Charles University Faculty of Science

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1 Charles University Faculty of Science Study programme: Chemistry Study branch: Physical Chemistry Jakub Kocák Nová metoda řešení Schrödingerovy rovnice A new method for the solution of the Schrödinger equation MASTER THESIS Supervisor: doc. RNDr. Filip Uhlík, Ph.D. Prague, 07

2 Prohlášení: Prohlašuji, že jsem závěrečnou práci zpracoval samostatně a že jsem uvedl všechny použité informační zdroje a literaturu. Tato práce ani její podstatná část nebyla předložena k získání jiného nebo stejného akademického titulu. V Praze dne... Podpis i

3 Název práce: Nová metoda řešení Schrödingerovy rovnice Autor: Jakub Kocák Katedra: Katedra fyzikální a makromolekulární chemie Vedoucí bakalářské práce: doc. RNDr. Filip Uhlík, Ph.D. Abstrakt: Tato práce se věnuje metodě řešení časově nezávislé Schrödingerovy rovnice pro základní stav. Vlnová funkce interpretována jako hustota pravděpodobnosti je reprezentovaná vzorky. V každé iteraci je aplikován aproximant propagátoru podél imaginárního času. Působení operátora je implementováno Monte Carlo simulací. Nemalá část práce se věnuje metodám výpočtu energie vlnové funkce reprezentované vzorky. Je rozebrána metoda na základě odhadu hodnoty vlnové funkce, metoda konvoluce s tepelným jádrem, metoda průměrné energie vážené vlnovou funkcí a metoda exponenciálního poklesu. Metoda řešení byla použita k nalezení základního stavu a enegie 6- dimenzionálního harmonického oscilátoru, anharmonického 3-dimenzionální oktického oscilátoru a atomu vodíku. Klíčová slova: propagace v imaginárním čase, Monte Carlo metoda, variační princip, základní stav Title: A new method for the solution of the Schrödinger equation Author: Jakub Kocák Department: Department of Physical and Macromolecular Chemistry Supervisor: doc. RNDr. Filip Uhlík, Ph.D. Abstract: In this thesis we study method for the solution of time-independent Schrödinger equation for ground state. The wave function, interpreted as probability density, is represented by samples. In each iteration we applied approximant of imaginary time propagator. Acting of the operator is implemented by Monte Carlo simulation. Part of the thesis is dedicated to methods of energy calculation from samples of wave function: method based on estimation of value of wave function, method of convolution with heat kernel, method of averaged energy weighed by wave function and exponential decay method. The method for the solution was used to find ground state and energy for 6-dimensional harmonic oscillator, anharmonic 3-dimensional octic oscillator and hydrogen atom. Keywords: imaginary time propagation, Monte Carlo method, variational principle, ground state ii

4 I would like to express my deep gratitude and appreciation to my supervisor doc. RNDr. Filip Uhlík, Ph.D. for his helpful and witty comments, useful discussions, his assistance with overcoming many obstacles and infinite patience. Special thanks to my family and friends, I would not be able to finish the work without their support. iii

5 Contents Introduction Theoretical Background 3. Quantum Mechanics The Imaginary Time Propagation Methods Approximations of the Imaginary Time Propagator Related Methods Comparison of Different Approximants Computational Methods 4. Calculation of Energy Method A Method B Method C Method D Implementation of the Operators Ĝi() Multiplication Convolution Optimization Golden-section search Results and Discussion Harmonic Oscillator Anharmonic Oscillator Hydrogen Atom Conclusion 4 A Appendices 43 A. The Matrix Elements of the Operator e z ˆT A. Estimate of the Value of Wave Function A.3 Analytically Solvable Potentials A.4 Approximants of the Kinetic Energy and the Unit Operator A.5 Estimators of the Kinetic Energy A.6 Gaussian Wave Function A.7 Relation Between Methods B and C A.8 Variational Principle and Estimators of Energy Bibliography 60

6 Introduction Chemistry on the scale of atoms and molecules can be successfully described by quantum mechanics. The evolution of quantum systems is determined by time-dependent Schrödinger equation. An important role in microworld is played by stationary states, which remain stationary in sense of any observable variable. However according to quantum field theory they are not stationary and spontaneously decay into ground state, the stationary state with the lowest energy. Stationary states can be found as solution of time-independent Schrödinger equation. In general it is difficult (if not impossible) to find analytical solution and we have to rely on computing methods. From the time of formulation of the quantum mechanics there have been developed wide spectrum of methods, each with its own limitation. A large portion of the methods is dependent on the choice of basis and can obtain accurate results using huge set of basis functions and sufficiently large computational time. In some sense the choice of basis brings a certain degree of arbitrariness. Therefore our attention was focused on methods, where choice of basis is not necessary and they are independent of basis. One example of another approach are methods based on the simulations of samples. In this thesis we have studied methods, where wave function is represented by samples. Even though the similar methods are well-studied, to our knowledge this specific approach is rare and we could not find any related literature. With Monte Carlo implementation of imaginary time propagation (ITP) methods we simulated propagation of initial wave function to the ground state. Inevitable part of the study is development of methods for calculation of energy of wave function represented by samples. Developed methods have been applied on systems with different difficulties: dimensionality and singular potential. The first chapter is dedicated to short introduction to key elements of the quantum mechanics and in more detail to ITP methods. We summarised results in this field and reconfirmed some of them. We compared various methods based on different order approximants in one dimension. The second chapter contains development of computational algorithms for calculation of energy, implementation of ITP methods for wave function represented by samples and brief insight into one-dimensional optimization. The third chapter summarises results of ITP method for 4 systems: 6-dimensional harmonic oscillator, anharmonic oscillator and hydrogen atom. We compared convergence for different systems and precision of energy calculation methods. Note to the reader: To achieve more fluent reading of main text the extensive derivations of formulas and mathematically rigorous justifications have been moved into Appendices and footnotes depending on the range of text. The variables in text are dimensionless and correspond to transformation of units to characteristic units for given system. This holds for explicitly time-independent Hamiltonian. For example, the characteristic units of atoms and molecules are atomic units.

