be non-degenerate if α(E) = 1. If

an eigenvalue E is degenerate, we may label its eigenkets by E, k with k = 1, . . . , α(E). Once a complete

set of orthonormal eigenkets of Hˆ has been found, we can expand any general state ψ as

α(E) X X ψ = E, k E, k ψ . (2.24) E k=1

The expansion coeﬃcient E, k ψ is the probability amplitude for ﬁnding the particle in the state E, k . Pα(E) 2 The probability that an energy measurement yields the value E is given by k=1 | E, k ψ | .

If we make an ordered list of all eigenkets E, k and relabel them by n with n = 1, 2,..., then n is an 0 0 eigenket of Hˆ with eigenvalue En. Note that, if there is degeneracy, then there will be n, n (n 6= n ) such that En = En0 . With this notation, we can write (2.24) as X ψ = n n ψ . (2.25) n

Quantum mechanics is deterministic in the sense that by knowing the state vector ψ, t0 at some time t0,

the state of the system ψ, t at any later time can be determined with certainty (provided we have not disturbed the system in any way, as happens e.g. in a measurement). The time evolution of the system is governed by the time-dependent Schr¨odingerequation

∂ Hˆ ψ, t = i ψ, t . (2.26) ~∂t In the position representation this becomes

2 ∂ − ~ ∇2 + V (x) ψ(x, t) = i ψ(x, t). (2.27) 2m ~∂t

If we know the state ψ, t0 at time t0, the time evolution can also be described by the equation ˆ ψ, t = U(t, t0) ψ, t0 , (2.28)

where the operator Uˆ(t, t0) is known as the time-evolution operator. For the time-independent Hamilto- nian (2.20) it is given by

i Uˆ(t, t0) = exp − Hˆ (t − t0) . (2.29) ~

6 3 Propagators

3.1 The Propagator and its Properties

Throughout this thesis, we will restrict our attention to quantum systems consisting of a single spinless particle of mass m, subjected to a time-independent potential V (x) in D dimensions. Thus we shall assume the Hamiltonian to be of the form p2 H(x, p) = + V (x) (3.1) 2m with the corresponding operator (2.20). The time evolution operator is then given by i Uˆ(t, t0) = exp − Hˆ (t − t0) . (3.2) ~ We deﬁne the propagator or time evolution amplitude of such a system by ˆ K(x, t; x0, t0) := x U(t, t0) x0 . (3.3) We interpret this quantity as the probability amplitude for the particle to be found at the point x at time t, given that it was known to be at the point x0 at time t0. By ﬁxing x0, t0 and viewing x, t as variables, the propagator is simply the wave function ψ(x, t) of the particle, valid for times t ≥ t0, given that the particle

was in the state x0 at time t0. We now show that the propagator not only determines the wave function for a particle starting in a state

x0 , but for any general state ψ, t0 . For t ≥ t0, the state of the particle is determined by applying the time-evolution operator: ˆ ψ; t = U(t, t0) ψ, t0 . (3.4)

The wave function corresponding to the state ψ; t may then be written as

ˆ ˆ D 0 0 0 ψ(x, t) = x ψ; t = x U(t, t0) ψ, t0 = x U(t, t0) d x x x ψ, t0 ˆ D 0 ˆ 0 0 = d x x U(t, t0) x ψ(x , t0). (3.5) ˆ 0 This shows that by knowing the propagator K(x, t; x , t0) and the wave function ψ(x, t0) at time t0, the wave function for times t ≥ t0 is determined from

ψ(x, t) = dDx0 K(x, t; x0, t ) ψ(x0, t ). (3.6) ˆ 0 0

Setting t = t0 in this equation suggests that the propagator for t = t0 serves as a Dirac delta function: 0 D 0 K(x, t0; x , t0) = δ (x − x ). (3.7)

Indeed, since Uˆ(t0, t0) = 1 it follows that 0 ˆ 0 0 D 0 K(x, t0; x , t0) = x U(t0, t0) x = x x = δ (x − x ). (3.8) Furthermore, using the basic property of the Dirac delta function as well as the unitarity of the time evolution operator, the calculation

D 0 D 0 D 0 0 D 0 0 ˆ † ˆ 1 = d x δ (x − x0) = d x x x0 = d x x U (t, t0)U(t, t0) x0 ˆ 0 0 ˆ 0 0 ˆ 0 0 D 0 D 0 ˆ † ˆ = d x d x x U (t, t0) x x U(t, t0) x0 ˆ 0 ˆ 0 D 0 D ˆ 0 ∗ ˆ = d x d x x U(t, t0) x x U(t, t0) x0 (3.9) ˆ 0 ˆ 0

7 shows that the propagator satisﬁes the normalisation condition

dDx0 dDx K∗(x, t; x0 , t ) K(x, t; x , t ) = 1 ∀ x , (3.10) ˆ 0 ˆ 0 0 0 0 0 ∗ valid for each starting point x0, with K denoting the complex conjugate of K.

For a general quantum state ψ, t , the wave function ψ(x, t) satisﬁes the Schr¨odingerequation (2.27). Since the propagator itself is a perfectly good wave function, it must satisfy this equation also: 2 ∂ − ~ ∇2 + V (x) K(x, t; x , t ) = i K(x, t; x , t ). (3.11) 2m x 0 0 ~∂t 0 0

Suppose we have a complete set of orthonormal energy eigenkets n (n = 1, 2,...) with corresponding energy eigenvalues En, where we allow for degeneracy. Using the completeness of this set, the propagator can be expanded as ˆ X i ˆ K(xb, tb; xa, ta) = xb U(tb, ta) xa = xb exp − H(tb − ta) n n xa n ~ X i = xb n n xa exp − En(tb − ta) . (3.12) n ~

Thus, by knowing a complete set of normalised energy eigenfunctions ψn(x) ≡ x n with energy eigenvalues En, the propagator can be determined from X ∗ i K(xb, tb; xa, ta) = ψn(xb)ψn(xa) exp − En(tb − ta) , (3.13) n ~ called the spectral representation of the propagator. Conversely, if we know the propagator and can write it in the form (3.13), we can extract the energy eigenfunctions and the energy eigenvalues [4]. Since the trace of the time evolution operator is given by

ˆ D ˆ Tr U(t, t0) = d x x U(t, t0) x (3.14) ˆ it can be obtained from the propagator by setting xa = xb and integrating:

Tr Uˆ(t, t ) = dDx K(x, t; x, t ). (3.15) 0 ˆ 0 Using the expansion (3.13), the trace can be expressed as X i Tr Uˆ(t, t ) = dDx K(x, t; x, t ) = dDx ψ (x)ψ∗ (x) exp − E (t − t ) 0 ˆ 0 ˆ n n n 0 n ~ X i X i = exp − E (t − t ) dDx ψ (x)ψ∗ (x) = exp − E (t − t ) , (3.16) n 0 ˆ n n n 0 n ~ n ~ which is simply the sum of eigenvalues of Uˆ(t, t0), in agreement with a basic fact from linear algebra.

The Fourier transform of Tr Uˆ(t, t0) with respect to ∆t ≡ t − t0 is given by +∞ i F Tr Uˆ(∆t, 0) (E) = d(∆t) exp E∆t Tr Uˆ(∆t, 0) ˆ−∞ ~ +∞ i X i = d(∆t) exp E∆t exp − E ∆t ˆ n −∞ ~ n ~ X +∞ = d∆t exp [i(E − E )∆t] . (3.17) ~ ˆ n n −∞

8 Using the formula

+∞ 0 0 dt exp [i(E − En)t ] = 2πδ(E − En) (3.18) ˆ−∞ we then have ˆ X F Tr U(∆t, 0) (E) = 2π~ δ(E − En). (3.19) n However, due to the delta functions, this result is not that useful. In the following subsection we will ﬁnd an improved version of this formula.

3.2 The Retarded Propagator and Fixed-Energy Amplitude

In calculations involving the propagator (3.3), we will always consider t ≥ t0. In what follows, it will be convenient to take the propagator to be zero for times t < t0. By making use of the Heaviside step function, defined as 0 t < 0 Θ(t) := (3.20) 1 t ≥ 0 we first define a retarded time evolution operator by

UˆR(t, t0) := Θ(t − t0)Uˆ(t, t0), (3.21) as well as a retarded Hamiltonian by

HR(x, p; t) := Θ(t − t0)H(x, p), (3.22) with the corresponding operator

HˆR := HR(xˆ, pˆ; t) = Θ(t − t0)H.ˆ (3.23)

We then define a retarded propagator by ˆ KR(x, t; x0, t0) := x UR(t, t0) x0 = Θ(t − t0)K(x, t; x0, t0). (3.24) Recalling that the propagator satisfies the Schrödingerequation (3.11), we can derive an analogous Schrödinger equation satisfied by the retarded propagator. By taking the standard viewpoint of the Dirac delta function as the ”derivative” of the Heaviside step function, we have ∂ ∂ i K (x, t; x , t ) = i Θ(t − t )K(x, t; x , t ) ~∂t R 0 0 ~∂t 0 0 0 d ∂ = i Θ(t − t ) K(x, t; x , t ) + i Θ(t − t ) K(x, t; x , t ) ~ dt 0 0 0 ~ 0 ∂t 0 0

= i~δ(t − t0)K(x, t; x0, t0) + Θ(t − t0)H(−i~∇, x)K(x, t; x0, t0) = i~δ(t − t0)K(x, t0; x0, t0) + Θ(t − t0)H(−i~∇, x)Θ(t − t0)K(x, t; x0, t0) D = i~δ(t − t0)δ (x − x0) + HR(−i~∇, x; t)KR(x, t; x0, t0). (3.25)

In the third line we have used (3.11), in the fourth line the fact that Θ(t − t0) = Θ(t − t0)Θ(t − t0), and in the fifth line the result (3.7). Thus the retarded propagator satisfies the Schrödingerequation ∂ H (−i ∇, x; t)K (x, t; x , t ) = i K (x, t; x , t ) − i δ(t − t )δD(x − x ). (3.26) R ~ R 0 0 ~∂t R 0 0 ~ 0 0

9 We now make use of the result [4] that for a function f(t) that vanishes for t < 0, the Fourier transform

+∞ i f˜(E) = dt exp Et f(t) (3.27) ˆ−∞ ~ is an analytic function in the upper half of the complex plane, and the inverse transform correctly gives

1 +∞ i dE exp − Et f˜(E) = f(t) ∀ t. (3.28) 2π~ ˆ−∞ ~ In particular, the retarded propagator (3.24), i ˆ KR(x, t; x0, t0) = Θ(t − t0) x exp − H(t − t0) x0 = KR(x, ∆t; x0, 0), (3.29) ~

depends only on ∆t ≡ t − t0 (for given xb, xa) and vanishes for ∆t < 0.

We then deﬁne the ﬁxed energy amplitude K˜ (x, x0; E) as the Fourier transform of KR(x, ∆t; x0, 0) with respect to ∆t, i.e.

+∞ i ∞ i K˜ (x, x0; E) := d(∆t) exp E∆t KR(x, ∆t; x0, 0) = d(∆t) exp E∆t K(x, ∆t; x0, 0). (3.30) ˆ−∞ ~ ˆ0 ~ The inverse transform is given by

1 +∞ i KR(x, t; x0, t0) = dE exp − E(t − t0) K˜ (xb, xa; E). (3.31) 2π~ ˆ−∞ ~ Obviously, the ﬁxed energy amplitude contains as much information as the retarded propagator.

Analogously, we deﬁne a resolvent operator Rˆ(E) as the Fourier transform of UˆR(t, t0) = UˆR(∆t, 0) with respect to ∆t:

+∞ i ∞ i Rˆ(E) := d(∆t) exp E∆t UˆR(∆t, 0) = d(∆t) exp E∆t Uˆ(∆t, 0). (3.32) ˆ−∞ ~ ˆ0 ~ Then its matrix elements in the position basis are +∞ 0 ˆ i 0 ˆ x R(E) x = d(∆t) exp E∆t x UR(∆t, 0) x ˆ−∞ ~ +∞ i 0 = d(∆t) exp E∆t KR(x , ∆t; x, 0), (3.33) ˆ−∞ ~ which is nothing but the ﬁxed energy amplitude:

0 0 K˜ (x , x; E) = x Rˆ(E) x . (3.34) Using the expansion (3.13) of the propagator in the energy eigenfunctions, we have ∞ 0 0 i 0 x Rˆ(E) x = K˜ (x , x; E) = d(∆t) exp E∆t K(x , ∆t; x, 0) ˆ0 ~ ∞ i X i = d(∆t) exp E∆t ψ (x0)ψ∗ (x) exp − E ∆t ˆ n n n 0 ~ n ~ ∞ X 0 i = x n n x dt exp (E − En)t . (3.35) ˆ n 0 ~

10 The integral over t in this expression is not convergent as it stands. To make it convergent, we instead evaluate it by replacing E with E + iη where η > 0 is inﬁnitesimal, eventually to be set to zero in all expressions for which this makes sense. Then

∞ ∞ " i # i exp (E + iη − En)t i ~ ~ dt exp (E + iη − En)t = i = , (3.36) ˆ0 (E + iη − En) E − En + iη ~ ~ 0 and (3.35) becomes

0 X 0 i~ X 0 i~ 0 i~ x Rˆ(E) x = x n n x = x n n x = x x . (3.37) E − E + iη ˆ ˆ n n n E − H + iη E − H + iη Since this holds for all x0, x we conclude that the resolvent operator is given by i Rˆ(E) = ~ (η inﬁnitesimal). (3.38) E − Hˆ + iη

This shows in particular that the expression on the right-hand side, with Hˆ in the denominator, makes sense. The calculation (3.37) also shows that

0 0 X 0 i~ K˜ (x , x; E) = x Rˆ(E) x = x n n x (3.39) E − E + iη n n and thus the ﬁxed energy amplitude can be expanded in the energy eigenfunctions as

X i K˜ (x0, x; E) = ψ (x0)ψ∗ (x) ~ (η infinitesimal). (3.40) n n E − E + iη n n This is called the spectral representation of the fixed energy amplitude. Knowing the fixed-energy amplitude, we can extract the energy eigenfunctions and energy eigenvalues from spectral analysis [4].

Since the trace of UˆR is given by

ˆ D ˆ Tr UR(t, t0) = d x x UR(t, t0) x (3.41) ˆ it is also obtained from the retarded propagator as

Tr Uˆ (t, t ) = dDx K (x, t; x, t ) = Θ(t − t ) Tr Uˆ(t, t ). (3.42) R 0 ˆ R 0 0 0

Using this result, the Fourier transform of Tr UˆR(t, t0) with respect to ∆t ≡ t − t0 gives +∞ ˆ i ˆ F Tr UR(∆t, 0) (E) = d(∆t) exp E∆t Tr UR(∆t, 0) ˆ−∞ ~ +∞ i D = d(∆t) exp E∆t d x KR(x, ∆t; x, 0) ˆ−∞ ~ ˆ +∞ D i = d x d(∆t) exp E∆t KR(x, ∆t; x, 0). (3.43) ˆ ˆ−∞ ~ From the deﬁnition (3.30) this in turn becomes

ˆ D ˜ F Tr UR(∆t, 0) (E) = d x K(x, x; E). (3.44) ˆ

11 Then using the expansion (3.40) of K˜ in the energy eigenfunctions, this becomes

ˆ D ˜ D X ∗ i~ F Tr UR(∆t, 0) (E) = d x K(x, x; E) = d x ψn(x)ψ (x) ˆ ˆ n E − E + iη n n X i = ~ dDx ψ (x)ψ∗ (x). (3.45) E − E + iη ˆ n n n n Here the integral is unity due to the normalisation of the eigenfunctions. Thus

ˆ X i~ F Tr UR(∆t, 0) (E) = (η inﬁnitesimal) (3.46) E − E + iη n n

(compare with the result (3.19)).

