Quantum Mechanics by Numerical Simulation of Path Integral Author: Bruno Gim´enezUmbert Facultat de F´ısica, Universitat de Barcelona, Diagonal 645, 08028 Barcelona, Spain.∗ Abstract: The Quantum Mechanics formulation of Feynman is based on the concept of path integrals, allowing to express the quantum transition between two space-time points without using the bra and ket formalism in the Hilbert space. A particular advantage of this approach is the ability to provide an intuitive representation of the classical limit of Quantum Mechanics. The practical importance of path integral formalism is being a powerful tool to solve quantum problems where the analytic solution of the Schr¨odingerequation is unknown. For this last type of physical systems, the path integrals can be calculated with the help of numerical integration methods suitable for implementation on a computer. Thus, they provide the development of arbitrarily accurate solutions. This is particularly important for the numerical simulation of strong interactions (QCD) which cannot be solved by a perturbative treatment. This thesis will focus on numerical techniques to calculate path integral on some physical systems of interest. I. INTRODUCTION [1]. In the first section of our writeup, we introduce the basic concepts of path integral and numerical simulation. Feynman's space-time approach based on path inte- Next, we discuss some specific examples such as the har- grals is not too convenient for attacking practical prob- monic oscillator. lems in non-relativistic Quantum Mechanics. Even for the simple harmonic oscillator it is rather cumbersome to evaluate explicitly the relevant path integral. However, II. QUANTUM MECHANICS BY PATH INTEGRAL his approach is extremely gratifying from a conceptual point of view. By imposing a certain set of sensible re- quirements on a physical theory, we are inevitably led to In this section, we recover the main ingredient of the a formalism equivalent to the usual formulation of Quan- path integral formalism for Quantum Mechanics intro- tum Mechanics. Methods based on path integrals have duced in [1]. Our purpose will be to show by concrete been found to be very powerful in other branches of mod- examples, such as the harmonic oscillator, that this for- ern physics, such as Quantum Field Theory or Statistical malism is equivalent to the Heisenberg and Shr¨odinger Mechanics. approach of Quantum Mechanics. In the Feynman formalism, the matrix elements of the operator in the Heisenberg representation at posi- tion xI and xF respectively at times tI and tF is ex- pressed in terms of multidimensional integrals in the x- space, namely Z tF −SE (x) hxF ; tF jO(X(t))jxI ; tI i = D[x]e (1) tI iHt where jx; ti = e jxiS is the position state at time t in the Heisenberg representation, SE(x) is the euclidean action and N−1 m N=2 Z Y D[x] = dx : (2) 2π∆t k k=1 FIG. 1: Path integral qualitative representation for the Here the time distance T = tF − tI has been divided in connection between the initial state with the final state. N intervals of longitude ∆t = (tF − tI )=N. For any time The red line represents the classical trajectory, while the slice tk = t0+k∆t (k = 0;:::;N, t0 = tI and tN = tF ) we dark lines are paths corresponding to quantum particles. integrate over xk = x(tk), which stands for the x-position of the path at time tk with x0 = xI and xN = xF . The above expression is understood in the limit N ! 1 and Our study is based on the seminal paper of Feynman ∆t ! 0 with T kept constant. The action of the system in units of } is Z tF S(x) = dtL ; (3) ∗Electronic address: [email protected] tI Quantum Mechanics by Numerical Simulation of Path Integral Bruno Gim´enezUmbert and the corresponding discretized Lagragian in the eu- at large time of eq. (8), clidean space is Z X −EnT T !1 −E0T dx0hx0; tF jx0; tI i = e ! e : (9) m x − x 2 x − x L = k+1 k + V k+1 k : (4) n 2 ∆t 2 Moreover, in this case eq. (6) can be simplified and at The basic difference between Classical Mechanics and large time can provide the ground state wave function Quantum Mechanics should now be apparent. In Classi- X −EnT 2 cal Mechanics, a definite path in the (x; t)-plane is associ- hx0; tF jx0; tI i = e j n(x0)j (10) ated with the particle's motion; in contrast, in Quantum n Mechanics all possible paths must play a role, including those which do not bear any resemblance to the classical T !1 −E0T 2 path. Yet we must somehow be able to reproduce Clas- hx0; tF jx0; tI i ! e j 0(x0)j : (11) sical Mechanics in a smooth manner in the limit h ! 