7 . Theoretical Background At the beginning it is instructive to provide a brief recapitulation of concepts and principles of the quantum mechanics [], which will be useful subsequently.. Quantum Mechanics In formalism of non-relativistic quantum theory the state of a system is completely described by the state vector ψ from the Hilbert space H. The state at instant t is denoted as ψ(t). The evolution of the state in the quantum mechanics is described by the time-dependent Schrödinger equation with the formal solution i d ψ(t) = Ĥ(t) ψ(t) = ψ(t) = Ĥ() e i 0 t ψ(0), (.) t =:Û(t) where Ĥ(t) is Hamiltonian of the system (in general time-dependent) and Û(t) the time evolution operator. We assume non-relativistic time-independent Hamiltonian Ĥ on the Hilbert space H = L (R N ), which is separable to the kinetic ˆT and potential energy ˆV Ĥ(t) = Ĥ = ˆT + ˆV, ˆT = N, ˆV = V (x), (.) where N = N x i= i is the Laplace operator in N-dimensional space and x R N is N-dimensional position vector 3. We denote the momentum eigenvectors p by N-dimensional momentum vector p R N. In position representation they can expressed as x p = (π) N / eip x. (.3) For explicitly time-independent Hamiltonian Ĥ there are important states E, α called the stationary states 4, which are eigenvectors of the Hamiltonian Ĥ (solution of the time-independent Schrödinger equation) Ĥ E, α = E E, α. (.4) In formalism of spectral decomposition the eigenvectors E, α of Hermitian operator Ĥ (or set of operators called complete set of commuting observables (CSCO)) 5 form In suitable units we can assume ħ =. The kinetic energy of particles with different masses m i can be linearly transformed into desired form as ˆT = i= m i x i mi x i x i ========= One has to bear in mind the change of coordinates also in the potential energy. 3 The position of M particles in 3D space can be denoted by 3M-dimensional position vector. 4 The symbol α denotes other necessary labels of different degenerate stationary states from same subspace. 5 For instance, in the Hilbert space H = L (R 3 ) the CSCO { ˆT, ˆL, ˆLz} forms an orthonormal basis E, l, m. i= x i. 3

8 orthonormal basis 6 E, α E, α = δ(e E )δ α α, Î = E,α de E, α E, α. (.5) The orthonormal basis (.5) guarantees unique decomposition of any state ψ into the eigenvectors E, α ψ = E,α de E, α ψ E, α (.6) and the evolution of the state ψ for time-independent Hamiltonian Ĥ can be described as Û(t) ψ = e itĥ ψ (.6) = de E, α ψ e ite E, α. E,α The variational principle in the quantum mechanics ensures that energy of any state ψ is greater or equal to the ground state energy E 0. Using decomposition (.6) we can write ψ Ĥ ψ ψ ψ (.6) = ψ ψ ψ ψ = E 0 ψ ψ de de ψ E, α E, α Ĥ E, α E, α ψ E,α E,α E δ(e E )δ α α de de ψ E, α E 0 δ(e E )δ α α E, α ψ E,α E,α de ψ E, α E, α ψ (.5) = E 0. (.7) E,α Let  be time-independent operator, then the expectation value of its commutator with time-independent Hamiltonian Ĥ for any eigenstate E, α is zero (the hypervirial theorem) 7 [Â, Ĥ] = 0. For the operator  = ˆp ˆx the commutator reads [ ˆp ˆx, Ĥ] = ˆp [ˆx, Ĥ] + [ ˆp, Ĥ] ˆx = i( ˆT V ˆx), iˆp/m i V and we obtain special case of the hypervirial theorem virial theorem V ˆx = ˆT. For homogeneous potential 8 with degree of homogeneity n the relation becomes 6 The symbol n V = ˆT. de denotes summation over bound states and integration over scattering states. E,α 7 On eigenstates the Hamiltonian behaves like multiplicative constant, which is always commutative with anything. 8 Examples of homogeneous potentials: N-dimensional harmonic oscillator (n = ) and Coulomb potential (n = ). In case of Coulomb potential the number of particles or different charges does not matter. 4

9 . The Imaginary Time Propagation Methods The formal substitution it in time-dependent Schrödinger equation (.) for timeindependent Hamiltonian Ĥ leads to heat or diffusion type equation with analogous formal solution ψ() = Ĥ ψ() = ψ() = e Ĥ ψ. The unique decomposition (.6) into the orthonormal basis (.5) is still valid and we can write for imaginary time evolution of the state ψ e Ĥ ψ (.6) = E,α de E, α ψ e E E, α. We can see, that with increasing parameter the coefficients of the excited states are exponentially decreasing relatively to the coefficient of the ground state E 0 E i, α ψ e E i E 0 ψ e E 0 = E i, α ψ E 0 ψ e (E i E 0 ). The basic idea of the imaginary time propagation (ITP) methods is to apply (repetitively) the operator (or its approximation) e Ĥ on the initial wave function ψ 0 with non-zero overlap with the ground state E 0 ψ 0 0. In each iteration the ground state component in wave function is relatively to other components amplified. Repeating this process the wave function converges to the ground state. Similarly formulated problems arise in many fields of mathematics and physics, e.g. already mentioned quantum mechanics [ 5], classical mechanics [6 0] or statistical mechanics [, ]. The common element is the evolution operator e (ˆT +ˆV ), where ˆT and ˆV are non-commuting operators. 9 Thus the development of the ITP methods is beneficial for several areas. There have been attempts to use the imaginary time evolution operator e Ĥ directly, e.g. [3]. But generally the eigenvectors and eigenvalues of the operator Ĥ are not known, therefore the direct formalism of spectral decomposition cannot be applied. Instead we are forced to use approximations of the imaginary time propagator... Approximations of the Imaginary Time Propagator In position representation, the operator e ˆV can be computed exactly. The operator e ˆT is exactly known in momentum representation, but can be easily transformed into position representation, more in Appendix A.. Hence it is convenient to build approximations using operators e ˆT and e ˆV. But there is restriction Re() 0, because otherwise the operator e ˆT is ill-defined. Also, the condition Re() 0 for the operator e ˆV is often necessary, otherwise the operator e ˆV causes infinities and instabilities 0. 9 The meaning of terms, ˆT and ˆV can differ from field to field. For instance, in quantum statistical mechanics the variable = β = /(k B T ) has meaning of the inverse temperature. 0 For instance, the operator e ˆV for LHO potential V (x) = x and Re() < 0 goes rapidly to infinity for x ± and produces vector outside of the Hilbert space. 5