12 4 Path Integrals

4.1 The Short-time Propagator

In this section we will derive expressions for the propagator corresponding to the time evolution during an infinitesimal time interval δt. To start off, we note the following fact. For arbitrary operators Aˆ, Bˆ and infinitesimal , we have h i h i h i exp Aˆ exp Bˆ = 1 + Aˆ + O(2) 1 + Bˆ + O(2) = 1 + Aˆ + Bˆ + O(2) = exp Aˆ + Bˆ (4.1) to first order in , even if Aˆ and Bˆ do not commute. Thus for infinitesimal time evolution δt the time evolution operator may be written as i i pˆ2 i i pˆ2 U(t + δt, t) = exp − Hδtˆ = exp − V (xˆ) + δt = exp − V (xˆ)δt exp − δt . (4.2) ~ ~ 2m ~ ~ 2m The corresponding short-time propagator then becomes 2 0 0 0 i i pˆ K(x , t + δt; x, t) = x U(t + δt, t) x = x exp − V (xˆ)δt exp − δt x ~ ~ 2m 2 D 0 i i pˆ = d p x exp − V (xˆ)δt exp − δt p p x ˆ ~ ~ 2m 2 D i 0 i p 0 = d p exp − V (x )δt exp − δt x p p x , (4.3) ˆ ~ ~ 2m D where we have inserted the identity operator d p p p on the second line. Using the momentum eigen- function (2.5), this becomes ´ i p2 exp i p · x0 exp − i p · x K(x0, t + δt; x, t) = dDp exp − + V (x0) δt ~ ~ ˆ ~ 2m (2π~)D/2 (2π~)D/2 dDp i p2 = exp p · (x0 − x) − + V (x0) δt (4.4) ˆ (2π~)D ~ 2m or dDp0 i x0 − x K(x0, t + δt; x, t) = exp p0 · − H(x0, p0) δt . (4.5) ˆ (2π~)D ~ δt We recognise the exponent as the short-time canonical action for a path connecting the points x and x0. Setting ∆x0 ≡ x0 − x for notational convenience, we now proceed to integrate out the momentum variable: dDp i ∆x0 p2 K(x0, t + δt; x, t) = exp p · − − V (x0) δt ˆ (2π~)D ~ δt 2m exp − i V (x0)δt i δt = ~ dDp exp − p2 + ∆x0 · p (4.6) (2π~)D ˆ ~ 2m Using (A.23), the integral on the right evaluates to " # i δt 2π mD/2 i 1 ∆x0 2 dDp exp − p2 + ∆x0 · p = ~ exp m δt (4.7) ˆ ~ 2m iδt ~ 2 δt and the short-time propagator (4.6) becomes " ! # m D/2 i 1 x0 − x2 K(x0, t + δt; x, t) = exp m − V (x0) δt . (4.8) 2πi~δt ~ 2 δt Here we recognise the exponent as the short-time Lagrangian action for a path connecting x and x0.

13 4.2 The Finite-time Propagator From the Short-time Propagator

The propagator K(xb, tb; xa, ta) corresponding to the time evolution during a ﬁnite time interval ∆t ≡ tb −ta may be obtained from the short-time propagator as follows. We ﬁrst note that the time-evolution operator (3.2) may be written as N i i N Uˆ(tb, ta) = exp − Hˆ ∆t = exp − Hˆ ∆t/N = Uˆ (δt, 0) (4.9) ~ ~

tb−ta with δt ≡ N and N ≥ 1 an integer. Consequently, we can write the propagator as ˆ ˆ N K(xb, tb; xa, ta) = xb U(tb, ta) xa = xb U (δt, 0) xa . (4.10)

By expressing the operator Uˆ N (δt, 0) as a product of N operators Uˆ(δt, 0) and inserting N − 1 copies of the D identity operator d x x x between these, this becomes ´ D D ˆ ˆ ˆ K(xb, tb; xa, ta) = d xN−1 ··· d x1 xb U(δt, 0) xN−1 xN−1 U(δt, 0) ··· x1 x1 U(δt, 0) xa ˆ ˆ N Y = dDx ··· dDx K(x , δt; x , 0), (4.11) ˆ N−1 ˆ 1 k k−1 k=1

with x0 ≡ xa and xN ≡ xb. This equation holds for any integer N ≥ 1. By taking N → ∞, we can write

(N) K(xb, tb; xa, ta) = lim K (xb, tb; xa, ta) (4.12) N→∞ with

N Y K(N)(x , t ; x , t ) := dDx ··· dDx K(x , δt; x , 0). (4.13) b b a a ˆ N−1 ˆ 1 k k−1 k=1

tb−ta where δt ≡ N is now small enough so that K(xk, δt; xk−1, 0) becomes the short-time propagator given by (4.5) or (4.8).

4.3 The Phase Space Path Integral

We now make use of the result (4.5) for the short-time propagator without the momentum integrated out, and plug it into (4.13). We then ﬁnd

N D Y d pk i ∆xk K(N)(x , t ; x , t ) = dDx ··· dDx exp p · − H(x , p ) δt , (4.14) b b a a ˆ N−1 ˆ 1 ˆ (2π )D k δt k k k=1 ~ ~

tb−ta where ∆xk ≡ xk − xk−1 and δt ≡ N . After expanding the product, this becomes

D D (N) D D d pN d p1 K (xb, tb; xa, ta) = d xN−1 ··· d x1 ··· × ˆ ˆ ˆ (2π~)D ˆ (2π~)D " N # i X ∆xk exp p · − H(x , p ) δt . (4.15) k δt k k ~ k=1

We now introduce a time-slicing of the interval tb − ta as

tk = ta + kδt (k = 0,...,N) with t0 ≡ ta and tN ≡ tb, (4.16)

14 and for each of the ordered sets {x1,..., xN−1} and {p1,..., pN }, we deﬁne piecewise linear paths

xk − xk−1 x{xi}(t) := xk−1 + (t − tk−1)(tk−1 ≤ t ≤ tk)(k = 1,...,N) (4.17) tk − tk−1 and

pk − pk−1 p{pi}(t) := pk−1 + (t − tk−1)(tk−1 ≤ t ≤ tk)(k = 2,...,N). (4.18) tk − tk−1

That way we have x{xi}(tk) = xk and p{pi}(tk) = pk and in the limit of large N the sum in the exponential of (4.15) becomes

N N X ∆xk X h i p · − H(x , p ) δt = p (t ) · x˙ (t ) − Hx (t ), p (t ) (t − t ) k δt k k {pi} k {xi} k {xi} k {pi} k k k−1 k=1 k=1 t b h i −→ dt p{pi}(t) · x˙ {xi}(t) − H x{xi}(t), p{pi}(t) ˆta = S x{xi}(t), p{pi}(t); ta, tb (4.19)

where S[x(t), p(t); ta, tb] is the classical canonical action (2.9) for the Hamiltonian. The propagator, being the limit of (4.15), then becomes

D D D D d pN d p1 i K(xb, tb; xa, ta) = lim d xN−1 ··· d x1 ··· exp S x{x }(t), p{p }(t); ta, tb . N→∞ ˆ ˆ ˆ (2π~)D ˆ (2π~)D ~ i i (4.20)

We interpret this as a sum over all paths in phase space connecting the conﬁguration space endpoints xa and xb. The following deﬁnition will give us a simpler way of writing this beast.

Deﬁnition: Let Q denote the space of functions q(t): R → RD and let F denote the space of functionals F : Q×Q → C. Deﬁne a functional integral on F,

x(t )=x b b D p(t) D x(t) : F → C ˆx(ta)=xa ˆ 2π~ by

x(tb)=xb Dp(t) Dx(t) F x(t), p(t) := ˆx(ta)=xa ˆ 2π~ D D D D d pN d p1 lim d xN−1 ··· d x1 ··· F xx1...xN−1 (t), pp1...pN (t) (4.21) N→∞ ˆ ˆ ˆ (2π~)D ˆ (2π~)D with

xk − xk−1 xx1...xN−1 (t) := xk−1 + (t − tk−1)(tk−1 ≤ t ≤ tk)(k = 1,...,N) tk − tk−1 and

pk − pk−1 pp1...pN (t) := pk−1 + (t − tk−1)(tk−1 ≤ t ≤ tk)(k = 2,...,N) tk − tk−1

tb−ta where x0 ≡ xa, xN ≡ xb and tk = ta + kδt (k = 0,...,N) with t0 ≡ ta, tN ≡ tb and δt ≡ N .

15 We can then write the propagator (4.20) as

x(tb)=xb D p(t) i K(xa, ta; xb, tb) = D x(t) exp S[x(t), p(t); ta, tb] . (4.22) ˆx(ta)=xa ˆ 2π~ ~ The expression on the right is called the phase space path integral. The deﬁnition above is not really mathematically rigorous, and it is hard to give (4.21) a precise mathematical meaning. Accordingly, (4.22) should be regarded as a formal expression that must be supplemented by a proper prescription to evaluate it. For our purposes, to calculate a phase space path integral we write it in the ﬁnite-N time-sliced form (4.15) as D D (N) D D d pN d p1 i (N) K (xb, tb; xa, ta) = d xN−1 ··· d x1 ··· exp S [x, p] (4.23) ˆ ˆ ˆ (2π~)D ˆ (2π~)D ~ with the time-sliced canonical action N X ∆xk S(N)[x, p] := p · − H(x , p ) δt, (4.24) k δt k k k=1 where ∆xk ≡ xk − xk−1, and then we take the limit N → ∞.

4.4 The Conﬁguration Space Path Integral

We can derive an analogous path integral in conﬁguration space by integrating out all momentum variables in (4.23). Equivalently, we can make use of the result (4.8) for the short-time propagator where the momentum has been integrated out, and plug it into formula (4.12). We then ﬁnd

N D/2 " 2 ! # Y m i 1 ∆xk K(N)(x , t ; x , t ) = dDx ··· dDx exp m − V (x ) δt , (4.25) b b a a ˆ N−1 ˆ 1 2πi δt 2 δt k k=1 ~ ~

tb−ta where ∆xk ≡ xk − xk−1 and δt ≡ N . After expanding the product, this becomes

DN/2 " N 2 ! # m i X 1 ∆xk K(N)(x , t ; x , t ) = dDx ··· dDx exp m − V (x ) δt . (4.26) b b a a 2πi δt ˆ N−1 ˆ 1 2 δt k ~ ~ k=1

As in the previous subsection, we introduce a time-slicing of the interval tb − ta as

tk = ta + kδt (k = 0,...,N) with t0 ≡ ta and tN ≡ tb, (4.27)

and for each ordered set {x1,..., xN−1} ≡ {xi}, we deﬁne a piecewise linear path

xk − xk−1 x{xi}(t) := xk−1 + (t − tk−1)(tk−1 ≤ t ≤ tk)(k = 1,...,N). (4.28) tk − tk−1

That way we have x{xi}(tk) = xk and the summand in the exponential of (4.26) becomes 2 1 ∆xk 1 2 m − V (xk) = m x˙ {x }(tk) − V x{x }(tk) = L x{x }(tk) , x˙ {x }(tk) (4.29) 2 δt 2 i i i i where L is the classical Lagrangian (2.1). The sum in (4.26) becomes, in the limit of large N, N " 2 # N X 1 ∆xk X m − V (xk) δt = L x{x }(tk) , x˙ {x }(tk) (tk − tk−1) 2 δt i i k=1 k=1 t b −→ dt L x{xi}(t) , x˙ {xi}(t) = S x{xi}(t); ta, tb (4.30) ˆta

16 where S[x(t); ta, tb] is the classical action (2.2). The propagator, being the limit of (4.26), then becomes

DN/2 m D D i K(xb, tb; xa, ta) = lim d xN−1 ··· d x1 exp S[x{x }(t); ta, tb] . (4.31) N→∞ 2πi~δt ˆ ˆ ~ i This is to be interpreted as a sum of the action-exponentials over all possible paths x(t) connecting the endpoints xa and xb. As in the previous subsection, we now give a more compact way of writing this result.

Deﬁnition: Let X denote the space of functions x(t): R → RD and let F denote the space of functionals F : X → C. Deﬁne a functional integral

x(t )=x b b D x(t) : F → C ˆx(ta)=xa by

x(tb)=xb DN/2 m D D D x(t) F x(t) := lim d xN−1 ··· d x1 F [xx ...x (t)] (4.32) N→∞ 1 N−1 ˆx(ta)=xa 2πi~δt ˆ ˆ with

xk − xk−1 xx1...xN−1 (t) := xk−1 + (t − tk−1)(tk−1 ≤ t ≤ tk)(k = 1,...,N) tk − tk−1

tb−ta where x0 ≡ xa, xN ≡ xb and tk = ta + kδt (k = 0,...,N) with t0 ≡ ta, tN ≡ tb and δt ≡ N .

We can then write the propagator (4.31) as

x(tb)=xb i K(xb, tb; xa, ta) = D x(t) exp S[x(t); ta, tb] . (4.33) ˆx(ta)=xa ~ The expression on the right is called the configuration space path integral. As in the previous subsection, the definition above is not really mathematically rigorous, and it is hard to give (4.32) a precise mathematical meaning. Accordingly, (4.33) should be regarded as a formal expression that must be supplemented by a proper prescription to evaluate it. For our purposes, to calculate a configuration space path integral we write it in the finite-N time-sliced form (4.26) as

DN/2 (N) m D D i (N) K (xb, tb; xa, ta) = d xN−1 ··· d x1 exp S [x] (4.34) 2πi~δt ˆ ˆ ~ with the time-sliced Lagrangian action

N " 2 # X 1 ∆xk S(N)[x] := m − V (x ) δt, (4.35) 2 δt k k=1 where ∆xk ≡ xk − xk−1, and then we take the limit N → ∞.

17 5 Finding a More Flexible Path Integral Formula

In Appendix B we solve the path integrals for the most simple physical systems – the free particle and the harmonic oscillator. The solutions are straightforward and without difficulties. For more complicated systems, however, this is not the case. In particular, for systems with a centrifugal barrier the path integrals (4.15) and (4.26) can be shown to diverge [4]. This also happens for the Coulomb potential and hence any atomic system, i.e. systems which are of much interest. The goal of this Section is therefore to find new, modified, path integral formulas that are free of this problem for singular potentials.

