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 Shrödinger Mechanics. approach of Quantum Mechanics. In the Feynman formalism, the matrix elements of the operator in the Heisenberg representation at position xI and xF respectively at times tI and tF is expressed in terms of multidimensional integrals in the x- space, namely

Z tF −SE (x) hxF , tF |O(X(t))|xI , tI i = D[x]e (1) tI

iHt where |x, ti = e |xiS 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 → ∞ 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 GiménezUmbert and the corresponding discretized Lagragian in the eu- at large time of eq. (8), clidean space is Z X −EnT T →∞ −E0T dx0hx0, tF |x0, 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 |x0, tI i = e |ψn(x0)| (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 →∞ −E0T 2 path. Yet we must somehow be able to reproduce Clas- hx0, tF |x0, tI i → e |ψ0(x0)| . (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 |x, tI i T →∞ 2 Z → |ψ0(x)| . (12) Here we show that by knowing the transition ampli- dxhx, tF |x, 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 Shrödingerrepresentation, the matrix Z dx hx , t |O(t)|x , 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 |x , t i hxF , tF |O(t)|xI , tI i = 0 0 F 0 I

−iH(tF −t) −iH(t−tI ) = hxF |e O(X)e |xI 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 |nihn| = 1 and H|ni = E |ni, namely 0 t N 0 n Z N−1 n Y −S(x) Z = dxke has been set to have hOi = 1. hx , t |O(t)|x , 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 |e |nihn|O|n ihn |e |xI i (6) T →∞ n,n0 hO(t)i → h0|O(x)|0i . (14)

X −iEn(tF −t) −iEn0 (t−tI ) 0 ∗ = e e hn|O|n iψn(xF )ψn0 (xI ) Interestingly, the expectation value of O(t0) = n,n0 δ(X(t0) − x) ≡ δ0 gives ∗ where hxF |ni = ψn(xF ) is the eigenstates wave function. Z Now, by setting xF = xI ≡ x0 and t → it and by inte- dx0δ0hx0, tF |x0, tI i hx, tF |x, tI i grating over x0 we get in the euclidean space Z = Z , (15) Z dx0hx0, tF |x0, tI i dxhx, tF |x, tI i X −En(tF −tI ) dx0hx0, tF |O(t)|x0, tI i = e hn|O|ni , (7) n which according to eq. (12) goes to |ψ(x)|2 at large time where we have used the orthonormality of the wave func- distance, T . R ∗ tions dx ψn(x)ψn0 (x) = δnn0 . With this change, the action 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 mω2x2 and we estimate the wave T →∞ −E0T 2 dx0hx0, tF |O(t)|x0, tI i → e h0|O(X)|0i. (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 |xI , 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(ωt)−2x x ) c. Then, the steps a. and b. are repeated even for hx , t |x , 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) mω 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(ωt) 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(ωt)−1) hxF , tF |xI , 0i = e , (17) V. HARMONIC OSCILLATOR VIA PATH 2π INTEGRALS: NUMERICAL SIMULATION mω where b(t) ≡ . Thus, by doing t → ∞ we get sinh(ωt) In this section, we simulate the harmonic oscillator with the Metropolis algorithm [3]. This means that the 1 mω t ln(K(t → ∞)) = ln( ) − ω(mx2 + ) = euclidean action corresponding to a single path is 2 π 2 r N 2 mω −ωmx2 ω X m xj − xj−1 2 mω (xj + xj−1) = ln( e ) − it , (18) SE(xj) ≡ [ ( ) + ] .(20) 2 ∆t 2 2 π 2 j=1 and from eq. (11) we obtain the analytic solution of In order to get reliable numerical results, it is needed 2 |ψ0(x)| for the harmonic oscillator. that our discrete simulation well approximates the con- tinuum limit, ∆t → 0, and the infinite time extension, T → ∞. Being ω related to the energy of our system, we IV. THE METROPOLIS ALGORITHM require ω∆t 1 and ωT 1, which corresponds in our simulation to m = 1/2, ω = 1, T = 100 and ∆t = 0.5. From eq. (13), we can infer that the expectation value Then the euclidean action reads as of an operator is a N-dimensional integral over a periodic N 2 X (xj − xj−1) (xj + xj−1) path ~x. Being Dµ(~x) defined positive and normalized, it S (x ) ≡ [ + ] . (21) E j 16 8 can be interpreted as a probability measure. Thus, we j=1 can estimate hO(t)i as a statistical average over a sample of paths, ~x(k) (k = 1, ··· ,M), extracted randomly with weight Dµ(~x): A. Wave Function of the ground state M 1 X (k) O¯ = O(~x ) . (19) M t In order to obtain the wave function of the ground k=1 state, we can use a time saving trick [4]. This trick con- ¯ sists of introducing a delta function into the probability For√ large M, hO(t)i ' O and the error goes to zero as 1/ M. The standard deviation of O¯ can be used to give integral an error on our estimation of hO(t)i at finite M. Let’s now discuss the technical part to generate a sample of Z 2 −ε(x,x1,...) paths ~x(k) distributed according Dµ(~x). This can be done |ψ0(x)| = dx1...dxN e , (22) by generating a Markov Chain of ~x(k) paths through the Metropolis algorithm. The Metropolis algorithm starts which allows us to calculate the wave function by only setting randomly an initial path ~x(0). In our simulation considering a concrete position of each path. For in- setup, we set ~x(0) = 0. Then, the following three steps stance, using the trick for the first position of the paths, come into play: we have: a. For each coordinate xi = x(ti) we consider a new proposal x0 randomly generated starting from the initial Z i 2 −ε(x,x1,...) path. Specifically, we introduce a gaussian variable ξ |ψ0(x)| = dx0...dxN δ(x − x0)e (23) 0 distributed in such a way that xi = xi + ξ. b. The Metropolis algorithm consists of accepting Hence, this value must be recovered by the expression: or rejecting the new path ~x0 in the following way. We 0 calculate the change induced in the action ∆S = S(~x ) − r r 2 2 mω −mωx2 1 − x |ψ (x)| = e → e 2 , (24) S(x) and if ∆S < 0 (namely the action is diminished by 0 π 2π the new path), we accept the new path ~x(1) = ~x0 with −∆S 1 1 probability e . In practice, we generate a random when m = 2 and ω = 1. In this case, En = n + 2 and number r uniformly distributed between 0 and 1. If r < n = 0, 1, 2, ...