10 Approximations without Gradient The general form of the approximants is a linear combination of products of operators e ˆT and e ˆV (sometimes denotes as multi-product expansion [4]) e Ĥ = c i e a i,j ˆT e b i,j ˆV + o( n ), (.8) i j where c i, a i,j and b i,j are fixed constants and n is the order of approximation in terms of parameter. However the most common approximants have form of a single product of operators e ˆT and e ˆV (sometimes denoted as factorization [6], decomposition scheme [9], or splitting [5]) e Ĥ = i e a i ˆT e b iˆv + o( n ). (.9) We tried to systematically find for given decomposition form the highest order approximants. We used program Mathematica [5] with extra packages: VEST [6] (Einstein summation convention, simplification of complex operator products), and NCAlgebra, Version [7] (non-commutative algebra, expansion and simplification of non-commutative terms). Our process of finding coefficient a i and b i can be described as follows:. We expressed operator products in terms of Einstein notation operators. For instance ˆT ˆV ψ = V,i ψ,i V ψ,ii.. We expanded the approximant (.9) in terms of (the n-th order) and operator products. For instance e a ˆT e b ˆV = ˆ a ˆT b ˆV + o( ). 3. We substituted operator products with Einstein notation operators, compared both sides term by term and obtained equations for coefficients a i and b i. 4. We solved equations. If there were left some free variables, we returned to the nd step for the (n + )-th order. For simple schemes V T and T V we obtain the simplest straightforward the st order approximation (called the Trotter decomposition or the Lie splitting) e Ĥ = e ˆT e ˆV + o( ) = e ˆV e ˆT + o( ). The schemes V T V and T V T lead to the nd order approximation (called Strang splitting) e Ĥ = e ˆT e ˆV e ˆT + o( ) = e ˆV e ˆT e ˆV + o( ). If we restrict ourselves to real positive coefficients a i > 0 and b i > 0 (to avoid illdefinedness and infinities), it have been shown, that the product can be at most the nd order approximation (non-existence theorem of positive decomposition) [8 0]. This is in agreement with our results. The higher schemes V T V... and T V T... produce negative or complex coefficients. For instance for the scheme V T V T V we obtained the 3rd order approximation with complex coefficients (agrees with []) e Ĥ = e b ˆV e a ˆT e b ˆV e a ˆT e b 3ˆV + o( 3 ) b = ( 3 ± ) 3i, a = ( 3 ± ) 3i, b = 6, a = ( 3 ) 3i, b 3 = 6 ( 3 ) 3i. For convenience we denote particular decomposition schemes by the order of operators in exponents, e.g. e a ˆT e b ˆV e a ˆT T V T. 6

11 It has been shown how to construct approximations with real coefficients (also negative) of any order (fractal decomposition, jump composition) [,3]. As mentioned above, negative imaginary time propagation is not well-defined and cannot be used. Similarly there have been studied complex coefficients schemes for Re(a i ) 0 [5,,4, 5]. Even though the single-product expansions are generally more common, there are studies of multi-product expansions [4, 6]. But Sheng showed that sum (.8) for positive coefficients a i,j > 0, b i,j > 0 and c i > 0 can be maximally the nd order approximation [8]. Approximations with Gradient When the kinetic ˆT and potential ˆV operators have form of the Laplace operator and multiplication with function V (x) (equation (.)), one may observe that the operator [ ˆV, [ ˆT, ˆV ]] is also multiplication with a function This also implies useful corollaries: [ ˆV, [ ˆT, ˆV ]] = V V. Some commutators in expansion of the operator e (ˆT +ˆV ) are zero. For instance [ ˆV, [ ˆV, [ ˆT, ˆV ]]] = 0. Besides operators e ˆT and e ˆV we can construct the operator e 3 [ˆV,[ˆT,ˆV ]]. To find any other useful operators we studied which linear combinations (LC) of products of operators ˆT and ˆV lead to operator with similar behaviour (multiplication with a function). Again we used program Mathematica [5] with package VEST [6]. For the LC of a single operators and products of operators we obtain only trivial cases ˆV = V, ˆV = V. For the LC of products of 3 operators we get the already known LC [ ˆV, [ ˆT, ˆV ]] = V,i V,i. The LC of products of 4 operators leads to obvious LCs ˆV [ ˆV, [ ˆT, ˆV ]] = V V,i V,i, [ ˆV, [ ˆT, ˆV ]] ˆV = V V,i V,i. The LC of products of 5 operators produces 7 obvious LCs ˆV [ ˆV, [ ˆT, ˆV ]] = V V,i V,i, ˆV [ ˆV, [ ˆT, ˆV ]] ˆV = V V,i V,i, [ ˆV, [ ˆT, ˆV ]] ˆV = V V,i V,i, and one new useful operator ˆT [ ˆV [ ˆV, [ ˆT, ˆV ]]] = 0, [ ˆV [ ˆV, [ ˆT, ˆV ]]] ˆT = 0, ˆV [ ˆV [ ˆV, [ ˆT, ˆV ]]] = 0, [ ˆV [ ˆV, [ ˆT, ˆV ]]] ˆV = 0, [ ˆV, [ ˆT, [ ˆV, [ ˆT, ˆV ]]]] = V,i V,j V,ij. We can see simple pattern and generate other operators by recursive definition 3 Ĉ 0 := ˆV, Ĉi+ := [ ˆV, [ ˆT, Ĉi]] = V C i. Or not. 3 In fact we can generate more operators by formula [Ĉ i, [ ˆT, Ĉj]] and plugging in any already generated operator. 7