5.1 The Pseudo-propagator

The starting point in the search for new path integral formulas is to consider the fixed-energy amplitude rather than the propagator itself, i.e. ∞ ˜ i ˆ K(x, x0; E) = d(∆t) exp E∆t K(x, ∆t; x0, 0) = xb R(E) xa (5.1) ˆ0 ~ with the resolvent operator i Rˆ(E) = ~ (η infinitesimal). (5.2) E − Hˆ + iη Since the propagator and the fixed-energy amplitude are obtained from one another through Fourier transforms, no information is lost. Now, if the system has a path integral formula for the propagator, it does also for the fixed-energy amplitude. To see this, we introduce a modified propagator and corresponding path integral by shifting the energy scale. For some fixed energy E, we first define an energy-shifted potential

VE(x) := V (x) − E. (5.3) The classical Hamiltonian and canonical action functional corresponding to this potential are pˆ2 H (x, p) := + V (x) = H(x, p) − E (5.4) E 2m E and

tb SE[x(t), p(t); ta, tb] := dt [p · x˙ − HE(x, p)] = S[x(t), p(t); ta, tb] + E(tb − ta) (5.5) ˆta while the classical Lagrangian and Lagrangian action functional for the potential (5.3) are

1 2 L E x, x˙ := mx˙ − VE(x) = L x, x˙ + E (5.6) 2 and t b SE[x(t); ta, tb] := dt L E x(t), x˙ (t) = S[x(t); ta, tb] + E(tb − ta). (5.7) ˆta The energy-shifted Hamiltonian operator, time-evolution operator, and propagator are then

HÊ := HE(xˆ, pˆ) = Hˆ − E (5.8) and i i UÊ(tb, ta) := exp − HÊ(tb − ta) = exp E(tb − ta) Uˆ(tb, ta) (5.9) ~ ~

18 and ˆ i KE(xb, tb; xa, ta) := xb UE(tb, ta) xa = exp E(tb − ta) K(xb, tb; xa, ta), (5.10) ~ respectively. All quantities and operators above merely correspond to a shift in the energy scale by E. In Section 4 we derived the path integral formalism for a general Hamiltonian of the form (3.1). Since HE has this form, all results from Section 4 also hold for KE(xb, tb; xa, ta). Thus the short-time propagator KE is given by

D 0 0 0 d p i 0 x − x 0 0 KE(x , t + δt; x, t) = exp p · − HE(x , p ) δt (5.11) ˆ (2π~)D ~ δt D/2 0 2 m i 1 x − x 0 = exp m − VE(x ) δt . (5.12) 2πi~δt ~ 2 δt

For a finite time-difference, KE can be written as the phase- and configuration-space path integrals

x(tb)=xb D p(t) i KE(xb, tb; xa, ta) = D x(t) exp SE[x(t), p(t); ta, tb] (5.13) ˆx(ta)=xa ˆ 2π~ ~

x(tb)=xb i = D x(t) exp SE[x(t); ta, tb] (5.14) ˆx(ta)=xa ~ (N) = lim KE (xb, tb; xa, ta), (5.15) N→∞

tb−ta where the ﬁnite-N time-sliced versions are (with δt ≡ N )

D D (N) D D d pN d p1 i (N) KE (xb, tb; xa, ta) = d xN−1 ··· d x1 ··· exp SE [x, p] (5.16) ˆ ˆ ˆ (2π~)D ˆ (2π~)D ~ DN/2 m D D i (N) = d xN−1 ··· d x1 exp SE [x] (5.17) 2πi~δt ˆ ˆ ~ with time-sliced canonical and Lagrangian actions (∆xk ≡ xk − xk−1, x0 ≡ xa, xb ≡ xN )

N (N) X ∆xk S [x, p] := p · − H (x , p ) δt (5.18) E k δt E k k k=1 and

N " 2 # (N) X 1 ∆xk S [x] := m − V (x ) δt, (5.19) E 2 δt E k k=1 respectively. The ﬁxed-energy amplitude (5.1) may now be written in terms of the energy-shifted propagator (5.10) as

∞ K˜ (xb, xa; E) = d(∆t) KE(xb, ∆t; xa, 0). (5.20) ˆ0 A ﬁnite-N version of this can be obtained by writing

∞ ∞ ˜ (N) (N) K(xb, xa; E) = d(∆t) lim KE (xb, ∆t; xa, 0) = lim d(∆t) KE (xb, ∆t; xa, 0), (5.21) ˆ0 N→∞ N→∞ ˆ0

19 implying that

(N) K˜ (xb, xa; E) = lim K˜ (xb, xa; E), (5.22) N→∞ where ∞ ∞ ˜ (N) (N) (N) K (xb, xa; E) := d(∆t) KE (xb, ∆t; xa, 0) = N d KE (xb, N; xa, 0). (5.23) ˆ0 ˆ0

In terms of the energy-shifted Hamiltonian HÊ, the resolvent operator (5.2) may be written i Rˆ(E) = ~ . (5.24) −HÊ + iη For the new yet-to-found path integral formulas, we would like to incorporate a functional degree of freedom through some arbitrary function of x that we can choose to our liking without changing the physical results. To proceed, it will be convenient with the following definition.

Deﬁnition: For a given function f : RD → C, depending on x, and for λ ∈ R, deﬁne functions

1−λ λ fl := f and fr := f .

These are called regulating functions, satisfying fl(x)fr(x) = f(x), and the parameter λ is called the splitting parameter.

Given a choice of f, we can incorporate the regulating functions into the resolvent operator using the operator identity

i~ i~ Rˆ(E) = = fr(xˆ) fl(xˆ). (5.25) −HÊ + iη fl(xˆ)(−HÊ + iη)fr(xˆ) We then define a pseudo-Hamiltonian

HˆE := fl(xˆ)HˆEfr(xˆ). (5.26) and a corresponding pseudo-time evolution operator

i UˆE(sb, sa) := fr(xˆ) exp − HˆE(sb − sa) fl(xˆ), (5.27) ~ and a pseudo-propagator ˆ KE(xb, sb; xa, sa) := xb UE(sb, sa) xa . (5.28)

In terms of this pseudo-propagator, the generalization of the formula (5.20) then reads [4].

∞ K˜ (xb, xa; E) = d(∆s) KE(xb, ∆s; xa, 0) (5.29) ˆ0 which is independent of the choice of f.

Classically, the splitting of f into fl and fr through the splitting parameter λ is of no speciﬁc interest, but quantum mechanically the factor ordering involving the corresponding operators is nontrivial. However, the

20 pseudo-time evolution operator is in fact independent of λ. By expanding the pseudo-time evolution operator (5.27) in its Taylor series and using frfl = f, it can be rewritten as i UˆE(sb, sa) = exp − f(xˆ)HˆE(sb − sa) f(xˆ). (5.30) ~ Consequently the pseudo-propagator (5.28) is independent of the choice of λ, too.

h i i Proceeding the same way as in Section 4.2, the operator exp − HÊ(sb − sa) in (5.27) is written as ~ N i i sb − sa exp − HÊ(sb − sa) = lim exp − HÊδs with δs ≡ . (5.31) ~ N→∞ ~ N Consequently, the pseudo-time evolution operator (5.27) can be expressed as

i i N UÊ(sb, sa) = fr(xˆ) exp − HÊ(sb − sa) fl(xˆ) = lim fr(xˆ) exp − HÊδs fl(xˆ), (5.32) ~ N→∞ ~ and the pesudo-propagator (5.28) then becomes

N ˆ i ˆ KE(xb, sb; xa, sa) = xb UE(sb, sa) xa = lim xb fr(xˆ) exp − HEδs fl(xˆ) xa N→∞ ~ N i ˆ = lim fr(xb)fl(xa) xb exp − HEδs xa . (5.33) N→∞ ~

N h i i h i i By writing the operator exp − HˆEδs as a product of N operators exp − HˆEδs and inserting N −1 ~ ~ D copies of the identity operator d x x x between these, this becomes ´ N D D Y i ˆ KE(xb, sb; xa, sa) = lim fr(xb)fl(xa) d xN−1 ··· d x1 xk exp − HEδs xk−1 . (5.34) N→∞ ˆ ˆ k=1 ~ In the limit of large N (i.e. small δs) we can write

i i i exp − HÊδs = 1 − HÊδs = 1 − fl(xˆ)HÊfr(xˆ)δs, (5.35) ~ ~ ~ so that i ˆ i ˆ xk exp − HEδs xk−1 = xk 1 − fl(xˆ)HEfr(xˆ)δs xk−1 ~ ~ i ˆ = xk 1 − fl(xk)HEfr(xk−1)δs xk−1 . (5.36) ~ By letting

δtk := fl(xk)fr(xk−1)δs (5.37) we then have, for small enough δs, i ˆ i ˆ i ˆ xk exp − HEδs xk−1 = xk 1 − HEδtk xk−1 = xk exp − HEδtk xk−1 ~ ~ ~ = KE(xk, δtk; xk−1, 0), (5.38)

21 giving

N D D Y KE(xb, sb; xa, sa) = lim fr(xb)fl(xa) d xN−1 ··· d x1 KE(xk, δtk; xk−1, 0), (5.39) N→∞ ˆ ˆ k=1 or

(N) KE(xb, sb; xa, sa) = lim KE (xb, sb; xa, sa), (5.40) N→∞ where

N (N) Y K (x , s ; x , s ) := f (x )f (x ) dDx ··· dDx K (x , δt ; x , 0) (5.41) E b b a a r b l a ˆ N−1 ˆ 1 E k k k−1 k=1

sb−sa with δtk ≡ fl(xk)fr(xk−1)δs and δs ≡ N .

5.2 New Path Integral Formula: Phase Space

We now derive a phase space path integral formula for the pseudo-propagator using the result (5.41). For large enough N (i.e. small δs), δtk is small enough so that, using (5.11),

D d pk i ∆xk KE(xk, δtk; xk−1, 0) = D exp pk · − HE(xk, pk) δtk ˆ (2π~) ~ δtk D d pk i ∆xk = D exp pk · − HE(xk, pk) fl(xk)fr(xk−1)δs ˆ (2π~) ~ fl(xk)fr(xk−1)δs D d pk i ∆xk = exp pk · − fl(xk)HE(xk, pk)fr(xk−1) δs . (5.42) ˆ (2π~)D ~ δs Equation (5.41) then becomes

N (N) Y K (x , s ; x , s ) = f (x )f (x ) dDx ··· dDx K (x , δt ; x , 0) E b b a a r b l a ˆ N−1 ˆ 1 E k k k−1 k=1 = f (x )f (x ) dDx ··· dDx × r b l a ˆ N−1 ˆ 1 N D Y d pk i ∆xk exp p · − f (x )H (x , p )f (x ) δs ˆ (2π )D k δs l k E k k r k−1 k=1 ~ ~ D D D D d pN d p1 = fr(xb)fl(xa) d xN−1 ··· d x1 ··· × (5.43) ˆ ˆ ˆ (2π~)D ˆ (2π~)D " N # i X ∆xk exp p · − f (x )H (x , p )f (x ) δs , k δs l k E k k r k−1 ~ k=1 or D D (N) D D d pN d p1 i ˜(N) KE (xb, sb; xa, sa) = fr(xb)fl(xa) d xN−1 ··· d x1 ··· exp SE [x, p] (5.44) ˆ ˆ ˆ (2π~)D ˆ (2π~)D ~ with the pseudo-time sliced canonical action N (N) X ∆xk S˜ [x, p] := p · − f (x )H(x , p ) − Ef (x ) δs. (5.45) E k δs l k k k r k−1 k=1

22 (N) Note that for finite N, we get different KE for different choices of the splitting parameter λ. In the continuum limit however, we have seen that KE is independent of λ. Thus, in taking the limit N → ∞, we can in particular set λ = 0 (giving fl ≡ f and fr ≡ 1). We then obtain

D D D D d pN d p1 i ˜(N) KE(xb, sb; xa, sa) = lim f(xa) d xN−1 ··· d x1 ··· exp SE [x, p] (5.46) N→∞ ˆ ˆ ˆ (2π~)D ˆ (2π~)D ~ with the pseudo-time sliced canonical action

N (N) X ∆xk S˜ [x, p] = p · − f(x )H(x , p ) − E δs. (5.47) E k δs k k k k=1 This may formally be written as a phase space path integral

x(sb)=xb D p(s) i KE(xb, sb; xa, sa) = f(xa) D x(s) exp S˜E[x(s), p(s); sa, sb] (5.48) ˆx(sa)=xa ˆ 2π~ ~ with the pseudo-action

sb h 0 i S˜E[x(s), p(s); sa, sb] = ds p(s) · x (s) − f x(s) H x(s), p(s) − E , (5.49) ˆsa where x0(s) denotes the derivative of x(s) with respect to pseudotime s.

5.3 New Path Integral Formula: Conﬁguration Space

The conﬁguration space path integral for the pseudo-propagator is obtained by integrating out the p-variables in the phase space path integral (5.44). Equivalently, we may again use the result (5.41). For large enough N (i.e. small δs), δtk is small enough so that, using (5.11),

D/2 " 2 ! # m i 1 ∆xk KE(xk, δtk; xk−1, 0) = exp m − VE(xk) δtk = 2πi~δtk ~ 2 δtk D/2 " 2 ! # m i 1 ∆xk = exp m − VE(xk) fl(xk)fr(xk−1)δs 2πi~fl(xk)fr(xk−1)δs ~ 2 fl(xk)fr(xk−1)δs " ! # 1 m D/2 i 1 ∆x 2 = exp m k − f (x )V (x )f (x ) δs . D/2 l k E k r k−1 [fl(xk)fr(xk−1)] 2πi~δs ~ 2fl(xk)fr(xk−1) δs (5.50)

Substituting this into (5.41), we obtain

N (N) Y K (x , s ; x , s ) = f (x )f (x ) dDx ··· dDx K (x , δt ; x , 0) = E b b a a r b l a ˆ N−1 ˆ 1 E k k k−1 k=1 N Y 1 m D/2 = f (x )f (x ) dDx ··· dDx × r b l a ˆ N−1 ˆ 1 [f (x )f (x )]D/2 2πi δs k=1 l k r k−1 ~ " 2 ! #! i 1 ∆xk exp m − fl(xk)VE(xk)fr(xk−1) δs (5.51) ~ 2fl(xk)fr(xk−1) δs

23 or, after expanding the product,

DN/2 N ! (N) m Y 1 K (x , s ; x , s ) = f (x )f (x ) dDx ··· dDx × E b b a a r b l a 2πi δs ˆ N−1 ˆ 1 [f (x )f (x )]D/2 ~ k=1 l k r k−1 " N 2 ! # i X 1 ∆xk exp m − fl(xk)VE(xk)fr(xk−1) δs . (5.52) 2fl(xk)fr(xk−1) δs ~ k=1

Using fr(x)fl(x) = f(x) we see that the product in fl, fr may be written as

N N−1 Y 1 1 Y 1 = , (5.53) [f (x )f (x )]D/2 [f (x )f (x )]D/2 D/2 k=1 l k r k−1 l b r a k=1 f(xk) giving

f (x )f (x ) m DN/2 dDx dDx i K(N)(x , s ; x , s ) = r b l a N−1 ··· 1 exp S˜(N)[x] , E b b a a [f (x )f (x )]D/2 2πi δs ˆ D/2 ˆ D/2 E l b r a ~ f(xN−1) f(x1) ~ (5.54)

with the pseudo-time sliced Lagrangian action

N " 2 # ˜(N) X 1 ∆xk SE [x] := m − fl(xk) V (xk) − E fr(xk−1) δs. (5.55) 2fl(xk)fr(xk−1) δs k=1

(N) As pointed out in the previous section, we get diﬀerent KE for diﬀerent choices of the splitting parameter λ, but in the continuum limit all choices should converge to the limit KE, independently of λ. Setting λ = 0, we obtain

DN/2 D D f(xa) m d xN−1 d x1 i ˜(N) KE(xb, sb; xa, sa) = lim ··· exp SE [x] N→∞ [f(x )]D/2 2πi δs ˆ D/2 ˆ D/2 b ~ f(xN−1) f(x1) ~ (5.56)

with the pseudo-time sliced Lagrangian action

N " 2 # ˜(N) X 1 ∆xk SE [x] = m − f(xk) V (xk) − E δs. (5.57) 2f(xk) δs k=1 This may formally be written as a conﬁguration space path integral

x(s )=x f(x ) b b D x(s) i K (x , s ; x , s ) = a exp S[x(s); s , s ] (5.58) E b b a a D/2 a b [f(xb)] ˆx(sa)=xa f(x(s)) ~ with the pseudo-action

sb " # 1 0 2 S[x(s); sa, sb] = ds m x (s) − f x(s) V x(s) − E , (5.59) ˆsa 2f x(s)

where x0(s) denotes the derivative of x(s) with respect to pseudo-time s.

