Numerical Methods for the Feynman’s path integrals Assignment for the doctoral course “Feynman Integral in Quantum Mechanics and Quantum Field Theory” held by prof. Saverio Pascazio Michele Tuttafesta1 1 Università degli Studi di Bari February 2011 Abstract A numerical method freely published in 1995 by Eduardo Mendel et al. [1] to calculate the Feynman path integrals has been re-implemented. First the systems analyzed by Mendel, harmonic oscillator and double well potential, have been tested in some different numerical conditions and after the method has been applied to a potential box and to a Morse and Lennard-Jones potential. The code has been written in C++ language and the computing time required by every single test ranged from one minute to about two hours, using a Pentium(R) Dual-Core CPU E5400 @ 2.7 GHz 2.7 GHz and a RAM of 4.00 GB. Introduction In order to numerically calculate the Feynman path integral many sophisticated methods have been developed since the early years of computer science. Today the most effective among them are based on the PIMC (Path Integral Monte Carlo) or using suitable strategies to find the most important class of paths, e.g. stationary phase approximations or to damp the integrand (e.g. imaginary time path integration) as used for finite temperature calculations. In the present work we will show the results of our personal implementation of the numerical method, described in 1995 by Eduardo Mendel et al. [1] which adopts a direct sum over all paths in configuration space and the 1 real time. So we first reproduce with some different numerical conditions the same systems analyzed by Mendel (harmonic oscillator and double well potential) and after we apply the method to calculate the energy spectrum and the wave function of other systems such as box potential, Morse and Lennard-Jones potential. Theoretical basis Let’s consider a quantum system described by a time-independent Hamiltonian H, then −iH(t−t0 ) U (t, t0 ) = e ~ is the system time evolution operator. So if |α, t0 i is the state of the system at the instant t0 then its evolution at the instant t is given by |α, ti = U (t, t0 )|α, t0 i In a one-dimensional system and in coordinate representation we can write R |α, ti = R dx0 U (t, t0 )|x0 ihx0 |α, t0 i ⇒ ⇒ hx|α, ti = dx0 hx|U (t, t0 )|x0 ihx0 |α, t0 i Defining the wavefunction Ψ(x, t) = hx|α, ti and the propagator G(x, t; x0 , t0 ) = hx|U (t, t0 )|x0 i then the last equation becomes Z Ψ(x, t) = dx0 G(x, t; x0 , t0 )Ψ(x0 , t0 ) (1) The trace (in the coordinate space) of the time evolution operator is equivalent to the space integral of G, setting the initial and final states equal and moreover it can be expressed in terms of the eigenvalues {En } of the system energy; in fact, setting t0 = 0 R trU (t, 0) = dxG(x, t; x, 0) R P −iEn t = dx n |hx|ni|2 e ~ (2) P −iEn t ~ = ne where {|ni} is a complete set of energy eigenstates. Since 1942, thanks to work [4] of the legendary american physicist R.P. Feynman, it is demonstrated that (see also [3] p.7) (one particle system with mass m) 2 0 G(x, t; x , t0 ) = lim N →∞ ∆t→0 N ∆t=t−t0 C N Z dx1 · · · dxN −1 N Y x e i∆t L ~ n +xn−1 xn −xn−1 , 2 ∆t (3) n=1 p m and L(x, ẋ) = 21 mẋ2 − V (x) is the system lagrangian where C = 2πi~∆t using the prescription of “midpoint rule” (see [3] p.23). An important condition we will have to take into account in our calculations is obtained by noting that the propagator is equal to the transition amplitude, that is (see [2] p.114) G(x, t; x0 , t0 ) = hx, t|x0 , t0 i Consequently we must have Z dxdx00 G∗ (x, t; x00 , t0 )G(x, t; x0 , t0 ) = 1 (4) for all x0 . The (4) meaning that the probability to go from one point to any possible point at a later time has to be 1. Numerical calculation of the propagator If we consider an equidistant discretization of time t0 , · · · , tN , ∆t = T /N , T = t − t0 and space