12 We may ask, how to define operators Ĉi for negative i? We will start with operator Ĉ. The straightforward condition on operator Ĉ is equation C 0 = V C. (.0) Let C be a solution to equation (.0). But this solution is not unique. Let W be function with gradient W orthogonal to the gradient of potential V everywhere V W = 0. Then the function C + αw for any coefficient α R is also a solution to equation (.0). One may notice a property of operators Ĉi, which is common for all operators with non-negative i and can help to define functions C i uniquely. Gradients C i are collinear with gradient of potential V. This implies, that any tangent gradient (like the gradient W ) should be zero. C i V C i = V C i = C i = C i V. (.) V V The equation (.) can be integrated back and the function C i is determined up to a constant. It is instructive to calculate a few functions Ĉi for some potential. For Coulomb potential V = /r in 3D space we get the following sequence (Table.). For Coulomb potential we have problem with operator e ˆV because it produces divergent state vector at origin r = 0, which cannot be normalised and thus is outside of the Hilbert space H C 3 C C C 0 C C C 3 C r8 /80 r5 /0 r / /r /r 4 4 /r 7 8 /r 0 80 /r 3... Table.: Functions C i. As before, the general form of approximation consists of products of operators e ˆT, e ˆV and e i+ Ĉ i. However, the nature of the operators e i+ Ĉ i implies that they can improve only approximations of the (i + )-th order or higher. It is important to remark, that for Coulomb potential the combination of operators e ˆV and e 3 Ĉ (see Table.) will avoid the infinity at origin and we obtain a normalisable state vector. Same as before, the most common approximants are single products. We use the same process to find the coefficients of different schemes. We get the following approx- 4 Let x ψ be spherically symmetric initial wavefunction with Taylor expansion at origin x ψ = + i=0 a i i! ri. Then the integral of e ˆV ψ over ball B R(0) with radius R and center at origin is convergent only if a i = 0 for all i N. This is not in general possible to ensure. 8

13 imations e Ĥ = e 6 ˆV e ˆT e 3 ˆV e ˆT e 6 ˆV e 7 3 Ĉ + o( 3 ), e Ĥ = e 48 3 Ĉ e 3 ˆT e 3 4 ˆV e 3 ˆT e 4 ˆV + o( 3 ), e Ĥ = e 6 ˆV e ˆT e 3 ˆV e 7 3 Ĉ e ˆT e 6 ˆV + o( 4 ), e Ĥ = e 6 ˆV e 44 3 Ĉ e ˆT e 3 ˆV e ˆT e 6 ˆV e 44 3 Ĉ + o( 4 ), e Ĥ = e 6 ˆV e c 3 Ĉ e ˆT e 3 ˆV e c 3 Ĉ e ˆT e 6 ˆV e c 3 Ĉ + o( 4 ), e Ĥ = e 3 3 ˆT 6 e ˆV e Ĉ e ˆT 3 e ˆV e Ĉ e 3 3 ˆT 6 + o( 4 ), ( where c = 6 9 ) 8c and c 0, 7. To have one pseudopotential V + 6c C we can choose c = 43 and c = 08. The 4th order approximations are in agreement with literature [9]. Chin showed that using the operators Ĉi and positive coefficients it is not possible to get higher than the 4th order approximation [7]. Combination of complex coefficients Re(a i ) > 0 and operators Ĉi was also studied [5]... Related Methods Besides the ITP methods there are other similar methods based on application of operators which amplify the ground state. One of the methods is the inverse iteration method applying operator (Ĥ λ). Repeated application of the operator converges to eigenvector with energy E i nearest to the parameter λ. Its convergence depends on how accurately we know the energy E i. Good estimation of E i ensures quicker convergence than ITP methods [8]. Other methods are based on application of operator e n Ĥ n, where n. We can carefully choose the origin of potential to shift energy spectrum so that energy of any state is positive (E 0 0). In the limit + the wave function converges to the ground state ψ E 0 ψ e n E n 0 E0. The approximation of the operator e n Ĥ n has to be a linear combination of products of operators e ˆT and e ˆV. 5 We found an approximant of the nd order for operator e Ĥ e Ĥ = e ˆT e ˆV e ˆT e ˆT e ˆV e ˆT + o( )...3 Comparison of Different Approximants To show how the order of approximants affects the convergence, we simulated numerically the ITP method on D grid. We used model potentials: linear harmonic oscilator (LHO) and double-well potential (WP). The ground state and its energy for these models are known. As the initial wave function ϕ 0 (x) we chose { ϕ 0 (x) = max 0, (x ) }. In the Table (.) we list the used approximants. 5 In the linear order of the expansion has to cancel out. In case of one product, this would lead to negative coefficients, which are not desirable. 9