24 Summary of Section 5

The ﬁxed-energy amplitude can be obtained from a pseudo-propagator KE(xb, sb; xa, sa) according to

∞ K˜ (xb, xa; E) = d(∆s) KE(xb, ∆s; xa, 0). (5.60) ˆ0

By choosing a suitable function f(x) depending on x, and a splitting parameter λ ∈ R, we deﬁne 1−λ λ regulating functions fl := f and fr := f . A corresponding pseudo-propagator is then obtained from the pseudo-time sliced path integral of phase space,

D D (N) D D d pN d p1 i ˜(N) KE (xb, sb; xa, sa) = fr(xb)fl(xa) d xN−1 ··· d x1 ··· exp SE [x, p] (5.61) ˆ ˆ ˆ (2π~)D ˆ (2π~)D ~ with the pseudo-time sliced canonical action

N (N) X ∆xk S˜ [x, p] := p · − f (x )H(x , p ) − Ef (x ) δs, (5.62) E k δs l k k k r k−1 k=1 or from the pseudo-time sliced path integral of conﬁguration space,

f (x )f (x ) m DN/2 dDx dDx i K(N)(x , s ; x , s ) = r b l a N−1 ··· 1 exp S˜(N)[x] , E b b a a [f (x )f (x )]D/2 2πi δs ˆ D/2 ˆ D/2 E l b r a ~ f(xN−1) f(x1) ~ (5.63)

with the pseudo-time sliced Lagrangian action

N " 2 # ˜(N) X 1 ∆xk SE [x] := m − fl(xk) V (xk) − E fr(xk−1) δs. (5.64) 2fl(xk)fr(xk−1) δs k=1

Taking the limit N → ∞ of (5.61) or (5.63) yields the pseudo-propagator,

(N) KE(xb, sb; xa, sa) = lim KE (xb, sb; xa, sa), (5.65) N→∞

which is independent of the splitting parameter λ. We then obtain the ﬁxed-energy amplitude from (5.60), the result being independent of the choice of function f. That is, provided that f has been suitably chosen such that the path integral formulas (5.61) and (5.63) are well deﬁned. Setting f(x) = 1 we recover the original path integrals in Section 4, which diverge for singular potentials such as the Coulomb potential. The presence of suitably chosen regulating functions is therefore essential to solving path integrals for many systems of interest.

25 6 Exact Solution for the Hydrogen Atom

6.1 The Hydrogenic Path Integral in D Dimensions

Using the formalism developed in section 5, we are now ready to tackle the Coulomb-problem and the hydrogen atom in particular. Consider a D-dimensional electron-proton system with Coulomb interaction, or, in the centre-of-mass system, an electron subjected to the potential

e2 V (x) = − , (6.1) (4π0)r

where e is the elementary charge, 0 the free space permittivity, and r ≡ |x|. To simplify the formulas, we shall work in atomic units in which 1 ~ = me = e = = 1 (6.2) 4π0 so that the potential (6.1) takes the simple form 1 V (x) = − . (6.3) r By choosing the function f(x) introduced in section 5 to be

f(x) := r (6.4) and deﬁning regulating functions

1−λ 1−λ λ λ fl(x) := f(x) = r and fr(x) := f(x) = r . (6.5) for arbitrary λ, the modified path integral formulas (5.61) and (5.63) become well-defined, as discovered by Duru and Kleinert in 1979 [4]. The pseudo-time sliced configuration space path integral for the pseudo propagator (5.54) then becomes

λ 1−λ DN/2 D D (N) rb ra 1 d xN−1 d x1 h (N) i K (xb, sb; xa, sa) = ··· exp iS˜ [x] (6.6) E [r1−λrλ]D/2 2πiδs ˆ D/2 ˆ D/2 E b a rN−1 r1 with the pseudo-time sliced action (5.55) given by

N " 2 # (N) X 1 ∆xk 1 S˜ [x] = − r1−λ − − E rλ δs. (6.7) E 1−λ λ k k−1 δs rk k=1 2rk rk−1 For our purposes in this Section the freedom in the value of the splitting parameter λ will not be needed. Setting λ = 0 yields

r 1 DN/2 dDx dDx h i K(N)(x , s ; x , s ) = a N−1 ··· 1 exp iS˜(N)[x] (6.8) E b b a a D/2 2πiδs ˆ D/2 ˆ D/2 E rb rN−1 r1 with

N " 2 # ˜(N) X 1 ∆xk SE [x] = + Erk + 1 δs. (6.9) 2rk δs k=1

26 6.2 Solution for the Two-Dimensional H-atom

Before solving the full three-dimensional problem, we will ﬁrst consider the simpliﬁed case of a two-dimensional Hydrogen atom. In two dimensions (D = 2), the pseudo-time sliced path integral (6.8) becomes

N 2 2 (N) ra 1 d xN−1 d x1 h ˜(N) i KE (xb, sb; xa, sa) = ··· exp iSE [x] (6.10) rb 2πiδs ˆ rN−1 ˆ r1 with the pseudo-time sliced action

N " 2 # ˜(N) X 1 ∆xk SE [x] = + Erk + 1 δs. (6.11) 2rk δs k=1 We shall see that this path integral can be transformed into that of the harmonic oscillator by making a coordinate transformation from {xi} to ”square root coordinates” {ui} satisfying u2 = r. This is accomplished by the Levi-Civita transformation ( x1 = (u1)2 − (u2)2 . (6.12) x2 = 2u1u2

Indeed, with this transformation we have

q 2 r = p(x1)2 + (x2)2 = (u1)2 − (u2)2 + 4(u1)2(u2)2 = (u1)2 + (u2)2 ≡ u2 (6.13)

as desired. Using the relation for x1 in (6.12) together with (6.13), we ﬁnd

( 1 2 1 1 (u ) = 2 (r + x ) 2 2 1 1 . (6.14) (u ) = 2 (r − x )

It is important to note that the transformation (6.12) is not a bijection, but two to one. An inverse can be found by restricting it to, say, u1 ≥ 0. The diﬀerentials of xi and ui are related by

[dx] = A(u) [du] (6.15) with the Jacobian matrix 2u1 −2u2 A(u) = . (6.16) 2u2 2u1

i The metric gij in u -coordinates then takes the simple form

2u1 2u2 2u1 −2u2 4u2 0 g(u) = AT A = = = 4u2I (6.17) −2u2 2u1 2u2 2u1 0 4u2 and in the continuum limit we have, to ﬁrst order in δs (summation convention implied):

2 i j T 2 2 (∆x) = gij∆u ∆u = [∆u] g[∆u] = 4u (∆u) , (6.18) or 1 (∆x)2 = 4(∆u)2. (6.19) r

27 Using (6.13) and (6.19), we can now write the pseudo-time sliced action (6.11) in terms of uk-variables as

N " 2 # N " 2 # (N) X 1 ∆uk X 1 ∆uk 1 −E S˜ [x] = · 4 + Eu2 + 1 δs = · 4 − · 4 u2 δs + ∆s (6.20) E 2 δs k 2 δs 2 2 k k=1 k=1 where ∆s = sb − sa = Nδs. Now we see that by letting

m := 4me = 4 (6.21) and r r −E −E ω := = (6.22) 2me 2 the pseudo-time sliced action (6.20) becomes

N " 2 # (N) X 1 ∆uk 1 S˜ [x] = m − mω2u2 δs + ∆s = S(N)[u] + ∆s, (6.23) E 2 δs 2 k osc k=1

(N) where Sosc [u] is the time-sliced action of the harmonic oscillator. Next, the determinant of A is

| det A| = pdet g = 4u2 = 4r, (6.24) so the volume elements d2x and d2u are related by

d2x = 4 d2u. (6.25) r

Using (6.25), we can now transform the integrals over xk in (6.10) to integrals over uk. The factor 4 from N−1 1 N N each integral combine to an overall factor 4 = 4 4 , and we can put the factor 4 inside the prefactor of (6.10), giving the correct prefactor for the harmonic oscillator path integral with m = 4. The result is

exp [i∆s]h i K(N)(x , s ; x , s ) = K(N)(u , s ; u , s ) + K(N)(−u , s ; u , s ) , (6.26) E b b a a 4 osc b b a a osc b b a a where m N h i K(N)(u , s ; u , s ) = d2u ··· d2u exp iS(N)[u] (6.27) osc b b a a 2πiδs ˆ N−1 ˆ 1 osc is the time-sliced path integral for the two-dimensional harmonic oscillator, and the integrals are over the 2 whole of R . The symmetrization in ub in (6.26) arises as a consequence of the mapping (6.12) being two to one. For each path going from xa to xb there are two paths in u-space, one going from ua to ub and another going from ua to −ub. By taking the limit N → ∞ of (6.26), we obtain directly

exp [i∆s]h i K (x , s ; x , s ) = K (u , s ; u , s ) + K (−u , s ; u , s ) (6.28) E b b a a 4 osc b b a a osc b b a a with the two-dimensional harmonic oscillator propagator given by (B.45) with D = 2, i.e.

mω mω K (u , s ; u , s ) = exp i (u2 + u2) cos(ω∆s) − 2u · u . (6.29) osc b b a a 2πi sin(ω∆s) 2 sin(ω∆s) b a b a

28 The pseudo-propagator (6.28) then becomes

exp [i∆s] mω mω K (x , s ; x , s ) = exp i (u2 + u2) cos(ω∆s) × E b b a a 4 2πi sin(ω∆s) 2 sin(ω∆s) b a mω mω exp i (−2u · u ) + exp i (+2u · u ) 2 sin(ω∆s) b a 2 sin(ω∆s) b a mω exp [i∆s] mω cos(ω∆s) mω = exp i (u2 + u2) cos u · u . (6.30) 4πi sin(ω∆s) 2 sin(ω∆s) b a sin(ω∆s) b a

After restoring SI units, this reads

h 2 i exp i e ∆s mω ~ 4π0 mω cos(ω∆s) 2 2 mω KE(xb, sb; xa, sa) = exp i (ub + ua) cos ub · ua . (6.31) 4πi~ sin(ω∆s) 2~ sin(ω∆s) ~ sin(ω∆s) By expressing the trigonometric functions as 1 1 cos(ω∆s) = exp [iω∆s] + exp [−iω∆s] = exp [iω∆s] 1 + exp [−i 2ω∆s] (6.32) 2 2 and 1 1 sin(ω∆s) = exp [iω∆s] − exp [−iω∆s] = exp [iω∆s] 1 − exp [−i 2ω∆s] , (6.33) 2i 2i and introducing the abbreviations r mω −2m E κ := = e (6.34) 2~ ~2 and

e2/(4π ) rm e4/(4π )2 ν := 0 = e 0 , (6.35) 2ω~ −2~2E we can rewrite (6.31) as

κ exp [−i 2ω∆s(−ν + 1/2)] 1 + exp [−i 2ω∆s] K (x , s ; x , s ) = exp −κ (u2 + u2) × E b b a a π 1 − exp [−i 2ω∆s] 1 − exp [−i 2ω∆s] b a 4iκ exp [−iω∆s] cos u · u . (6.36) 1 − exp [−i 2ω∆s] b a

Finally, we express the parameters ua,b in terms of the physical coordinates xa,b. From (6.12) and (6.14) we ﬁnd

r1 u2 = r , u · u = ± (r r + x · x ), (6.37) a,b a,b b a 2 b a b a and the pseudo-propagator (6.36) takes the ﬁnal form

κ exp [−i 2ω∆s(−ν + 1/2)] 1 + exp [−i 2ω∆s] K (x , s ; x , s ) = exp −κ (r + r ) × E b b a a π 1 − exp [−i 2ω∆s] 1 − exp [−i 2ω∆s] b a ! 4iκ exp [−iω∆s] r1 cos (r r + x · x ) . (6.38) 1 − exp [−i 2ω∆s] 2 b a b a

29 Having solved the pseudo-time sliced path integral and obtained the pseudo-propagator, the ﬁxed-energy amplitude (5.29) can now be found from ∞ K˜ (xb, xa; E) = ds KE(xb, s; xa, 0) (6.39) ˆ0 with κ exp [−i 2ωs(−ν + 1/2)] 1 + exp [−i 2ωs] K (x , s; x , 0) = exp −κ (r + r ) × E b a π 1 − exp [−i 2ωs] 1 − exp [−i 2ωs] b a ! 4iκ exp [−iωs] r1 cos (r r + x · x ) . (6.40) 1 − exp [−i 2ωs] 2 b a b a

When evaluating the integral (6.39), we have to pass around the singularities of (6.40) in the complex plane. We can invoke the residue theorem to evaluate (6.39) as an integral in the complex plane according to

K˜ (xb, xa; E) = ds KE(xb, s; xa, 0) (6.41) ˆC where the path C may be parametrized as s(σ) = σ − iη with σ ∈ (0, ∞) and η inﬁnitesimal. Since (6.40) is an analytic function in the domain beneath C, the integral is path-independent there, and we may write (6.41) as

K˜ (xb, xa; E) = ds KE(xb, s; xa, 0) + ds KE(xb, s; xa, 0) (6.42) ˆC1 ˆC2

where C1 is the negative imaginary axis, with parametrization s(σ) = −iσ, σ ∈ (0,R), and C2 the path with parametrization s(α) = R exp [iα], −π/2 ≤ α ≤ 0, and R → ∞. Now, it is readily veriﬁed that (6.40) vanishes for |s(α)| → ∞ so that the integral over C2 vanishes. Thus we are left with ∞ ds ∞ K˜ (xb, xa; E) = ds KE(xb, s; xa, 0) = dσ KE(xb, s(σ); xa, 0) = −i dσ KE(xb, −iσ; xa, 0) ˆC1 ˆ0 dσ ˆ0 κ ∞ exp [−2ωσ(−ν + 1/2)] 1 + exp [−2ωσ] = −i dσ exp −κ (rb + ra) × π ˆ0 1 − exp [−2ωσ] 1 − exp [−2ωσ] ! 4iκ exp [−ωσ] r1 cos (r r + x · x ) . (6.43) 1 − exp [−2ωσ] 2 b a b a We now change the integration variable to % := exp [−2ωσ] (6.44) so that 1 d% d% = −2ω exp [−2ωσ] dσ, dσ = − , (6.45) 2ω % and (6.43) becomes √ ! κ 1 %−ν−1/2 1 + % 4κ %r1 K˜ (xb, xa; E) = −i d% exp −κ (rb + ra) cosh (rbra + xb · xa) . (6.46) 2ωπ ˆ0 1 − % 1 − % 1 − % 2

From (6.21) and (6.22) we have ω = 2~κ = ~κ and thus the ﬁxed-energy amplitude of the two-dimensional m 2me Hydrogen atom takes the form

1 −ν−1/2 √ r ! me % 1 + % 4κ % 1 K˜ (xb, xa; E) = d% exp −κ (rb + ra) cosh (rbra + xb · xa) (6.47) iπ~ ˆ0 1 − % 1 − % 1 − % 2 with κ and ν given by (6.34) and (6.35), respectively.