x0 , · · · , xD , ∆x = (x − x0 )/D and defining the “key of the method” (D + 1) × (D + 1) matrix i∆t xm + x n xm − xn L , Kn,m (∆t) := C exp (5) ~ 2 ∆t then from equation (3) we can approximate G for a fixed N (T = N ∆t defined as a “characteristic time”) as follows G(x Z r , tk + N ∆t; xs , tk ) ≈ ≈ dx1 dx2 · · · dxN −2 dxN −1 Kr,1 K1,2 · · · KN −2,N −1 KN −1,s ≈ Z ≈ ∆x dx2 dx3 · · · dxN −2 dxN −1 (K 2 )r,2 K2,3 · · · KN −2,N −1 KN −1,s ≈ Z 2 ≈ (∆x) dx3 dx4 · · · dxN −2 dxN −1 (K 3 )r,3 K3,4 · · · KN −2,N −1 KN −1,s ≈ ≈ · · · (∆x)N −1 (K N (∆t))r,s := Gr,s (N ∆t) = Gr,s (T ) 3 (6) noting that the final discretized expression is independent from tk . So the propagator on a finite time T has became a (D + 1) × (D + 1) matrix, the scheme of such propagation is shown in Figure 1(a). In addition with a minimal computational effort one can obtain a propagator at a time nT multiple of T , recursively by the composition law G(nT ) = G(T )G((n − 1)T )∆x (7) whose conceptual scheme is shown in Figure 1(b). (a) Single path in a finite time transition described by the propagator Gr,s (T ). The initial and the final positions xs and xr are fixed. (b) Single path in a finite time transition described as a multiple of a base one. The propagator is obtained by the recursion G(nT ) = G(T )G((n − 1)T )∆x. Figure 1: Schemes of finite propagations In order to compensate for numerical errors we have to apply the constrain (4) which in our matrix approximation translates to P= D X (∆x)2 Gij∗ Gik −−−−→ 1 D→∞ ∆x→0 i,j=0 (8) for all k. Energy levels and FFT Through a Fourier transform of the time evolution operator trace, as one can deduce from the last right expression in (2), we are able to calculate the P −iEn t energy spectrum of the system. Setting f (t) = trU (t, 0) = n e ~ we will calculate the transform as Z ∞ 1 ˆ f (ω) = dtf (t)eiωt 2π −∞ 4 that is using the “positive convention” for the sign of the exponent in eiωt so we should find energy level values as peaks in the fˆ(ω) graph where ω = En /~. Actually we calculate the discretized time development of the trace in the interval [T, (NT + 1)T ] (T is our characteristic time, see equation 6) trU (nT, 0) = D X Gii (nT )∆x; n = 1, · · · , NT i=0 using the composition law (7). We then perform an FFT (Fast Fourier Transform) using the appropriate routine of the Gnu Scientific Library named gsl fft complex radix2 backward (which adopts the above required “positive convention”). The FFT is an optimized algorithm for the Discrete Fourier Transform (DFT) that, neglecting aliasing, provides N samples of fˆ(ω) in its major amplitude interval [0, ωp ] taking in input N samples of f (t) in its major amplitude interval [t0 , t0 + Tp ] so that Tp ωp = 2πN ; the samples are equidistant ∆ω in the ω domain and ∆t in the t domain so that ∆t = TNp = ω2πp and ∆ω = ωNp = 2π . Tp Therefore our recipe to obtain the energy spectrum is summarized as follows. 1. Set ~ and m to 1 in order to simplify the formalism and the results validation. 2. Choose the maximum value Emax of the system energy up to which perform the calculation, therefore must be ωp = Emax . 3. According to the chosen Emax we should be able to evaluate a spatial limitation for the quantum particle position, that is the interval [x0 , xD ]. 4. Set the numbers (quite arbitrarily): N for the evaluation of (6) (N should be a power of 2 in order to make recursive matrix multiplication possible), D for the spatial discretization, NT for the trace time sampling (NT should be a power of 2 in order to make FFT possible). Wavefunctions Through our numerical method the equation (1) can be discretized as follows Ψ(xi , nT ) = D X ∆xGij (nT )Ψ(xj , 0) j=0 with n = 1, · · · , NT and the (D + 1) vector Ψ(xj , 0) given as input. 