0. So dividing the propagator over its integral over all the initial states we get the ground state wave function A. Spectrum from a transition amplitude hx; tF jx; tI i T !1 2 Z ! j 0(x)j : (12) Here we show that by knowing the transition ampli- dxhx; tF jx; tI i tudes in eq. (1), all the physical properties of the system (namely eigenvalues and eigenstates) can be known. This It is useful to introduce the expectation value of a generic approach is an alternative to that of Quantum Mechan- observable O as ics where we directly diagonalize the Hamiltonian. In the most familiar Shr¨odingerrepresentation, the matrix Z dx hx ; t jO(t)jx ; t i element in eq. (1) reads as 0 0 F 0 I Z hO(t)i= Z = Dµ(~x)O(xt) ; (13) dx hx ; t jx ; t i hxF ; tF jO(t)jxI ; tI i = 0 0 F 0 I −iH(tF −t) −iH(t−tI ) = hxF je O(X)e jxI i : (5) 1 QN−1 −S(~x) where Dµ(~x) = dxk e and the path vari- Let's now insert in this expression a complete set of eigen- Z k=0 X ables are ~x = (x ; : : : ; x ; : : : ; x = x ). Additionally, states jnihnj = 1 and Hjni = E jni, namely 0 t N 0 n Z N−1 n Y −S(x) Z = dxke has been set to have hOi = 1. hx ; t jO(t)jx ; t i = k=0 F F I I According to eq. (9), at large time distance we have X −iH(tF −t) 0 0 −iH(t−tI ) = hxF je jnihnjOjn ihn je jxI i (6) T !1 n;n0 hO(t)i ! h0jO(x)j0i : (14) X −iEn(tF −t) −iEn0 (t−tI ) 0 ∗ = e e hnjOjn i n(xF ) n0 (xI ) Interestingly, the expectation value of O(t0) = n;n0 δ(X(t0) − x) ≡ δ0 gives ∗ where hxF jni = n(xF ) is the eigenstates wave function. Z Now, by setting xF = xI ≡ x0 and t ! it and by inte- dx0δ0hx0; tF jx0; tI i hx; tF jx; tI i grating over x0 we get in the euclidean space Z = Z ; (15) Z dx0hx0; tF jx0; tI i dxhx; tF jx; tI i X −En(tF −tI ) dx0hx0; tF jO(t)jx0; tI i = e hnjOjni ; (7) n which according to eq. (12) goes to j (x)j2 at large time where we have used the orthonormality of the wave func- distance, T . R ∗ tions dx n(x) n0 (x) = δnn0 . With this change, the ac- tion of the system matches the hamiltonian, and is known III. HARMONIC OSCILLATOR VIA PATH as the euclidean action S (x). At large time T = t − t E F I INTEGRALS: ANALYTIC SOLUTION only the ground state will contribute to the above sum, namely Now we consider the case of the harmonic oscillator Z with potential V (x) = 1 m!2x2 and we estimate the wave T !1 −E0T 2 dx0hx0; tF jO(t)jx0; tI i ! e h0jO(X)j0i: (8) function and the energy-level of the ground state by eval- uating eq. (1) for the operator O(t) = 1 ≡ K(t). The ana- By studying the special case O(X) = 1 , we can extract lytic solution for K(xF ; tF ; xI ; tI ) ≡ hxF ; tF jxI ; tI i when the energy-level of the ground state from the behaviour tI = 0 can be expressed as [2]: Treball de Fi de Grau 2 Barcelona, 2017 Quantum Mechanics by Numerical Simulation of Path Integral Bruno Gim´enezUmbert e−∆S then the algorithm accepts the new path, otherwise r it returns to the old value. a(t) ia(t)((x2 +x2)cos(!t)−2x x ) c: Then, the steps a: and b: are repeated even for hx ; t jx ; 0i = e F I F I ; (16) F F I πi the path ~x(1), and it continues with ~x(2) and so forth. (k) m! The algorithm stops when the S(~x ) values on the (k) where a(t) ≡ . Therefore, by setting xF = xI ≡ updated ~x chain converge to a stable value, the so- 2sin(!t) x and t ! it, we obtain called thermalization phase (FIG. 2). In the next section, we illustrate the algorithm with a concrete example. r b(t) ib(t)x2(cosh(!t)−1) hxF ; tF jxI ; 0i = e ; (17) V. HARMONIC OSCILLATOR VIA PATH 2π INTEGRALS: NUMERICAL SIMULATION m! where b(t) ≡ . Thus, by doing t ! 1 we get sinh(!t) In this section, we simulate the harmonic oscillator with the Metropolis algorithm [3]. This means that the 1 m! t ln(K(t ! 1)) = ln( ) − !(mx2 + ) = euclidean action corresponding to a single path is 2 π 2 r N 2 m! −!mx2 ! X m xj − xj−1 2 m! (xj + xj−1) = ln( e ) − it ; (18) SE(xj) ≡ [ ( ) + ] :(20) 2 ∆t 2 2 π 2 j=1 and from eq.