Treball de Fi de Grau 3 Barcelona, 2017 Quantum Mechanics by Numerical Simulation of Path Integral Bruno Gim´enezUmbert

Action Thermalization Action Thermalization

60 50 L L i i x x H H 40 S S : : 40 30 Action Action 20 20

0 0 0 2000 4000 6000 8000 10 000 0 50 100 150 200

Number of Paths: xi Number of Paths: xi FIG. 2: Behaviour of the harmonic oscillator action as a function of sample of paths generated by the Metropolis algorithm. Left: We consider a large simulation of 10000 paths. Right: We focus on the ﬁrst 200 paths of the same sample as the plot on the left. After 50 iterations, an asymptotic value is reached.

B. Expectation Values C. Symmetries and improvement of the estimations

From the numerical simulation we can calculate the One could observe that the estimated values for X and following matrix elements of the position operator and X3 have to be exactly zero, since they are odd and the ac- their respective errors (σ) with the formulas introduced tion is symmetric for parity. However, in the last section in the previous section. The values obtained are: we have seen that the estimated values are not exactly zero, with a small error. The reason why is because the sample of paths is not symmetrical under parity transfor- hψ0|X|ψ0i = 0.00 ± 0.03 (25) mation. This happens since Metropolis algorithm takes 2 hψ0|X |ψ0i = 1.01 ± 0.02 (26) paths randomly, thus it is not guaranteed that each path 3 would have a symmetrical one in the sample because hψ0|X |ψ0i = 0.0 ± 0.09 (27) 4 the algorithm breaks the parity symmetry. Nonetheless, hψ0|X |ψ0i = 3.12 ± 0.09 (28) since two symmetrical paths give the same value for the euclidean action SE(x) = SE(−x), we can create an im- Whereas the analytic estimates from standard Quantum proved sample with symmetrical paths that implies hav- Mechanics are: ing these values exactly zero, in addition to enhancing 2 the statistics. Therefore, these new values are: hψ0|X |ψ0i = 1 (29) 4 −17 hψ0|X |ψ0i = 3 (30) hψ0|X|ψ0i = (0.0 ± 1)10 (34) hψ |X3|ψ i = (0.0 ± 1)10−17 (35) Notice that we can also estimate the previous matrix ele- 0 0 ments by knowing the wave function of the ground state, 2 4 namely while the estimations of X and X are the same, beacuse of the symmetry of the problem. In fact, from eq. (23) Z we have made the calculations of (25)··· (28), (32) and 2 hψ0|O(x)|ψ0i = dx|ψ0(x)| O(x) ' (33) by considering the first position of each path, i.e. by fixing t at the initial time. However, time reversal X 2 ' |ψ0(xk)| O(xk)∆x . (31) invariance allows to calculate the same wave function at k any time, since the fluctuations are the same. The estimated values in this way are:

2 hψ0|X |ψ0i = 0.98 (32) D. Ground state energy Level 4 hψ0|X |ψ0i = 3.11 (33) From Virial Theorem, we can estimate the ground 2 where the diﬀerence is an estimate of the systematic error state energy through the calculation of hψ0|X |ψ0i. In in the approach, produced due to the fact that in the fact, in a stationary state in Quantum Mechanics the right-hand side we have a discrete sum. Virial Theorem is

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VI. CONCLUSIONS dV 2hT i = hx i . (36) dx In this thesis the goal has been studying one- In the harmonic oscillator case, since the potential is V = dimensional quantum systems using the formulation of 2 mω2 hx i , we obtain that path integrals, both analytically and numerically. We 2 have seen that Feynman’s path integral formalism is equivalent to the Schr¨odingerand Heisenberg interpre- dV tations for Quantum Mechanics. In fact, by using path hx i = mω2hx2i (37) dx integral formalism some expected values in Quantum Me- hx2i chanics can be obtained analytically in diﬀerent ways. hT i = mω2 . (38) 2 Furthermore, we have seen that Markov chains are useful for generating paths with the desired probability distri- bution and the Metropolis algorithm is suitable for the numerical simulations, which by using path integral formalism enables to solve systems where the analytic solution is unknown. With regards to the particular case of the harmonic oscillator, it can be solved analytically with the path integral formalism, which provides a semi- classical approach. However, it can also be easily solved by numerical simulations. For instance, it is useful to take advantage of the symmetries of the problem in order to obtain more accurate results. In general, the results presented in this thesis are well compared with the theoretical values or with other results presented in the literature.

FIG. 3: Ground state wave function obtained through the time saving trick and with the symmetric sample. The red line represents the gaussian function of eq. (24).

Acknowledgments 1 Therefore, by setting m = 2 and ω = 1, we can estimate the ground state energy as First of all, I would like to thank my advisor Federico 2 Mescia for his valuable guidance during the realization hψ0|x |ψ0i E0 = . (39) of this project and for always being there when I needed. 2 I would also like to thank the High Energy Astrophysics That gives us a result of E0 = 0.51, which compared to Group for letting me collaborate with them during this the analytic value E0 = 0.5, we can infer that is a good last year. Finally, I would like to thank my parents for estimation with a systematic error. their outright support over the years.

[1] R. P. Feynman, A. R. Hibbs (1965), “ Quantum Mechan- A. H. Teller and E. Teller, “Equation of state calculations ics and Path Integrals,” McGraw-Hill, New York. For a by fast computing machines,” J. Chem. Phys. 21, 1087 pedagogical discussion see also J. J. Sakurai and J. Napoli- (1953). tano, “Modern quantum physics,” Boston, USA: Addison- [4] R. H. Landau, M. J. Paez, and C. C. Bordeianu, “Compu- Wesley (2011) 550 p tational Physics: Problem Solving with Computers”, 3rd [2] J. J. Sakurai, “Modern Quantum Mechanics”, Revised Edition, ISBN: 978-3-527-40626-5. Edition, ISBN: 0-201-53929-2. [3] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth,

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