30 The integral in (6.47) converges only for ν < 1/2, but we can ﬁnd another integral representation that converges for all ν 6= 1/2, 3/2,... by changing the integration variable to 1 + % ζ := . (6.48) 1 − % with ζ going from 1 to ∞ as % goes from 0 to 1. We then have ζ − 1 2 % = , d% = dζ (6.49) ζ + 1 (ζ + 1)2 so that 2 1 − % = , (6.50) ζ + 1 √ s % ζ + 1 ζ − 1 1p = = ζ2 − 1, (6.51) 1 − % 2 ζ + 1 2 %−ν−1/2 2 ζ + 1 ζ − 1−ν−1/2 (ζ + 1)ν−1/2 d% = dζ = dζ , (6.52) 1 − % (ζ + 1)2 2 ζ + 1 (ζ − 1)ν+1/2

and the ﬁxed-energy amplitude (6.47) becomes

∞ ν−1/2 r ! me (ζ + 1) p 1 K˜ (x , x ; E) = dζ exp [−κζ(r + r )] cosh 2κ ζ2 − 1 (r r + x · x ) . (6.53) b a ν+1/2 b a b a b a iπ~ ˆ1 (ζ − 1) 2

Figure 1. The integration contour C in the complex plane. The integrand of (6.53) has branch cuts in the complex ζ-plane extending from ζ = −1 to −∞ and from ζ = 1 to ∞, the integral running along the latter cut. By invoking the residue theorem, we can evaluate the integral in the complex plane according to [4]

∞ dζ π exp [iπ(ν + 1/2)] 1 dζ 1 dζ ··· = ··· = ··· ν+1/2 ν+1/2 ν+1/2 ˆ1 (ζ − 1) sin[π(ν + 1/2)] 2πi ˆC (ζ − 1) 1 + exp [−i 2πν] ˆC (ζ − 1) (6.54)

along the contour C encircling the right-hand cut clockwise (see fig. 1). The fixed-energy amplitude (6.53) then finally becomes

m 1 (ζ + 1)ν−1/2 K˜ (x , x ; E) = e dζ exp [−κζ(r + r )] × b a ν+1/2 b a iπ~ 1 + exp [−i 2πν] ˆC (ζ − 1) r ! p 1 cosh 2κ ζ2 − 1 (r r + x · x ) , (6.55) 2 b a b a

where this integral representation converges for all ν 6= 1/2, 3/2,....

31 6.3 Solution for the Three-Dimensional H-atom

The two-dimensional hydrogen atom is of course a toy model, the real world being three-dimensional. In three dimensions (D = 3), the pseudo-time sliced path integral (6.8) becomes

r 1 3N/2 d3x d3x h i K(N)(x , s ; x , s ) = a N−1 ··· 1 exp iS˜(N)[x] (6.56) E b b a a 3/2 2πiδs ˆ 3/2 ˆ 3/2 E rb rN−1 r1 with the pseudo-time sliced action

N " 2 # ˜(N) X 1 ∆xk SE [x] = + Erk + 1 δs. (6.57) 2rk δs k=1 As in two dimensions, we shall see that we can transform this path integral into that of the harmonic oscillator by going over to ”square root coordinates” uµ whose sum of squares equals r. This can be done for three dimensions by introducing a mapping from a four-dimensional {uµ} space to the three-dimensional {xi} space by

xi = z†σiz (6.58) with u1 + iu2 z := (6.59) u3 + iu4 and the Pauli spin matrices 0 1 0 −i 1 0 σ1 = σ2 = σ3 = . (6.60) 1 0 i 0 0 −1

With this transformation we indeed have r = (u1)2 + (u2)2 + (u3)2 + (u4)2 ≡ ~u2, as shown in Appendix C. The mapping (6.58) is obviously not invertible, so the inverse relationship will be multivalued. By expressing the xi in terms of spherical coordinates r, θ, φ, we ﬁnd (see Appendix C)  √ u1 = r cos θ cos φ+γ  2 2  √  2 θ φ+γ  u = − r cos 2 sin 2 √ (6.61) u3 = r sin θ cos φ−γ  2 2  √  4 θ φ−γ  u = r sin 2 sin 2 with γ ∈ (0, 4π). The parameter γ is compliments r, θ, φ as coordinates for the four-dimensional {uµ} space. Accordingly, we introduce a fourth coordinate x4 and extend the mapping from uµ to xi by the diﬀerential relation

dx4 = 2u2 du1 − 2u1 du2 + 2u4 du3 − 2u3 du4 = r cos θ dφ + r dγ. (6.62)

The diﬀerentials dxµ and duν are then related by

[d~x] = A(~u)[d~u] (6.63) where the Jacobian matrix A, given by (C.23), has the determinant

| det A(~u)| = 16r2, (6.64)

32 µ and the metric gµν in u coordinates takes the simple form

gµν = 4r δµν . (6.65) Equation (6.63) deﬁnes a mapping between the four-dimensional {xµ} and {uµ} spaces. This mapping µ µ becomes bijective once it has been speciﬁed at an initial point u (~xa) = ua . We now incorporate the fourth dummy dimension x4 into the path integral (6.56). First note that r is 4 4 4 4 4 4 4 4 independent of x . Writing ∆xk ≡ xk − xk−1, we have (with x0 ≡ xa and arbitrary xN ≡ xb ):

N/2 " N # 1 1 +∞ dx4 +∞ dx4 +∞ X 1 ∆x4 2 N−1 ··· 1 dx4 exp i k δs = 1/2 1/2 1/2 0 2πiδs ˆ ˆ ˆ 2rk δs rb −∞ rN−1 −∞ r1 −∞ k=1 1 N/2 +∞ d(∆x4 ) 1 ∆x4 2 +∞ d(∆x4) 1 ∆x4 2 = N exp i N δs ··· 1 exp i 1 δs 1/2 1/2 2πiδs ˆ 2rN δs ˆ 2r1 δs −∞ rN −∞ r1 N 1/2 +∞ N 1/2 1/2 Y 1 4 1 4 2 Y 1 iπ = d(∆xk) exp i (∆xk) = 2πiδsrk ˆ 2rkδs 2πiδsrk 1/(2rkδs) k=1 −∞ k=1 = 1 (6.66) where we have used (A.22) for the integrals in the third line. By inserting this identity into the integrand of the pseudo-time sliced path integral (6.56) and changing the order of the integrals, we get

r 1 4N/2 +∞ K(N)(x , s ; x , s ) = a dx4 × E b b a a 4/2 2πiδs ˆ 0 rb −∞ d3x +∞ dx4 d3x +∞ dx4 h i N−1 N−1 ··· 1 1 exp i S˜(N)[~x] ˆ 3/2 ˆ 1/2 ˆ 3/2 ˆ 1/2 E rN−1 −∞ rN−1 r1 −∞ r1 2 2N +∞ 4 4 4 ra 1 dxa d ~xN−1 d ~x1 h ˜(N) i = 2 2 ··· 2 exp i SE [~x] (6.67) rb 2πiδs ˆ−∞ ra ˆ rN−1 ˆ r1 with the deﬁnition

N 4 2 N " 2 # ˜(N) ˜(N) X 1 ∆xk X 1 ∆~xk SE [~x] := SE [x] + δs = + Erk + 1 δs. (6.68) 2rk δs 2rk δs k=1 k=1

1 2 3 4 1 3 3 Here ~xk denotes the four-vector (xk, xk, xk, xk), not to be confused with the three-vector xk = (xk, xk, xk), for which we still denote |xk| ≡ rk. With ∆~xk ≡ ~xk − ~xk−1 we can rewrite (6.67) as

2 2N +∞ 4 4 4 (N) ra 1 dxa d (∆~xN ) d (∆~x2) h ˜(N) i KE (xb, sb; xa, sa) = 2 2 ··· 2 exp i SE [~x] rb 2πiδs ˆ−∞ ra ˆ rN−1 ˆ r1 2N +∞ 4 4 4 2 1 dxa d (∆~xN ) d (∆~x2) ra h ˜(N) i = 2 ··· 2 2 exp i SE [~x] . (6.69) 2πiδs ˆ−∞ ra ˆ rN ˆ r2 r1 Since, in the continuum limit, the dominant contributions comes from the continuous paths [5], we can approximate ra/r1 = 1 to ﬁrst order in δs, giving

2N +∞ 4 4 4 (N) 1 dxa d (∆~xN ) d (∆~x2) h ˜(N) i KE (xb, sb; xa, sa) = 2 ··· 2 exp i SE [~x] . (6.70) 2πiδs ˆ−∞ ra ˆ rN ˆ r2 Having incorporated the fourth variable x4 into the path integral, we can now go over to the uµ variables. The 4 ν µ 4 2 integral over xa provides a unique mapping between x and u for each value of xa. With | det A(~uk)| = 16rk, the path integral (6.70) transforms as [4]

33 2N +∞ 4 4 (N) 1 dxa d (∆~uN ) KE (xb, sb; xa, sa) = 2 | det A(~uN )| · · · 2πiδs ˆ−∞ ra ˆ rN 4 d (∆~u2) h ˜(N) i ··· 2 | det A(~u2)| exp i SE [~x] + SJ ˆ r2 2N +∞ 4 1 4 dxa 4 4 h ˜(N) i = d ~uN−1 ··· d ~u1 exp i SE [~x] + SJ . (6.71) 16 2πiδs ˆ−∞ ra ˆ ˆ

The quantity SJ appearing here is called the Jacobian action. It arises, for a generic variable transformation {xµ} → {uν }, from correction terms due to the finite time-slicing and finite coordinate differences ∆uµ (the technical details of which is beyond the scope of this thesis). For our purposes it suffices to note that it can be expressed as [4]

N X SJ = SJ,k(δs) (6.72) k=1 where (summation convention implied) 1 1 iS (δs) = Γ µ ∆uν − iδs δαβ Γ µ Γ λ (6.73) J,k 2 µ ν k 8 µ α λ β with the aﬃne connection ∂uµ ∂2xi Γ µ = (6.74) λκ ∂xi ∂uλ∂uκ i evaluated at ~u = ~uk. Luckily, in the case of our interest, x has no second derivatives with respect to any uλ, uκ so the Jacobian action vanishes identically and (6.71) simpliﬁes to

2N +∞ 4 (N) 1 4 dxa 4 4 h ˜(N) i KE (xb, sb; xa, sa) = d ~uN−1 ··· d ~u1 exp i SE [~x] . (6.75) 16 2πiδs ˆ−∞ ra ˆ ˆ Next we need to write the time-sliced pseudo-action (6.68) in terms of uµ. To lowest order in δs we have

2 µ ν T 2 (∆~x) = gµν ∆u ∆u = [∆~u] [gµν ][∆~u] = 4r(∆~u) . The four-dimensional pseudo-action (6.68) then becomes

N " 2 # N " 2 # ˜(N) X 1 ∆~uk 2 X 1 ∆~uk 1 2 SE [~x] = 4rk + E~uk + 1 δs = 4 − 4(−E/2)~uk δs + ∆s (6.76) 2rk δs 2 δs 2 k=1 k=1 with ∆s ≡ sb − sa = Nδs. Now we see that by letting

m := 4me = 4 (6.77) and p p ω := −E/2me = −E/2 (6.78) we can write (6.76) as

N " 2 # (N) X 1 ∆~uk 1 S˜ [~x] = m − mω2~u2 δs + ∆s = S(N)[~u] + ∆s, (6.79) E 2 δs 2 k osc k=1

(N) where Sosc [~u] is the time-sliced action for the harmonic oscillator. The path integral (6.75) then becomes

34 +∞ 4 2N (N) exp [i∆s] dxa m 4 4 h (N) i KE (xb, sb; xa, sa) = d ~uN−1 ··· d ~u1 exp i Sosc [~u] 16 ˆ−∞ ra 2πiδs ˆ ˆ +∞ 4 exp [i∆s] dxa (N) = Kosc (~ub, sb; ~ua, sa) (6.80) 16 ˆ−∞ ra

(N) where Kosc (~ub, sb; ~ua, sa) is the corresponding time-sliced path integral of the four-dimensional harmonic oscillator. By taking the limit N → ∞ of (6.80) we obtain directly

+∞ 4 exp [i∆s] dxa KE(xb, sb; xa, sa) = Kosc(~ub, sb; ~ua, sa), (6.81) 16 ˆ−∞ ra with the four-dimensional harmonic oscillator propagator given by (B.65) with D = 4, i.e.

mω 2 mω K (~u , s ; ~u , s ) = exp i (~u2 + ~u2) cos(ω∆s) − 2~u · ~u . (6.82) osc b b a a 2πi sin(ω∆s) 2 sin(ω∆s) b a b a

µ The u variables may be expressed directly in terms of r, θ, φ, γ using (6.61). Since xa is ﬁxed in (6.81), (ra, θa, φa) are all ﬁxed and (6.62) gives us

4 dxa = ra dγa. (6.83) Recalling that γ ∈ (0, 4π), we can then write (6.81) as

exp [i∆s] 4π KE(xb, sb; xa, sa) = dγa Kosc(~ub, sb; ~ua, sa) 16 ˆ0 4π 2 exp [i∆s] mω mω 2 2 = dγa exp i (~ub + ~ua) cos(ω∆s) − 2~ub · ~ua 16 ˆ0 2πi sin(ω∆s) 2 sin(ω∆s) (6.84)

2 or, after restoring SI units, substituting ~ua,b = ra,b and rearranging, as

h i e2 i 2 exp ∆s 1 mω ~ 4π0 mω cos(ω∆s) KE(xb, sb; xa, sa) = − exp i (rb + ra) × 16π2 2~ sin2(ω∆s) 2~ sin(ω∆s) 4π mω 2 dγa exp −i ~ub · ~ua . (6.85) ˆ0 2~ sin(ω∆s) As in the two-dimensional problem, we introduce the abbreviations r mω −2m E κ := = e (6.86) 2~ ~2 and

e2/(4π ) rm e4/(4π )2 ν := 0 = e 0 , (6.87) 2ω~ −2~2E and write (6.85) a little more compactly as

κ2 exp [i 2ων∆s] cos(ω∆s) 4π −i 2κ KE(xb, sb; xa, sa) = − 2 2 exp iκ (rb + ra) dγa exp ~ub · ~ua . 16π sin (ω∆s) sin(ω∆s) ˆ0 sin(ω∆s) (6.88)

35 We proceed to evaluate the integral over γa in (6.88). Using (6.61) we ﬁnd, after some algebra and trigonometric rearrangements, r1 γ − γ − β ~u · ~u = (r r + x · x ) cos a b , (6.89) b a 2 b a b a 2

where β is an angle independent of γa, deﬁned by β cos θb+θa sin φb−φa tan := 2 2 . (6.90) 2 θb−θa φb−φa cos 2 cos 2

The integral over γa in (6.88) then becomes 4π 4π " r # −i 2κ −i 2κ 1 γa − γb − β dγa exp ~ub · ~ua = dγa exp (rbra + xb · xa) cos ˆ0 sin(ω∆s) ˆ0 sin(ω∆s) 2 2 (−γ −β)/2+2π " r # b −i 2κ 1 = 2 dγ exp (r r + x · x ) cos γ . (6.91) ˆ b a b a (−γb−β)/2 sin(ω∆s) 2 By using the Jacobi–Anger expansion [6] ∞ X −n exp [iz cos γ] = i Jn(z) exp [inγ] (6.92) n=−∞

where the Jn are the Bessel functions of the ﬁrst kind, we have for arbitrary z and α, ∞ α+2π X α+2π exp [iz cos γ] dγ = i−nJ (z) exp [inγ] dγ = 2πJ (z), (6.93) ˆ n ˆ 0 α n=−∞ α due to the integral in the sum being zero for n 6= 0. The integral on the right-hand side of (6.91) has precisely the form of (6.93), and thus evaluates to ! 4π −i 2κ −2κ r1 dγa exp ~ub · ~ua = 4πJ0 (rbra + xb · xa) . (6.94) ˆ0 sin(ω∆s) sin(ω∆s) 2 Substituting this result into (6.88), the pseudo-propagator now becomes ! κ2 exp [i 2ων∆s] cos(ω∆s) −2κ r1 KE(xb, sb; xa, sa) = − exp iκ (rb + ra) J0 (rbra + xb · xa) . 4π sin2(ω∆s) sin(ω∆s) sin(ω∆s) 2 (6.95) As in the two-dimensional problem, we express the trigonometric functions as 1 cos(ω∆s) = exp [iω∆s] 1 + exp [−i 2ω∆s] (6.96) 2 and 1 sin(ω∆s) = exp [iω∆s] 1 − exp [−i 2ω∆s] , (6.97) 2i whereby we can write the pseudo-propagator (6.95) in the ﬁnal form κ2 exp [−i 2ω∆s(1 − ν)] 1 + exp [−i 2ω∆s] K (x , s ; x , s ) = exp −κ (r + r ) × E b b a a π (1 − exp [−i 2ω∆s])2 1 − exp [−i 2ω∆s] b a ! 4κ exp [−iω∆s] r1 I (r r + x · x ) (6.98) 0 1 − exp [−i 2ω∆s] 2 b a b a

where I0(z) = I0(−z) = J0(iz) is the zeroth order modiﬁed Bessel function of the ﬁrst kind.