5 (9) Harmonic oscillator We apply the numerical method described in the previous sections to the harmonic oscillator, that is at the system described by the lagrangian L(x, ẋ) = 1 mẋ2 − 12 kx2 . Actually setting m = k = 1 we consider 2 1 1 L(x, ẋ) = ẋ2 − x2 2 2 (10) We suppose (remember the setting ~ = 1) the system spatially confined with En ≤ Emax = ωp = 8. Furthermore supposing Emax = Vmax = 12 x2max then it must be −4 ≤ x ≤ 4. The results are grouped in Figure 2. The location of the peaks in the energy spectrum is seen to be in excellent agreement with the theoretical energies 1 n = 0, 1, · · · En = n + 2 Double well potential Another interesting application for our method is describing a quantum particle in the following potential V (x) = α(x − xmin )2 (x + xmin )2 also known as “mexican hat potential”. From the calculated energy spectrum in Figures 3(a,b) one can infer that the system has two lowest and closed energy levels, named ES and EA , that are probably associated to a symmetric and antisymmetric states respectively. Such levels are below the threshold in the potential center (Figure 3(c)) then a classical particle with that energy is not able to leave one of the wells. But this is possible in the quantum case through the so called “tunnel effect” as shown in Figure 3(d). The results in Figure 4 show that in this test case enhancing computational effort does not provides additional significant informations. 6 (a) Energy spectrum - linear scale (b) Energy spectrum - log scale (c) Transition amplitude (d) Time evolution of wavefunction norm |Ψ(x, t)|2 and potential energy V (x) Tp NT xmin xmax D ωp N NT T = -4 4 100 8 4 256 0.785 = 2π ωp V (x) 1 2 x 2 Ψ(x, 0) 1 2 α 4 −α e 2 (x−xs ) ; α = 2; xs = 0.8 π Figure 2: Harmonic oscillator. 7 (a) Energy spectrum - linear scale. (b) Energy spectrum - log scale. (c) Potential energy (d) Time evolution of wavefunction norm xmin xmax D ωp N NT T -5 5 150 7.5 2 128 0.838 V (x) α(x − xm )2 (x + xm )2 α = 0.02; xm = 2.5 Ψ(x, 0) 1 2 α 4 −α e 2 (x−xs ) π α = 2; xs = 2 Figure 3: Double well potential - Test 1: minimal computational effort. 8 (a) Energy spectrum - linear scale. (b) Energy spectrum - log scale. (c) Potential energy (d) Time evolution of wavefunction norm xmin xmax D ωp N NT T -6 6 200 16 2 512 0.393 V (x) α(x − xm )2 (x + xm )2 α = 0.02; xm = 2.5 Ψ(x, 0) 1 2 α 4 −α e 2 (x−xs ) π α = 2; xs = 2 Figure 4: Double well potential - Test 2: enhanced computational effort. 9 Particle in a box As a further validation of the method we consider a particle in a one-dimensional box of length a. The potential energy of the system is 0 |x| < a/2 V (x) = +∞ |x| ≥ a/2 The theoretical energy levels are given by En = and a = 32 π then we have ~2 n2 π 2 . 2ma2 Setting ~ = m = 1 2 En = n2 n = 1, 2, 3, · · · 9 so the first five values are 0.22, 0.89, 2.0, 3.55, 5.55. We chose to perform our calculations up to ωp = 70. The results are grouped in Figure 5. The location of the calculated energy peaks in Figure 5(a) are in good agreement with the theoretical ones. Morse potential It is a convenient model for the potential energy of a diatomic molecule and has the following expression 2 V (r) = d 1 − e−a(r−re ) where r is the distance between the atoms, re is the equilibrium bond distance, d is the well depth (defined relative to the dissociated atoms), and a controls the “width” of the potential (the smaller a the larger the well). In order to a comparison with the harmonic oscillator we set r − re = x so we have V (x) = d 1 − e−ax 2 Expanding the last potential expression up to second order then one obtains V (x) ≈ da2 x2 , which suggests us to set d = 8 and a = 1/4 so that V (x) ≈ 21 x2 . Thus our Morse potential is x 2 V (x) = 8 1 − e− 4 10 Moreover it is known that the theoretical energy levels for such system are 2 p [~ω0 (n+ 21 )] 1 En = ~ω0 (n + 2 ) − 2d/m. In our case, setting , with ~ω = a 0 4d ~ = m = 1, we have 2 (n + 12 ) 1 En = n + − 2 32 n En 0 1 2 3 4 5 6 0.49 1.43 2.3 3.12 3.87 4.55 5.18 7 5.74 ··· ··· Table 