14 V LHO (x) = x ψ 0,LHO (x) = (π) e x /4 E 0,LHO = 0 V LHO (x) ψ 0,LHO (x) x V WP (x) = (x 6 3x 3) V WP (x) ψ 0,WP (x) ψ 0,WP (x) = Ne x4 4 E 0,WP = x Figure.: Potential V LHO/WP (x), ground state ψ 0,LHO/WP (x) and the ground state energy E 0,LHO/WP. The normalizing constant is N i n scheme G i () T V e ˆTe ˆV V T e ˆVe ˆT 3 T V T e ˆTe ˆVe ˆT 4 V T V e ˆVe ˆTe ˆV 5 4 V T V CT V e 6 ˆVe ˆTe 3 ˆVe 7 3 Ĉ e ˆTe 6 ˆV 6 4 V CT V T V C e 6 ˆVe 44 3 Ĉ e ˆTe 3 ˆVe ˆTe 6 ˆVe 44 3 Ĉ 7 4 V CT V CT V C e 6 ˆVe 43 3 Ĉ e ˆTe 3 ˆVe 08 3 Ĉ e ˆTe 6 ˆVe 8 4 T V CT V CT e 3 ˆTe 6 ˆVe Ĉ e 3 ˆTe ˆVe 3 Table.: Reference number i, order n, scheme of approximant G i () Ĉ 48 3 Ĉ e 3 3 ˆT 6 We used grid with 000 equidistant points on the interval 6, 6. The operators have been applied numerically: operators e ˆV and e 3 Ĉ i by multiplying in each point, operator e ˆT using the formula x e ˆT ϕ i = + dx x e ˆT x x ϕ i (A.8) = + dx (x x ) (π) e ϕ / i (x ). In each iteration we varied to find the minimal energy, which is according to vari- 0 0 E E i = 0. i = Figure.: Energy difference E E 0 of state ϕ i+ (x) = Ĝ6()ϕ i (x) for LHO (i-th iteration). 0

15 E E0 ational principle (.7) an upper bound for the ground state energy E 0. The principle of the method can be seen in Figure.. In each iteration the wave function is normalised. The evolution of the wave function ϕ i (x) after each iteration can be seen in Figure.5 (LHO) and.6 (WP). We can clearly distinguish between approximants of different order. The 4th order operators Ĝ5() to Ĝ8() are better than the nd order operators Ĝ 3 () and Ĝ4(), which are better than the st order operators Ĝ() and Ĝ(). This property is visible for both potentials LHO WP 4 8 Ĝ () Ĝ () Ĝ 3 () Ĝ 4 () Ĝ 5 () Ĝ 6 () Ĝ 7 () Ĝ 8 () i i Figure.3: Conv. (log log plot) of the energy difference E E 0 depending on i number of iterations for operators Ĝj() (LHO left, WP right). The lines serve as a visual aid to compare convergence of different operators. 0 0 E E i = 0. i = Figure.4: Energy difference E E 0 of state ϕ i+ (x) = Ĝ8()ϕ i (x) for LHO (i-th iteration). To visualise the rate of convergence we plotted dependence of the energy difference E E 0 on the number of iterations i into log log plot (Figure.3). We can see different convergence rate for different order operators. We can see unusual behaviour for operator Ĝ8() in LHO potential. To understand this behaviour we replotted Figure. for our case in Figure.4. We can see different trend of lines in Figures. and.4. This is special case of LHO. The ground state wave function is Gaussian function, the the operators e ˆV and e Ĉ are Gaussian functions. Also the operator e ˆT produces from Gaussian functions Gaussian functions of different width. Therefore it is possible for same bigger to obtain ground state Gaussian function. But this problem occurs only for purely quadratic potentials. This explains the first rapid convergence. The second constant trend can be explained as effect of the rounding error at level 0 9.

16 Ĝ () Ĝ () ϕi(x) ϕi(x) x 4.0 Ĝ 3 () x 4.0 Ĝ 4 () 0.8 ϕi(x) ϕi(x) x 4.0 Ĝ 5 () x 4.0 Ĝ 6 () 0.8 ϕi(x) ϕi(x) x 4.0 Ĝ 7 () x 4.0 Ĝ 8 () 0.8 ϕi(x) ϕi(x) x x 4 i = 0. i = 9 ψ 0,LHO (x) Figure.5: Progress of states ϕ i+ (x) = Ĝj()ϕ i (x) for different operators Ĝj() for LHO.

17 Ĝ () Ĝ () ϕi(x) ϕi(x) x Ĝ 3 () x Ĝ 4 () ϕi(x) ϕi(x) x Ĝ 5 () x Ĝ 6 () ϕi(x) ϕi(x) x Ĝ 7 () x Ĝ 8 () ϕi(x) ϕi(x) x x i = 0. i = 9 ψ 0,WP (x) Figure.6: Progress of states ϕ i+ (x) = Ĝj()ϕ i (x) for different operators Ĝj() for WP. 3

18 . Computational Methods If the Hamiltonian Ĥ has the form like in equation (.) with real-valued potential V (x), the solution of the time-independent Schrödinger equation can be restricted to real-valued wave functions. Let be ψ E,α (x) complex-valued solution of the timeindependent Schrödinger equation (.4) with energy E. Then the function ψe,α (x) is also solution ) (ĤψE,α (x) = (EψE,α (x)) Ĥψ E,α(x) = Eψ E,α(x). Then we can replace complex-valued solutions ψ E,α (x) and ψe,α (x) with real-valued solutions, ψ E,α (x) + ψe,α (x) ψ E,α (x) ψe,α, (x). i This is useful property, because we can restrain ourselves onto real-valued wave function without loss of generality. The Copenhagen interpretation of quantum mechanics shows standard probabilistic interpretation of squared absolute value of wave function ψ(x) as probability density ρ(x) d (probability) = ψ(x) dv = ρ(x)dv. This allows to represent function ψ(x) with randomly generated samples {x i }. 3 For some potentials the ground state wave function is not only real-valued, but also positivevalued function. This is not general rule. As a counterexample we can use any system with more than fermions (with spin /). The wave function of fermions is antisymmetric in any permutation of particles. However if ψ 0 (x) is positive-valued function, we could ask if we can use not the function ψ 0 (x), but the function ψ 0 (x) as probability density. 4 And samples will represent the function ψ 0 (x) and not the function ψ 0 (x). The function ψ 0 (x) can represent probability density iff R N dr N ψ 0 (x) = I < +. (.) To show rough justification that the integral (.) is finite, we will first focus on the D potential V (x). For the potential V (x) we require the condition 5 lim V (x) E 0 = C ± > 0, x ± where E 0 is ground state energy. We can apply WKB approximation for region where V (x) E 0 < 0 (classical motion) ψ 0 (x) A 0 e +i dx (E 0 V (x)) [(E 0 V (x))] /4 (.) We need to renormalise new solutions. One can notice that if the solution ψ E,α(x) is real-valued, we obtain pair ψ E,α(x) and 0. 3 This is for example used in some Monte Carlo methods. 4 This probability density does not correspond to real probability of occurrence of particle in infinitesimal space dv. 5 The constant C ± can be +. 4