36 Having solved the pseudo-time sliced path integral and obtained the pseudo-propagator, the ﬁxed-energy amplitude (5.29) can now be found from

∞ K˜ (xb, xa; E) = ds KE(xb, s; xa, 0) (6.99) ˆ0 with κ2 exp [−i 2ωs(1 − ν)] 1 + exp [−i 2ωs] K (x , s; x , 0) = exp −κ (r + r ) × E b a π (1 − exp [−i 2ωs])2 1 − exp [−i 2ωs] b a ! 4κ exp [−iωs] r1 I (r r + x · x ) . (6.100) 0 1 − exp [−i 2ωs] 2 b a b a

When evaluating the integral (6.99), we have to pass around the singularities of (6.100) in the complex plane. We can invoke the residue theorem to evaluate (6.99) as an integral in the complex plane according to

K˜ (xb, xa; E) = ds KE(xb, s; xa, 0) (6.101) ˆC where the path C may be parametrized as s(σ) = σ − iη with σ ∈ (0, ∞) and η inﬁnitesimal. As for the two-dimensional problem, we convince ourselves that (6.100) vanishes for |s| → ∞ in the fourth quadrant, so that the integral (6.101) may be evaluated along the negative imaginary axis. With the parametrization s(σ) = −iσ, we then have

∞ ds ∞ K˜ (xb, xa; E) = dσ KE(xb, s(σ); xa, 0) = −i dσ KE(xb, −iσ; xa, 0) ˆ0 dσ ˆ0 κ2 ∞ exp [− 2ωσ(1 − ν)] 1 + exp [− 2ωσ] = −i dσ 2 exp −κ (rb + ra) × π ˆ0 (1 − exp [− 2ωσ]) 1 − exp [− 2ωσ] ! 4κ exp [−ωσ] r1 I (r r + x · x ) . (6.102) 0 1 − exp [− 2ωσ] 2 b a b a

After changing the integration variable to 1 d% % := exp [−2ωσ] , dσ = − , (6.103) 2ω % and substituting ω = 2~κ = ~κ , the ﬁxed-energy amplitude (6.102) takes the form m 2me

1 −ν √ r ! ˜ meκ % 1 + % 4κ % 1 K(xb, xa; E) = d% 2 exp −κ (rb + ra) I0 (rbra + xb · xa) . (6.104) iπ~ ˆ0 (1 − %) 1 − % 1 − % 2

The integral in (6.104) converges only for ν < 1, but we can ﬁnd another integral representation that converges for all ν 6= 1, 2,... by changing the integration variable to 1 + % ζ := . (6.105) 1 − % as in the two-dimensional case. We then have ζ − 1 2 % = , d% = dζ, (6.106) ζ + 1 (ζ + 1)2

37 so that 2 1 − % = , (6.107) ζ + 1 √ % 1p = ζ2 − 1, (6.108) 1 − % 2 %−ν 2 (ζ + 1)2 ζ − 1−ν 1 ζ + 1ν d% = dζ = dζ , (6.109) (1 − %)2 (ζ + 1)2 4 ζ + 1 2 ζ − 1 and the ﬁxed-energy amplitude (6.104) becomes

∞ ν r ! meκ ζ + 1 p 2 1 K˜ (xb, xa; E) = dζ exp [−κζ(rb + ra)] I0 2κ ζ − 1 (rbra + xb · xa) . (6.110) 2πi~ ˆ1 ζ − 1 2

As in the two-dimensional case, the integrand has branch cuts extending from −1 to −∞ and from 1 to ∞, with the integral running along the second cut. We again transform this integral into an integral over a contour C encircling the right-hand branch cut in the clockwise sense, as in ﬁgure 1, Section 6.2. The replacement rule is now [4]

∞ dζ π exp [iπν] 1 dζ 1 dζ ν ··· = ν ··· = ν ··· , (6.111) ˆ1 (ζ − 1) sin πν 2πi ˆC (ζ − 1) 1 − exp [−i 2πν] ˆC (ζ − 1) and the ﬁxed-energy amplitude (6.110) ﬁnally becomes ν meκ 1 ζ + 1 K˜ (xb, xa; E) = dζ exp [−κζ(rb + ra)] × 2πi~ 1 − exp [−i 2πν] ˆC ζ − 1 r ! p 1 I 2κ ζ2 − 1 (r r + x · x ) , (6.112) 0 2 b a b a where this integral representation converges for all ν 6= 1, 2,....

38 7 Conclusion

We have given analytical solutions of path integrals for the two– and three-dimensional Hydrogen atom, thereby obtaining integral representations for the corresponding fixed-energy amplitudes. To do so, we had to construct a new path integral formula for an auxiliary quantity called the pseudo-propagator, from which the fixed-energy amplitude is obtained. The new path integral formula incorporates a functional degree of freedom that can be exploited when dealing with singular potentials to bring the path integral to a form that is easier to deal with. For the two- and three-dimensional Hydrogen atoms, the resulting path integrals could then, by means of a coordinate transformation, be transformed into the Gaussian form of a harmonic oscillator, whereby the solution was readily obtained. We now summarize the main results. For the two-dimensional Hydrogen atom, we found integral representations for the fixed-energy amplitude given by (6.47) and (6.55), namely,

1 −ν−1/2 √ r ! me % 1 + % 4κ % 1 K˜ (xb, xa; E) = d% exp −κ (rb + ra) cosh (rbra + xb · xa) , iπ~ ˆ0 1 − % 1 − % 1 − % 2 which converges for ν < 1/2, and m 1 (ζ + 1)ν−1/2 K˜ (x , x ; E) = e dζ exp [−κζ(r + r )] × b a ν+1/2 b a iπ~ 1 + exp [−i 2πν] ˆC (ζ − 1) r ! p 1 cosh 2κ ζ2 − 1 (r r + x · x ) , 2 b a b a which converges for all ν 6= 1/2, 3/2,.... The integration contour C encircles the branch cut of the integrand from 1 to ∞ in the clockwise sense (see fig. 1, Section 6.2), and the quantities ν and κ are defined by r rm e4/(4π )2 −2m E ν := e 0 and κ := e . −2~2E ~2 For the three-dimensional Hydrogen atom, we found integral representations for the fixed-energy amplitude given by (6.104) and (6.112), namely,

1 −ν √ r ! ˜ meκ % 1 + % 4κ % 1 K(xb, xa; E) = d% 2 exp −κ (rb + ra) I0 (rbra + xb · xa) , iπ~ ˆ0 (1 − %) 1 − % 1 − % 2 which converges for ν < 1, and ν meκ 1 ζ + 1 K˜ (xb, xa; E) = dζ exp [−κζ(rb + ra)] × 2πi~ 1 − exp [−i 2πν] ˆC ζ − 1 r ! p 1 I 2κ ζ2 − 1 (r r + x · x ) , 0 2 b a b a which converges for all ν 6= 1, 2,..., and where I0(z) is the zeroth order modified Bessel function of the first kind. The contour C is the same as for the two-dimensional case, as well as the quantities ν and κ. The success in calculating the path integrals demonstrates the power of the new path integral formulas, developed in Section 5, involving the regulating functions fl and fr. In fact, this method has made it possible to solve a large class of previously unsolvable Feynman path integrals [4]. Having obtained the fixed-energy amplitude of the Hydrogen atom, the next step is to extract from it the various physical quantities. The integral representations are enough to obtain the well known energy eigenvalues and eigenfunctions. This procedure is done in the Duru–Kleinert article [2] as well as in the book [4] by Kleinert. To take the study of the Hydrogen atom one step further, we may take into account relativistic effects. The corresponding relativistic path integral is solved in Kleinert’s 1996 article [7].

39 A Gaussian Integrals

When solving some standard path integrals, we often encounter D-dimensional integrals of the form i dDq exp (αq2 + q0 · q) ˆ ~ where α 6= 0 is a real parameter not depending on q, and the integral is over the whole q-space. To solve this integral, we begin by completing the square in the exponential as

q0 q0 2 q0 2 αq2 + q0 · q = α q2 + 2 · q = α q + − α . (A.1) 2α 2α 2α We then have " # " # i i q0 2 i q0 2 dDq exp (αq2 + q0 · q) = exp − α dDq exp α q + ˆ ~ ~ 2α ˆ ~ 2α " # i q0 2 i = exp − α dDq exp αq2 ~ 2α ˆ ~ " # D i q0 2 +∞ i = exp − α dq exp αq2 . (A.2) ~ 2α ˆ−∞ ~

We ﬁrst treat the case α < 0. Then the integral on the right-hand side of (A.2) can be written as

+∞ i ∞ i 1/2 ∞ dq exp αq2 = 2 dq exp − |α|q2 = 2 ~ dξ exp −iξ2 . (A.3) ˆ−∞ ~ ˆ0 ~ |α| ˆ0 We can write the integral over ξ in (A.3) as an integral in the complex plane,

∞ dξ exp −iξ2 = dz exp −iz2 , (A.4) ˆ0 ˆC over the contour C : z(ξ) = ξ, 0 ≤ ξ < ∞. By invoking the residue theorem, we can integrate along a diﬀerent contour according to

∞ " # dξ exp −iξ2 = lim dz exp −iz2 + dz exp −iz2 , (A.5) R→∞ ˆ0 ˆC1(R) ˆC2(R)

where C1 and C2 are given by

C1 : z(ξ) = exp [−iπ/4] ξ, 0 ≤ ξ < R (A.6) and

C2 : z(θ) = R exp [iθ] , −π/4 ≤ θ ≤ 0, (A.7)

respectively. The integral along C2 becomes 0 dz dz exp −iz2 = dθ exp −iz(θ)2 . (A.8) ˆC2(R) ˆ−π/4 dθ

dz The derivative dθ is linear in R, whereas exp −iz(θ)2 = exp −iR2(cos 2θ + i sin 2θ) = exp R2 sin 2θ exp −iR2 cos 2θ . (A.9)

40 dz 2 For θ ∈ (−π/4, 0) we have sin 2θ < 0. Therefore the product dθ exp −iz(θ) → 0 as R → ∞ so that the integral (A.8) vanishes, and (A.5) reduces to

∞ R dz dξ exp −iξ2 = lim dz exp −iz2 = lim dξ exp −iz(ξ)2 R→∞ R→∞ ˆ0 ˆC1(R) ˆ0 dξ ∞ ∞ = dξ exp [−iπ/4] exp −i(−iξ2) = exp [−iπ/4] dξ exp −ξ2 ˆ0 ˆ0 1 +∞ = exp [−iπ/4] dξ exp −ξ2 . (A.10) 2 ˆ−∞ We now use the well known result for the Gaussian integral, ∞ √ dξ exp −ξ2 = π, (A.11) ˆ−∞ giving

∞ 1 √ 1 π 1/2 dξ exp −iξ2 = exp [−iπ/4] π = , (A.12) ˆ0 2 2 i √ where we use the branch i ≡ exp [iπ/4]. Using this result, (A.3) becomes

+∞ i 1/2 1 π 1/2 π 1/2 iπ 1/2 dq exp αq2 = 2 ~ = ~ = ~ , (A.13) ˆ−∞ ~ |α| 2 i i|α| α and (A.2) becomes " # i iπ D/2 i q0 2 dDq exp (αq2 + q0 · q) = ~ exp − α , (A.14) ˆ ~ α ~ 2α valid for α < 0. Next, we treat the case α > 0. Then the integral on the right-hand side of (A.2) can be written as

+∞ i 1/2 ∞ dq exp αq2 = 2 ~ dξ exp iξ2 . (A.15) ˆ−∞ ~ α ˆ0 We can again invoke the residue theorem and integrate according to

∞ " # dξ exp iξ2 = lim dz exp iz2 + dz exp iz2 , (A.16) R→∞ ˆ0 ˆC1(R) ˆC2(R) where this time we take C1 and C2 to be

C1 : z(ξ) = exp [iπ/4] ξ, 0 ≤ ξ < R (A.17) and

C2 : z(θ) = R exp [iθ] , π/4 ≥ θ ≥ 0, (A.18) respectively. The integral along C2 becomes

0 dz dz exp iz2 = dθ exp iz(θ)2 . (A.19) ˆC2(R) ˆπ/4 dθ

41 dz The derivative dθ is linear in R, whereas

exp iz(θ)2 = exp iR2(cos 2θ + i sin 2θ) = exp −R2 sin 2θ exp iR2 cos 2θ . (A.20)

dz 2 For θ ∈ (0, π/4) we have sin 2θ > 0. Therefore the product dθ exp iz(θ) → 0 as R → ∞ so that the integral (A.19) vanishes, and (A.16) reduces to

∞ R dz dξ exp iξ2 = lim dz exp iz2 = lim dξ exp iz(ξ)2 R→∞ R→∞ ˆ0 ˆC1(R) ˆ0 dξ ∞ ∞ = dξ exp [iπ/4] exp i(iξ2) = exp [iπ/4] dξ exp −ξ2 ˆ0 ˆ0 1 +∞ 1 √ 1 = exp [iπ/4] dξ exp −ξ2 = exp [iπ/4] π = (iπ)1/2 , (A.21) 2 ˆ−∞ 2 2 √ again using the branch i ≡ exp [iπ/4]. Using this result, (A.15) becomes

+∞ i 1/2 1 iπ 1/2 dq exp αq2 = 2 ~ (iπ)1/2 = ~ , (A.22) ˆ−∞ ~ α 2 α and (A.2) becomes " # i iπ D/2 i q0 2 dDq exp (αq2 + q0 · q) = ~ exp − α , (A.23) ˆ ~ α ~ 2α

valid for α > 0. This is the same as the formula (A.14) and is therefore valid for both α < 0 and α > 0.