1: Energy levels for Morse potential The results of our numerical method applied to this case are shown in Figure 6 and for instance we note an excellent agreement between the locations of the computed energy peaks (Figure 6(b)) and the theoretical ones in Table 1. Lennard-Jones potential Now we test the Lennard-Jones potential (also referred to as the L-J potential, 6-12 potential, or 12-6 potential) that is a mathematically simple model that approximates the interaction between a pair of neutral atoms or molecules. The base expression from which we start is (see Figure on the right) x 6 xm 12 m −2 V (x) = d x x where x is the distance between the atoms, xm is the equilibrium bond distance, d is the well depth (defined relative to the dissociated atoms); 1 moreover one can write xm = 2 6 σ, where σ is a “characteristic” dimension of the atoms. Such model is empiric and (we know) nobody has a finite expression for the associated propagator. In order to compare the results of the simulation we will perform on this system with the ones we obtained about harmonic oscillator and Morse potential then we transform the above expression of the 11 potential so to have the minimum in the origin of our reference frame. So we will refer to the following form " 12 6 # xm xm V (x) = d −2 +d x + xm x + xm 1 1 72 d 2 Moreover V (x) ≈ V (0) + V 0 (0)x + V 00 (0)x2 = x. 2 2 x2m Now we require that V (x) → 12 x2 for small x and the well depth d be the same of the one we have just used √ for the Morse potential. d = 1 ⇒ x = So d = 8 ∧ 72 576 = 24. m x2m In the results shown in Figure 7 we emphasize the comparison with the ones obtained in the Morse potential case. Conclusions The numerical method applied in this work is certainly not “optimized” in the matter of required computational resources. In particular there is a severe limitation in order of the spatial dimension of the system one can consider. However the results obtained from the examined simple quantum systems are accurate enough, then we hope that in a not so far future an improving in the computer capability and parallel architectures (GPU with CUDA for simple personal computers) will give new importance to such method. References [1] Mendel E. Dullweber A., Hilf E.R. Simple quantum mechanical phenomena and the feynman real time path integral. quant-ph/9511042, november 1995. [2] Sakurai J.J. Modern Quantum Mechanics. Addison-Wesley Publishing Company, Inc., 1994. [3] Schulman L.S. Techniques and Applications of Path Integration. John Wiley & Sons, New York 1981. [4] Feynman R.P. A New Approach to Quantum Theory. PhD thesis, Princeton University, 1942. 12 (a) Energy spectrum - log scale. The vertical arrows show theoretical peaks at 0.22, 0.89, 2.0, 3.55, 5.55. (c) First steps in the time evolution of wavefunction norm. (b) Transition amplitude (d) Time evolution of wavefunction norm up to a nearly stationary state starting at t = 435 T xmin xmax D ωp N NT T − 34 π 3 π 4 100 70 2 512 0.0897 V (x) 0 |x| < 34 π +∞ |x| ≥ 43 π Figure 5: Particle in a box. 13 Ψ(x, 0) 1 2 α 4 −α e 2 (x−xs ) π α = 10; xs = 0.5 (a) Energy spectrum - linear scale. (b) Energy spectrum - log scale. Theoretical numeric data are on graph top. (c) Morse potential compared with the harmonic one. (d) Time evolution of wavefunction norm xmin xmax D ωp N NT T V (x) -2.5 8 100 6 2 256 1.05 8 1 − e− 4 x 2 Figure 6: Morse potential. 14 Ψ(x, 0) 1 2 α 4 −α e 2 (x−xs ) π α = 2; xs = 0.8 (a) Energy spectrum - linear scale. (b) Energy spectrum - log scale. Dash line and numeric data on graph top refer to Morse potential. (c) Lennard-Jones potential compared with the harmonic one. (d) Time evolution of wavefunction norm xmin xmax D ωp N NT -2.37 9.55 100 6 2 256 T V (x) d [p(x)12 − 2p(x)6 ] + d 1.05 xm p(x) = x+x ; d = 8; xm = 24 m Figure 7: Lennard-Jones potential. 15 Ψ(x, 0) 1 2 α 4 −α e 2 (x−xs ) π α = 2; xs = 0.8