19 and for regions where V (x) E 0 > 0 (quantum tunneling regions) ψ 0 (x) A +e + dx (V (x) E 0 ) + A e dx (V (x) E 0 ) [(V (x) E 0 )] /4. (.3) The divergence of the integral I can be caused by problems: infinite wave function at region of classical motion 6, or too slow decay of wave function into infinity x ±. For reasonable potential V (x) the wave function (.) is finite and hence does not create infinity in integral I. 7 In quantum tunneling regions (.3) there are solution, one growing to infinity for x ± and one decaying to zero for x ±. The growing term has constant A ± equal to zero (otherwise the wave function could not be normalised). The condition (.) for finite C ± guarantees at least exponential decay e C ± x in marginal regions, which is sufficient for integral I to be finite. The infinite C ± guarantees even quicker decay. Similar reasoning can be used for N-dimensional spherically symmetric potentials. The N-dimensional Schrödinger equation can be transformed into one-dimensional Schrödinger equation (more in Appendix A.3). For general case it is harder to obtain justification. Even though this is not rigorous justification, it gives some qualitative insight that it is possible for function ψ 0 (x) to represent probability density. But this can be overcame with introduction of positive and negative samples.. Calculation of Energy When we are applying operator Ĝi() on the wave function, we want to get closest to ground state as possible. Good indicator is energy of state E = ψ Ĥ ψ ψ ψ (variational principle, equation (.7)). It can be also used to set optimal parameter in one iteration. However the wave function 8 ψ(x) is represented by n point-like samples {x i } n i=, effectively represented by wave function ψ(x) n δ N (x x i ), (.4) n i= where δ N ( ) is N-dimensional Dirac delta distribution. To evaluate energy E we need to evaluate three integrals: ψ ψ, ψ ˆT ψ and ψ ˆV ψ. 9 Application of the kinetic energy ˆT = N on the N-dimensional Dirac delta distribution δ N (x x i ) is not welldefined action. To overcome this problem there have been developed several different approaches... Method A This method is based on estimation of value of wave function and can be used to evaluate the norm ψ ψ and the potential energy integral ψ ˆV ψ. 6 The infinite value is not sufficient, it has to be infinite integral on finite interval. 7 The finite wave function integral over classical motion interval I C can be dominated as dx ψ(x) dx max{ ψ(x) } = max{ ψ(x) }λ(i C ) < +, x I I C I C x I C C where λ(i C ) is Lebesgue measure of the interval I C (fancy word for length). 8 We denote the wave function as ψ(x) and not ψ 0(x), because in following sections we will talk about energy evaluation on positive-valued function in general. Not only for the ground state. 9 We need to remember that the samples are samples of function ψ(x) not the function ψ (x). This means that the function ψ(x) is normalised and the integral ψ ψ is not in general equal to. 5

20 In position representation the potential energy can be evaluated as ψ ˆV ψ = dx V (x)ψ(x). R N In standard Monte Carlo simulation for samples {x i } n i= of function ψ(x) we could estimate the potential energy integral ψ ˆV ψ as ψ ˆV ψ n V (x i ). n i= However in case of samples of function ψ(x) we need the value of the wave function ψ(x i ) ψ ˆV ψ n V (x i )ψ(x i ). n i= In Appendix A. we showed that ψ(x i ) can be estimated by ψ(x i ) as (equation (A.)) ψ(x i ) = n(x i, r)n n S N r N, (.5) + (V (x i ) E) r N+ where the radius of neighbourhood r was left as free parameter. In Figure. we demonstrate how to choose radius r optimally. The quantity ψ(x i ) is expected value of estimator ψ(x i ). We want radius r to be as big as possible to get large n(x i, r), because the relative error of estimate is approximately / n(x i, r) (reddish band shows the error of ψ(x i ) ). On the other hand we want radius r to be as small as possible because the value ψ(x) is changing from place to place and we are effectively calculating averaged wave function ψ(x). 0 ψ(x i ) ψ(x i ) 0 0 r opt r Figure.: Choice of optimal radius r opt. We set some maximal number of samples n max and find n max closest samples, the furthest in radius r max. The number n(x i, r) is approximately scaling as r N n(x i, r) = rn n max rmax N. The condition of optimal radius is relative error / n(x i, r) equal to relative first correction ropt = V (x i ) E n(x i, r opt ) N +. 0 This can be understood in term of Taylor series (A.9) in Appendix A., where residual term is rising with radius r. 6