42 B Exact Solutions for some Simple Path Integrals

B.1 The Free Particle

We will now derive the propagator for a free particle in D dimensions by solving the conﬁguration space path integral

x(tb)=xb i K(xb, tb; xa, ta) = D x(t) exp S[x(t); ta, tb] (B.1) ˆx(ta)=xa ~ with the action integral

tb 1 2 S[x(t); ta, tb] = dt mx˙ . (B.2) ˆta 2 The time-sliced form of (B.1) reads

DN/2 (N) m D D i (N) K (xb, tb; xa, ta) = d xN−1 ··· d x1 exp S [x] (B.3) 2πi~δt ˆ ˆ ~ with the time-sliced action N 2 N X 1 ∆xk X S(N)[x] = m δt = a (x − x )2 (B.4) 2 δt k k−1 k=1 k=1 where m a := . (B.5) 2δt Letting

m D/2 a D/2 N := = (B.6) 2πi~δt iπ~ we can write (B.3) as

" N # i X K(N)(x , t ; x , t ) = N N dDx ··· dDx exp a (x − x )2 b b a a ˆ N−1 ˆ 1 k k−1 ~ k=1 D i 2 = d xN−1 N exp a(xN − xN−1) ··· ˆ ~ D i 2 i 2 ··· d x1 N exp a(x2 − x1) N exp a(x1 − x0) . (B.7) ˆ ~ ~

The integral over x1 in (B.7) is a special case of the following integral in which we have replaced a in the last 0 exponential with an arbitrary constant b, and x2 with x . We solve this case instead, for later convenience: D i 0 2 i 2 d x N exp a(x − x) N exp b(x − x0) = ˆ ~ ~ 2 D i 02 2 0 2 2 = N d x exp a(x + x − 2x · x) + b(x + x0 − 2x · x0) ˆ ~ 2 i 02 2 D i 2 0 = N exp ax + bx0 d x exp (a + b)x − 2(ax + bx0) · x . (B.8) ~ ˆ ~

43 Using (A.23), the integral on the right-hand side evaluates to

D/2 " 0 2# D i 2 0 iπ~ i −2(ax + bx0) d x exp (a + b)x − 2(ax + bx0) · x = exp − (a + b) ˆ ~ a + b ~ 2(a + b) iπ D/2 i (ax0 + bx )2 = ~ exp − 0 (B.9) a + b ~ a + b so that (B.8) becomes D i 0 2 i 2 d x N exp a(x − x) N exp b(x − x0) = ˆ ~ ~ D/2 0 2 2 i 02 2 iπ~ i (ax + bx0) = N exp ax + bx0 exp − ~ a + b ~ a + b D/2 2 iπ~ i 1 02 2 0 2 = N exp (a + b)(ax + bx0) − (ax + bx0) a + b ~ a + b D/2 D/2 a iπ~ i 1 2 02 2 2 2 02 2 2 0 = N exp (a + ab)x + (ab + b )x0 − (a x + b x0 + 2abx · x0) iπ~ a + b ~ a + b D/2 a i 1 02 2 0 = N exp abx + abx0 − 2abx · x0 a + b ~ a + b D/2 a i ab 0 2 = N exp (x − x0) . (B.10) a + b ~ a + b

0 Using this result with b = a and x = x2, the integral over x1 in (B.7) becomes

D/2 D i 2 i 2 1 i a 2 d x1 N exp a(x2 − x1) N exp a(x1 − x0) = N exp (x2 − x0) (B.11) ˆ ~ ~ 2 ~ 2

and using this result, the integral over x2 in (B.7) becomes D i 2 D i 2 i 2 d x2 N exp a(x3 − x2) d x1 N exp a(x2 − x1) N exp a(x1 − x0) ˆ ~ ˆ ~ ~ D/2 1 D i 2 i a 2 = d x2 N exp a(x3 − x2) N exp (x2 − x0) 2 ˆ ~ ~ 2 D/2 D/2 1 a i aa/2 2 = N exp (x3 − x0) 2 a + a/2 ~ a + a/2 D/2 1 i a 2 = N exp (x3 − x0) (B.12) 3 ~ 3 where we have again used (B.10). Thus for n − 1 = 1, 2 we have D i 2 D i 2 i 2 d xn−1 N exp a(xn − xn−1) ··· d x1 N exp a(x2 − x1) N exp a(x1 − x0) = ˆ ~ ˆ ~ ~ D/2 1 i a 2 = N exp (xn − x0) . (B.13) n ~ n

44 Suppose (B.13) holds for n − 1 = 1, 2, . . . , k − 1 for some k. Then by integrating k times we get D i 2 D i 2 d xk N exp a(xk+1 − xk) d xk−1 N exp a(xk − xk−1) ··· (B.14) ˆ ~ ˆ ~ D i 2 i 2 ··· d x1 N exp a(x2 − x1) N exp a(x1 − x0) = ˆ ~ ~ D/2 D i 2 1 i a 2 = d xk N exp a(xk+1 − xk) N exp (xk − x0) ˆ ~ k ~ k D/2 D/2 1 a i aa/k 2 = N exp (xk+1 − x0) k a + a/k ~ a + a/k D/2 1 i a 2 = N exp (xk+1 − x0) (B.15) k + 1 ~ k + 1 where we have used (B.10) once again. Thus (B.13) holds for n − 1 = k as well, and by induction must hold for all n − 1 = 1, 2,.... After N − 1 integrations we therefore get D/2 (N) 1 i a 2 K (xb, tb; xa, ta) = N exp (xN − x0) N ~ N D/2 D/2 1 m i m/2δt 2 = exp (xN − x0) N 2πi~δt ~ N m D/2 i m (x − x )2 = exp N 0 2πi~Nδt ~ 2 Nδt m D/2 i m (x − x )2 = exp b a (B.16) 2πi~∆t ~ 2 ∆t with ∆t ≡ tb − ta = Nδt. Note that this result is independent of the number of time slices. Thus the D-dimensional free-particle propagator is given by

D/2 2 m i m (xb − xa) K(xb, tb; xa, ta) = exp . (B.17) 2πi~∆t ~ 2 ∆t

B.2 The Harmonic Oscillator

We will now derive the propagator for a particle in D dimensions subjected to a harmonic oscillator potential 1 V (x) = mω2x2 (B.18) 2 by solving the conﬁguration space path integral

x(tb)=xb i K(xb, tb; xa, ta) = D x(t) exp S[x(t); ta, tb] (B.19) ˆx(ta)=xa ~ with the action integral

tb 1 2 1 2 2 S[x(t); ta, tb] = dt mx˙ − mω x . (B.20) ˆta 2 2 The time-sliced form of (B.19) reads

DN/2 (N) m D D i (N) K (xb, tb; xa, ta) = d xN−1 ··· d x1 exp S [x] . (B.21) 2πi~δt ˆ ˆ ~

45 Instead of using the expression (4.35) for the time-sliced action, we write it as N " 2 2 2 # X 1 ∆xk 1 x + x S(N)[x] = m − mω2 k k−1 δt, (B.22) 2 δt 2 2 k=1 1 which diﬀers from (4.35) in that we have replaced V (x(tk)) with the average 2 V (x(tk)) + V (x(tk−1)) , the large N limit still being (B.20). We can then rewrite it as N X m x2 + x2 S(N)[x] = (x − x )2 − ω2δt2 k k−1 δt 2δt2 k k−1 2 k=1 N X m 1 = 1 − ω2δt2 (x2 + x2 ) − 2x · x 2δt 2 k k−1 k k−1 k=1 N X 2 2 = a1(xk + xk−1) − 2b1xk · xk−1 (B.23) k=1 with m 1 m a := 1 − ω2δt2 and b := . (B.24) 1 2δt 2 1 2δt Also letting m D/2 N1 := (B.25) 2πi~δt the time-sliced path integral (B.21) becomes " N # i X K(N)(x , t ; x , t ) = N N dDx ··· dDx exp a (x2 + x2 ) − 2b x · x b b a a 1 ˆ N−1 ˆ 1 1 k k−1 1 k k−1 ~ k=1 N Y i = dDx ··· dDx N exp a (x2 + x2 ) − 2b x · x ˆ N−1 ˆ 1 1 1 k k−1 1 k k−1 k=1 ~ D i 2 2 = d xN−1 N1 exp a1(xN + xN−1) − 2b1xN · xN−1 ··· ˆ ~ D i 2 2 ··· d x1 N1 exp a1(x2 + x1) − 2b1x2 · x1 × ˆ ~ i 2 2 N1 exp a1(x1 + x0) − 2b1x1 · x0 . (B.26) ~

The integral over x1 in (B.26) is a special case of the following integral in which we have replaced a1, b1 in 0 the last exponential with arbitrary nonzero constants a, b, and x2 with x . We solve this case instead, for later convenience: D i 02 2 0 i 2 2 d x exp a1(x + x ) − 2b1x · x exp a(x + x0) − 2bx · x0 = ˆ ~ ~ i 02 2 D i 2 0 = exp a1x + ax0 d x exp (a1 + a)x − 2(b1x + bx0) · x . (B.27) ~ ˆ ~ Using (A.23), the integral on the right-hand side evaluates to

D/2 " 0 2# D i 2 0 iπ~ i −2(b1x + bx0) d x exp (a1 + a)x − 2(b1x + bx0) · x = exp − (a1 + a) ˆ ~ a1 + a ~ 2(a1 + a) iπ D/2 i (b x0 + bx )2 = ~ exp − 1 0 (B.28) a1 + a ~ a1 + a

46 so that (B.27) becomes D i 02 2 0 i 2 2 d x exp a1(x + x ) − 2b1x · x exp a(x + x0) − 2bx · x0 = ˆ ~ ~ D/2 0 2 i 02 2 iπ~ i (b1x + bx0) = exp a1x + ax0 exp − ~ a1 + a ~ a1 + a D/2 iπ~ i 1 02 2 0 2 = exp (a1 + a)(a1x + ax0) − (b1x + bx0) a1 + a ~ a1 + a D/2 iπ~ i 1 02 2 2 02 2 2 0 = exp (a1 + a)(a1x + ax0) − b1x − b x0 − 2b1bx · x0 a1 + a ~ a1 + a D/2 iπ~ i 1 2 02 2 2 0 = exp (a1 + a)a1 − b1 x + (a1 + a)a − b x0 − 2b1bx · x0 a1 + a ~ a1 + a D/2 2 2 2 2 iπ~ i a1 − b1 + a1a 02 a − b + a1a 2 b1b 0 = exp x + x0 − 2 x · x0 . (B.29) a1 + a ~ a1 + a a1 + a a1 + a

0 Using this result (with a = a1, b = b1, x = x2) the integral over x1 in (B.26) becomes D i 2 2 i 2 2 d x1 N1 exp a1(x2 + x1) − 2b1x2 · x1 N1 exp a1(x1 + x0) − 2b1x1 · x0 = ˆ ~ ~ D/2 2 2 2 2 2 2 iπ~ i 2a1 − b1 2 2a1 − b1 2 b1 = N1 exp x2 + x0 − 2 x2 · x0 2a1 ~ 2a1 2a1 2a1 i 2 2 = N2 exp a2(x2 + x0) − 2b2x2 · x0 (B.30) ~ with

2 2 2 D/2 2a1 − b1 b1 2 iπ~ a2 := , b2 := and N2 := N1 . (B.31) 2a1 2a1 2a1

Using (B.30) the integral over x2 in (B.26) then becomes D i 2 2 d x2 N1 exp a1(x3 + x2) − 2b1x3 · x2 × ˆ ~ D i 2 2 i 2 2 d x1 N1 exp a1(x2 + x1) − 2b1x2 · x1 N1 exp a1(x1 + x0) − 2b1x1 · x0 = ˆ ~ ~ D i 2 2 i 2 2 = d x2 N1 exp a1(x3 + x2) − 2b1x3 · x2 N2 exp a2(x2 + x0) − 2b2x2 · x0 ˆ ~ ~ D/2 2 2 2 2 iπ~ i a1 − b1 + a1a2 2 a2 − b2 + a1a2 2 b1b2 = N2N1 exp x3 + x0 − 2 x3 · x0 . (B.32) a1 + a2 ~ a1 + a2 a1 + a2 a1 + a2

0 where in the last step we have used the result (B.29) with x = x3, x = x2, a = a2 and b = b2. From (B.31) we see that a2 = a1 − b2 and

2 2 2 2 2 2 2 a2 − b2 = (a1 − b2) − b2 = a1 − 2a1b2 = a1 − b1 (B.33)

so that (B.32) becomes

47 D i 2 2 d x2 N1 exp a1(x3 + x2) − 2b1x3 · x2 × ˆ ~ D i 2 2 i 2 2 d x1 N1 exp a1(x2 + x1) − 2b1x2 · x1 N1 exp a1(x1 + x0) − 2b1x1 · x0 = ˆ ~ ~ i 2 2 = N3 exp a3(x3 + x0) − 2b3x3 · x0 (B.34) ~ with

2 2 D/2 a1 − b1 + a1a2 b1b2 iπ~ a3 := , b3 := and N3 := N2N1 . (B.35) a1 + a2 a1 + a2 a1 + a2 Thus for n − 1 = 1, 2 we have D i 2 2 d xn−1 N1 exp a1(xn + xn−1) − 2b1xn · xn−1 ··· ˆ ~ D i 2 2 i 2 2 ··· d x1 N1 exp a1(x2 + x1) − 2b1x2 · x1 N1 exp a1(x1 + x0) − 2b1x1 · x0 = ˆ ~ ~ i 2 2 = Nn exp an(xn + x0) − 2bnxn · x0 (B.36) ~ with

2 2 2 2 a1 − b1 + a1an−1 an−1 − bn−1 + a1an−1 an = = , (B.37) a1 + an−1 a1 + an−1 b1bn−1 bn = , (B.38) a1 + an−1 D/2 iπ~ Nn = Nn−1N1 . (B.39) a1 + an−1

Suppose (B.36) holds for n − 1 = 1, 2, . . . , k − 1 for some k. Then by integrating k times we get D i 2 2 d xk N1 exp a1(xk+1 + xk) − 2b1xk+1 · xk ··· ˆ ~ D i 2 2 i 2 2 ··· d x1 N1 exp a1(x2 + x1) − 2b1x2 · x1 N1 exp a1(x1 + x0) − 2b1x1 · x0 = ˆ ~ ~ D i 2 2 i 2 2 = d xk N1 exp a1(xk+1 + xk) − 2b1xk+1 · xk Nk exp ak(xk + x0) − 2bkxk · x0 ˆ ~ ~ D/2 2 2 2 2 iπ~ i a1 − b1 + a1ak 2 ak − bk + a1ak 2 b1bk = NkN1 exp xk+1 + x0 − 2 xk+1 · x0 (B.40) a1 + ak ~ a1 + ak a1 + ak a1 + ak

where in the last step we have again used (B.29). Now, since ak, bk by assumption satisfy (B.37)–(B.38), we have

2 2 2 2 2 2 (a1 − b1 + a1ak−1) − (b1bk−1) ak − bk = 2 (a1 + ak−1) 4 4 2 2 2 2 3 2 2 2 2 2 a1 + b1 − 2a1b1 + a1ak−1 + 2a1ak−1 − 2a1ak−1b1 + b1(a1 − b1 − ak−1) = 2 (B.41) (a1 + ak−1)

48 2 2 2 2 where in the second line we have substituted −bk−1 = a1 − b1 − ak−1 from (B.37). After some algebra, this reduces to