21 From last two equations we can express the optimal radius r opt as r opt = [ (N + )r N/ max V (x i ) E n / max ] 4+N. (.6) We can see that for V (x i ) E the optimal radius goes to infinity. We can expect that in regions where the potential energy V (x) is further from value E we need smaller number of samples. We define estimators of norm integral Ñ, potential integral and calculated potential energy V cal as Ñ := n Ṽ := n n ψ(x i ), i= n i= V cal := Ṽ Ñ. V (x i ) ψ(x i ), To demonstrate this method we sampled few analytic potentials (more in Appendix A.3): gamma distribution function and Gaussian distribution function. In N = dimensions we generated 000 independent samples for gamma (a =, b = ) and Gaussian distribution function (a =, σ = ). The maximal number of samples was set n max = 00. In Figure. we plotted calculated and exact value of wave function ψ(x i ) with and without the correction (equation (.5)). The samples are coloured according to number of samples in neighbourhood used to calculation n s. This number is also important in view of that the estimated relative error is / n s. The correction improved results, but the difference is not very noticeable. We can also observe one important property. For gamma distribution the corresponding potential is Coulomb potential. Because it is unbound in origin we observe drop in number n s. This is in agreement with equation (.6). This can be seen in Figure. in top right plot. This means that the samples with the highest weight (ψ(x i )) have great error. On the other hand for Gaussian distribution the corresponding potential is (in origin) finite LHO and this behaviour is not present. The quadratic potential of LHO is also unbounded and this causes error for big radii, but these samples have small weight (ψ(x i )). In same Figure we can see calculated potential energy V cal with estimated error compared to exact ratio V ψ ψ. Error was calculated from error of each sample / n s. We can expect the error to be approximately / n max or higher. In N = 3 dimensions we generated 000 samples for gamma (a = 3, b = ) and Gaussian distribution function (a = 3, σ = ). The maximal number of samples was set n max = 00. In Figure.3 we plotted analogous graphs to Figure.. The correction strongly improved results in contrast to D case. In case of 3 dimensions we observe same problem with Coulomb potential as in case of dimensions. For 4-dimensional and 5-dimensional case we generated 000 samples for gamma (a = 4, b = ) and Gaussian distribution function (a = 4, σ = ) with n max = 00. The graphs in Figure.4 are analogous to previous graphs. The correction helps to improve results, but they are scattered. However this seems does not effect the calculated potential energy. The samples are mostly under the precise value therefore they does not effect the calculated potential energy very much. This method has drawbacks: computational time rises as n with number of samples and this method does not provide calculation for the kinetic energy. Therefore there have been developed other methods. 7

22 with correction without correction Gamma distribution r r x y Gaussian distribution r r 0 x 0 y D Gamma n s D Gauss 0.5 Vcal 0.50 V ψ ψ Vcal 0.50 V ψ ψ 0.75 # 0.75 Figure.: The dependance of calculated (coloured dots) and exact (black line) value of wave function ψ(x i ) on radius r with and without correction for gamma distribution (Coulomb potential) and Gaussian distribution (LHO), samples in D plane ( right graphs) and calculated potential energy V cal with estimated error compared to ratio V ψ ψ for gamma and Gaussian distribution. The symbol # is run number. The average result with extimated error band is shown (green line and band). # 8

23 with correction without correction Gamma distribution Gaussian distribution r r r r 6 z z x 0 x y 0 y D Gamma n s 3D Gauss 0.50 Vcal 0.50 V ψ ψ Vcal 0.75 V ψ ψ 0.75 #.00 Figure.3: The dependance of calculated (coloured dots) and exact (black line) value of wave function ψ(x i ) on radius r with and without correction for gamma distribution (Coulomb potential) and Gaussian distribution (LHO), samples in 3D space ( right graphs) and calculated potential energy V cal with estimated error compared to ratio V ψ ψ for gamma and Gaussian distribution. # 9

24 Gamma distribution with correction r without correction r Vcal D Gamma # V ψ ψ Gaussian distribution r r Vcal D Gauss # V ψ ψ with correction without correction Gamma distribution r r Vcal D Gamma # V ψ ψ Gaussian distribution r r Vcal D Gauss # V ψ ψ n s Figure.4: The dependance of calculated (coloured dots) and exact (black line) value of wave function ψ(x i ) on radius r with and without correction for gamma distribution (Coulomb potential) and Gaussian distribution (LHO) in 4D and 5D, the calculated potential energy V cal with estimated error compared with exact value V ψ ψ. 0

25 .. Method B The Method B employs the convolution of sampled wave function with heat kernel and can evaluate the norm ψ ψ, the kinetic energy integral ψ ˆT ψ and the potential energy integral ψ ˆV ψ. Kinetic Energy Usually to calculate the kinetic energy we need the second derivative of wave function, but we represent the wave function with point-like objects: samples. Thus at first glance it looks hopelessly. But the operator e ˆT will help to overcome this obstacle. The basic idea is as follows: the wave function represented by samples can be understood as linear combination of Dirac delta functions (equation (.4)). Applying the operator e ˆT on the linear combination of Dirac delta functions we will get linear combination of Gaussian functions. We can calculate the kinetic energy of this linear combination and using the limit 0 + we can get the kinetic energy of the wave function ψ(x). The kinetic energy of the vector e ˆT ψ can be formally calculated as d d ( ) e ˆT ψ = ˆT e ˆT ψ. This is justified in momentum representation as d d p e ˆT ψ = d d e p p ψ = p In position representation we get p e p ψ = p ˆT e ˆT ψ. x ˆT e ˆT ψ = d d x e ˆT ψ = d dx x e ˆT x x ψ d R ( N = dx d ) R N d x e ˆT x = dx x ˆK 0 () x x ψ, R N where in step we used Lebesgue s dominated convergence theorem to justify the The operator e ˆT is heat distribution propagator. The exchange is justified if the derivative of integrated function is dominated by some integrable function. If the function x ψ is finite everywhere, we can write ( ) d e x x d (π) N ψ(x ) / N x x (π) N e ψ(x ) / + x x x x (π) N e ψ(x ) / ( max ψ(x N ) x R N (π) N / + x ) x e x x. The dominating function is integrable ( dx N + x ) ( ) x e x x t=x x R N ======= dt + t e N R t N If the function x ψ is infinite at some compact region C, we can imply ψ(x ) L (R N ) = ψ(x ) L (C) = ψ(x ) L (C). Therefore we can choose dominating integrable function Mψ(x ), where ( M = max N x C (π) N / + x ) x e x x. = N (π)n / < +.