2 2 2 2 ak − bk = a1 − b1 (B.42)

and (B.40) becomes D i 2 2 d xk N1 exp a1(xk+1 + xk) − 2b1xk+1 · xk ··· ˆ ~ D i 2 2 i 2 2 ··· d x1 N1 exp a1(x2 + x1) − 2b1x2 · x1 N1 exp a1(x1 + x0) − 2b1x1 · x0 = ˆ ~ ~ D/2 2 2 iπ~ i a1 − b1 + a1ak 2 2 b1bk = NkN1 exp (xk+1 + x0) − 2 xk+1 · x0 a1 + ak ~ a1 + ak a1 + ak i 2 2 = Nk+1 exp ak+1(xk+1 + x0) − 2bk+1xk+1 · x0 (B.43) ~ with

2 2 2 2 D/2 a1 − b1 + a1ak ak − bk + a1ak b1bk iπ~ ak+1 = = , bk+1 = , Nk+1 = NkN1 , (B.44) a1 + ak a1 + ak a1 + ak a1 + ak

so that (B.36) holds for n − 1 = k as well. By induction, then, (B.36) must hold for all n − 1 = 1, 2,.... After N − 1 integrations, we therefore have (N) i 2 2 K (xb, tb; xa, ta) = NN exp aN (xN + x0) − 2bN xN · x0 (B.45) ~

where we need to ﬁnd aN , bN , NN recursively from (B.37)–(B.39). From (B.37) we have, for arbitrary n, q 2 2 2 an = bn − (b1 − a1) (B.46)

and the recursion formula for bn may then be written

b1bn−1 b1bn−1 b1 bn = = q = r (B.47) a1 + an−1 2 2 2 2 2 a + b − (b − a ) a1 b1−a1 1 n−1 1 1 + 1 − 2 bn−1 bn−1 or

s 2 2 ! 1 1 a1 b1 − a1 = + 1 − 2 . (B.48) bn b1 bn−1 bn−1

By introducing an auxiliary frequencyω ˜ deﬁned such that ωδt˜ ωδt sin := , (B.49) 2 2

we can express a1 as

m (ωδt)2 m ωδt˜ m a = 1 − = 1 − 2 sin2 = cos(˜ωδt). (B.50) 1 2δt 2 2δt 2 2δt

49 The relation (B.48) then becomes  s  m m2 m2 2 1 1 2δt cos(˜ωδt) 4δt2 − 4δt2 cos (˜ωδt) = m  + 1 − 2  bn 2δt bn−1 bn−1 s cos(˜ωδt) 2δt m2 sin2(˜ωδt) = + 1 − 2 2 . (B.51) bn−1 m 4δt bn−1

For notational convenience, we now introduce reduced quantities

bn 2δt βn := = bn. (B.52) b1 m

Then the recursion formula for βn reads s 1 cos(˜ωδt) sin2(˜ωδt) = + 1 − 2 (B.53) βn βn−1 βn−1

with β1 = 1. For n = 2, 3 we get 1 q sin(2˜ωδt) = cos(˜ωδt) + 1 − sin2(˜ωδt) = (B.54) β2 sin(˜ωδt) and

s 2 1 sin(2˜ωδt) 2 sin (2˜ωδt) sin(3˜ωδt) = cos(˜ωδt) + 1 − sin (˜ωδt) 2 = , (B.55) β3 sin(˜ωδt) sin (˜ωδt) sin(˜ωδt) so in general we expect that

1 sin(nωδt˜ ) = . (B.56) βn sin(˜ωδt)

Suppose (B.56) holds for all n = 1, 2, . . . , k for some k. Then

s 2 1 sin(kωδt˜ ) 2 sin (kωδt˜ ) sin((k + 1)˜ωδt) = cos(˜ωδt) + 1 − sin (˜ωδt) 2 = (B.57) βk+1 sin(˜ωδt) sin (˜ωδt) sin(˜ωδt) so that (B.56) holds for k + 1 as well. By induction it must hold for all n = 1, 2,... and hence solves the recursion relation (B.53). Thus we have

m m sin(˜ωδt) b = β = . (B.58) n 2δt n 2δt sin(nωδt˜ )

From (B.46) and (B.50) we then get s q 2 2 2 2 2 2 2 m sin (˜ωδt) m m 2 an = b − (b − a ) = − − cos (˜ωδt) n 1 1 2δt sin2(nωδt˜ ) 2δt 2δt s m sin2(˜ωδt) m cos(nωδt˜ ) = − sin2(˜ωδt) = sin(˜ωδt) . (B.59) 2δt sin2(nωδt˜ ) 2δt sin(nωδt˜ )

50 We have now determined the an and the bn. Finally, we need to determine the normalisation constant (B.39). For n = 2, 3 we get D/2 D/2 D/2 iπ~ 2 iπ~ sin(˜ωδt) N2 = N1N1 = N1 m = N1 (B.60) a1 + a1 2 2δt cos(˜ωδt) sin(2˜ωδt) and  D/2 iπ D/2 sin(˜ωδt) D/2 iπ N = N N ~ = N 2 ~ 3 2 1 1  cos(2˜ωδt)  a1 + a2 sin(2˜ωδt) m 2δt cos(˜ωδt) + sin(˜ωδt) sin(2˜ωδt) sin(˜ωδt) D/2 = N (B.61) 1 sin(3˜ωδt) so we expect the general result

sin(˜ωδt) D/2 N = N . (B.62) n 1 sin(nωδt˜ ) Suppose (B.62) holds for all n = 1, 2, . . . , k for some k. Then

 D/2 iπ D/2 sin(˜ωδt) D/2 iπ N = N N ~ = N 2 ~ k+1 k 1 1  cos(kωδt˜ )  a1 + ak sin(kωδt˜ ) m 2δt cos(˜ωδt) + sin(˜ωδt) sin(kωδt˜ ) sin(˜ωδt) D/2 = N , (B.63) 1 sin((k + 1)˜ωδt) so that (B.62) holds for k + 1 as well. By induction it must hold for all n = 1, 2,... and hence solves the recursion relation (B.39). Having obtained the constants (B.59), (B.58) and (B.62), we now plug them into (B.45) and get

(N) K (xb, tb; xa, ta) = i 2 2 = NN exp aN (xN + x0) − 2bN xN · x0 ~ D/2 sin(˜ωδt) i m cos(Nωδt˜ ) 2 2 m sin(˜ωδt) = N1 exp sin(˜ωδt) (xN + x0) − 2 xN · x0 sin(Nωδt˜ ) ~ 2δt sin(Nωδt˜ ) 2δt sin(Nωδt˜ ) D/2 D/2 m sin(˜ωδt) i m sin(˜ωδt) 2 2 = exp (xN + x0) cos(Nωδt˜ ) − 2xN · x0 2πi~δt sin(Nωδt˜ ) ~ 2δt sin(Nωδt˜ ) D/2 m ω˜ sin(˜ωδt) i mω˜ sin(˜ωδt) 2 2 = exp (xN + x0) cos(Nωδt˜ ) − 2xN · x0 2πi~ sin(Nωδt˜ ) ωδt˜ ~ 2 sin(Nωδt˜ ) ωδt˜ D/2 m ω˜ sin(˜ωδt) i mω˜ sin(˜ωδt) 2 2 = exp (xb + xa) cos(˜ω∆t) − 2xb · xa (B.64) 2πi~ sin(˜ω∆t) ωδt˜ ~ 2 sin(˜ω∆t) ωδt˜

sin(˜ωδt) with ∆t ≡ tb − ta = Nδt. In taking the limit N → ∞, δt → 0, we have ωδt˜ → 1 and the auxiliary frequencyω ˜ deﬁned by (B.49) simply becomes the oscillator frequency ω. We then ﬁnally obtain the following expression for the propagator of the D-dimensional harmonic oscillator: D/2 mω i mω 2 2 K(xb, tb; xa, ta) = exp (xb + xa) cos(ω∆t) − 2xb · xa . (B.65) 2πi~ sin(ω∆t) ~ 2 sin(ω∆t)

51 C Square-root Coordinates for the 3-D H-atom

For the solution of the path integral for the three-dimensional Hydrogen atom, we introduced a mapping from a four-dimensional {uµ} space to the three-dimensional {xi} space by

xi = z†σiz (C.1) with

1 2 z1 u + iu z = := 3 4 (C.2) z2 u + iu and the Pauli spin matrices

0 1 0 −i 1 0 σ1 = σ2 = σ3 = . (C.3) 1 0 i 0 0 −1

Explicitly, the transformation (C.1) reads

 1  † 1   ∗ ∗   ∗   1 3 2 4  x z σ z z1 z2 + z2 z1 2 Re(z1 z2) 2u u + 2u u 2 † 2 ∗ ∗ ∗ 1 4 2 3 x  = z σ z = −iz1 z2 + iz2 z1 =  2 Im(z1 z2)  =  2u u − 2u u  (C.4) 3 † 3 ∗ ∗ 2 2 1 2 2 2 3 2 4 2 x z σ z z1 z1 − z2 z2 |z1| − |z2| (u ) + (u ) − (u ) − (u ) and the relations between the diﬀerentials are du1 dx1 2u3 2u4 2u1 2u2  du2 dx2 = 2u4 −2u3 −2u2 2u1   . (C.5)     du3 dx3 2u1 2u2 −2u3 −2u4   du4

The transformation (C.1) has been chosen so that r = ~u2. Indeed, we have

2 X i 2 ∗ ∗ 2 ∗ ∗ 2 ∗ ∗ 2 2 2 2 r = (x ) = (z1 z2 + z2 z1) + (−iz1 z2 + iz2 z1) + (z1 z1 − z2 z2) = (|z1| + |z2| ) (C.6) i so that

2 2 1 2 2 2 3 2 4 2 2 r = |z1| + |z2| = (u ) + (u ) + (u ) + (u ) ≡ ~u . (C.7)

The mapping (C.4) is obviously not invertible, so the inverse relationship will be multivalued. To ﬁnd an inverse relationship, we ﬁrst express the xi in terms of spherical coordinates

 1  x = r sin θ cos φ x2 = r sin θ sin φ . (C.8)  x3 = r cos θ

Next, by writing

z1 = |z1| exp [iθ1] , z2 = |z2| exp [iθ2] (C.9) we have, from (C.4) and (C.7),

( 2 2 ( 2 1 2 |z1| + |z2| = r |z1| = 2 r(1 + cos θ) = r cos (θ/2) 2 2 or 2 1 2 (C.10) |z1| − |z2| = r cos θ |z2| = 2 r(1 − cos θ) = r sin (θ/2)

52 giving √ z = r cos(θ/2) exp [iθ ] 1 √ 1 . (C.11) z2 = r sin(θ/2) exp [iθ2] To ﬁnd the phase angles, we calculate 1 z∗z = r cos(θ/2) sin(θ/2) exp [i(θ − θ )] = r sin θ exp [i(θ − θ )] (C.12) 1 2 2 1 2 2 1 and use, from (C.4),

1 ∗ r sin θ cos φ = x = 2 Re(z1 z2) = r sin θ cos(θ2 − θ1) (C.13) and

2 ∗ r sin θ sin φ = x = 2 Im(z1 z2) = r sin θ sin(θ2 − θ1), (C.14) to ﬁnd that

θ2 − θ1 = φ + 2πn. (C.15) Letting φ + γ θ = − , with γ ∈ , (C.16) 1 2 R then gives us φ + γ φ − γ θ = − + φ + 2πn = + 2πn. (C.17) 2 2 2 Thus √ z = r cos(θ/2) exp [−i(φ + γ)/2] 1 √ (C.18) z2 = r sin(θ/2) exp [i(φ − γ)/2] or  √ u1 = r cos θ cos φ+γ  2 2  √  2 θ φ+γ  u = − r cos 2 sin 2 √ . (C.19) u3 = r sin θ cos φ−γ  2 2  √  4 θ φ−γ  u = r sin 2 sin 2 For ﬁxed (r, θ, φ), this describes a curve in {uµ} space parametrized by γ. Each point on this curve maps to the same point (r, θ, φ) in {xi} space. Note that the curve is closed, since uµ(γ + 4π) = uµ(γ). Thus we can restrict γ to the interval [0, 4π). We can then interpret γ as an additional angle that compliments r, θ, φ as coordinates for the four-dimensional {uµ} space. Accordingly, we introduce a fourth coordinate x4 and extend the mapping from uµ to xi by the diﬀerential relation dx4 = 2u2 du1 − 2u1 du2 + 2u4 du3 − 2u3 du4 (C.20) so that dx1 2u3 2u4 2u1 2u2  du1 dx2 2u4 −2u3 −2u2 2u1  du2   =     (C.21) dx3 2u1 2u2 −2u3 −2u4 du3 dx4 2u2 −2u1 2u4 −2u3 du4

53 or

[d~x] = A(~u)[d~u] (C.22)

with the Jacobian matrix 2u3 2u4 2u1 2u2  2u4 −2u3 −2u2 2u1  A(~u) =   . (C.23) 2u1 2u2 −2u3 −2u4 2u2 −2u1 2u4 −2u3

The relation (C.20) has been chosen such that the metric in uµ coordinates,

∂xα ∂xβ g = δ (summation convention implied), (C.24) µν ∂uµ ∂uν αβ takes the simple form

u3 u4 u1 u2  u3 u4 u1 u2  ~u2 0 0 0  4 3 2 1 4 3 2 1 2 T u −u u −u  u −u −u u   0 ~u 0 0  g(~u) = A A = 4     = 4   u1 −u2 −u3 u4  u1 u2 −u3 −u4  0 0 ~u2 0  u2 u1 −u4 −u3 u2 −u1 u4 −u3 0 0 0 ~u2 = 4rI (C.25) so that the determinant of A is

| det A| = pdet g = p(4r)4 = 16r2. (C.26)

Note that the relation (C.20) is not integrable since the mixed partial derivatives don’t commute, e.g.:

∂2x4 ∂2x4 = − . (C.27) ∂u2∂u1 ∂u1∂u2 Nevertheless, the relation between {xµ} and {uµ} becomes bijective once it has been specified at an initial µ µ point u (~xa) = ua . Using the relations in (C.19) we can express the differentials duµ in terms of (dr, dθ, dφ, dγ) and substitute these into (C.20) to find

dx4 = r cos θ dφ + r dγ. (C.28)

54 References

[1] Wikipedia contributors. Path integral formulation [Internet]. Wikipedia, The Free Encyclopedia; 2016 May 7 [cited 2015 May 23]. Available from: https://en.wikipedia.org/wiki/Path_integral_ formulation [2] Duru, I. H. and Kleinert, H., Solution of the path integral for the H-atom, Physics Letters vol. 84B, number 2, 185–188 (1979) [3] Duru, I. H. and Kleinert, H., Quantum Mechanics of H-atom from Path Integrals, Fortschritte der Physik 30, 401–435 (1982) [4] Kleinert, H., 2004, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 3rd ed., World Scientiﬁc, 1468 p. [5] Glimm, J. and Jaﬀe, A., 1981, Quantum Physics, A Functional Integral Point of View, 2nd ed., Springer, 535 p. [6] Wikipedia contributors. Jacobi–Anger expansion [Internet]. Wikipedia, The Free Encyclopedia; 2015 May 8 [cited 2015 May 23]. Available from: https://en.wikipedia.org/wiki/Jacobi-Anger_ expansion [7] Kleinert, H., Path Integral of Relativistic Coulomb System, Physics Letters vol. A 212, number 15, 15–21 (1996)