26 interchange of derivative and integral. The matrix element x ˆK 0 () x is x e x ˆK 0 () x = (π) N / ( ) N x, where x = x x. Then in the limit 0 + we can write ψ ˆT ψ = lim e ˆT ψ = lim dx R N dx ψ x x ˆK 0 () x x ψ. (.7) R N For wave function ψ(x) represented by samples {x i } n i= we can write estimator of kinetic energy as ψ ˆT n n ψ lim 0 + n K 0 (, x ij ), i= j= where we used more convenient notation K 0 (, x ij ) = x i ˆK 0 () x j. However we can notice that the operator ˆK 0 () in power expansion of ˆK 0 () = ˆT e ˆT = ˆT ˆT +! ˆT 3 3! 3 ˆT can be improved. In same matter as we could express ψ ˆT e ˆT ψ, we can express any power ψ ˆT n e ˆT ψ and make correction to operator ˆK 0 (). We define the higher order operators ˆK m () as ˆK m () := ˆT e ˆT T m (e ˆT ), (.8) where T m ( ) is Taylor series of the m-th order in variable. In Appendix A.4 we calculated spacial matrix elements x ˆK m () x for m > 0. Same procedure used before for ˆK 0 () can be used to obtain more general form of equation (.7) 3 ψ ˆT ψ = lim 0 + dx R N dx ψ x x ˆK m () x x ψ. R N and estimator ψ ˆT n n ψ lim 0 + n K m (, x ij ). (.9) i= j= However the estimator (.9) is biased and we would like to estimate the error of the estimator. In Appendix A.5 we showed that the estimator ˆT is unbiased (equation (A.9)) estimator of the integral ψ ˆK m () ψ ˆT := n(n ) n K m (, x ij ), (.0) and the estimator V ˆ T is unbiased estimator of the variance Var[ ˆT ] Vˆ 4 n T := n(n )(n )(3n 5) 4 n K m (, x ij )K m (, x jk ) n(n )(4n 5) ( ˆT ) i= j>i i= k= j>i k j n K m(, x ij ). (.) i= j>i 3 The justification for interchange of derivative and integral is similar. The dominating integrable function will have the form P ( x )e x, where P ( ) is polynomial.

27 Potential Energy and Norm In same manner we can use this technique to find integral ψ ψ and ψ ˆV ψ. define the approximants of unit operator Ĵm() We and approximants of potential operator ˆV m () as 4 Ĵ m () := e ˆT T m (e ˆT ), (.) ˆV m () := ˆV e ˆT T m (e ˆT ). In Appendix A.4 we calculated spacial matrix elements x Ĵm() x. matrix elements x ˆV m () x are equal to The spacial x ˆV m () x = V (x) x Ĵm() x. We can generalise the formulas (.0) and (.) ˆX := n(n ) n X m (, x ij ), i= j>i Vˆ 4 X := n n(n )(n )(3n 5) 4 n X m (, x ij )X m (, x jk ) i= k= j>i k j n(n )(4n 5) ( ˆX n ) X m(, x ij ), i= j>i where for X {J, T, V } we introduce estimators ˆX { J ˆ, ˆT, ˆV } and estimators of variance V ˆ X { V ˆ J, VT ˆ, VV ˆ }. The energy can be estimated as H := T + V J. To demonstrate this method we will use analytic solvable system: Gaussian distribution function. For N = dimensional Gaussian function (σ =, equation (A.)) we generated 4000 samples. In Figure.5 we plotted dependence of calculated integral ˆK m () (coloured, equation (.0)) with estimated error (equation (.)) and exact values of integrals ˆK m () (black, equation (A.3)) for wide range of imaginary time. For large the calculation is more precise, but the calculated value is further from value ψ ˆT ψ. For small the error is enormous. To choose optimal we propose following procedure: Set large and estimate integrals ψ ˆK m () ψ and ψ ˆK m+ () ψ. Decrease until the estimates of ψ ˆK m () ψ and ψ ˆK m+ () ψ are within the estimated error. This procedure was used to find optimal and calculate the norm ψ ψ, the kinetic energy integral ψ ˆT ψ and the potential energy integral ψ ˆV ψ (Figure.6). 4 This definition is asymmetric, because the operators ˆV and ˆT are not commutative in general. One might want to define another approximants as e ˆT T m(e ˆT )ˆV. The difference between spacial matrix elements would be V (x) x Ĵ m() x vs. V (x ) x Ĵ m() x. But when the integral will be evaluated on samples and summed over pair of samples, the result will be same because the matrix elements x Ĵ m() x are symmetric in exchange of variables x x. 3

28 ˆKm() N = ˆK 0 () ˆK () ˆK () ˆK 3 () ˆK 4 () Figure.5: The -dependence of estimated (coloured with error band) and exact integrals ψ ˆK m () ψ for N = (4000 samples) ˆVm() Ĵm() ˆKm() Ĵ0() Ĵ() Ĵ() Ĵ3() Ĵ4() ˆK 0 () ˆK () ˆK () ˆK 3 () ˆK 4 () ˆV 0 () ˆV () ˆV () ˆV 3 () ˆV 4 () Figure.6: The choice of optimal and calculation of the norm ψ ψ, the kinetic energy integral ψ ˆT ψ and the potential energy integral ψ ˆV ψ for N = (000 samples). 4

29 We expect that when the calculated kinetic energy and real value are not within the error, then we need more samples of wave function for better representation of wave function. 5 Also when the sampled wave function and number of samples N does not change a lot, we need to find optimal only once. However there is a catch. With n samples the estimators ˆX and V ˆ X need approximately n operations, because the estimators are evaluated on n(n )/ pairs of samples. With rising number of samples the computational time rises unbearably. Also the pairs of samples are not independent, because each sample is in (n ) pairs. To solve this problem we need to generate independent pairs of samples. From n samples we have n/ pairs of samples. The corresponding estimators are defined as ˆX := n X m (, x i ), n i= Vˆ n [ X := X m (, x i ) n(n ) ˆX ], i= where for X {J, T, V } we estimators estimators ˆX { J ˆ, ˆT, ˆV } and estimators of variance V ˆ X { V ˆ J, VT ˆ, VV ˆ }. We will not use all information from samples, but the estimators can be calculated quicker. For the same computational time the estimator ˆX from larger set of pairs of samples gives more precise value with smaller error. The energy can be estimated as H := T + V J. In Appendix A.8 we discuss if and when does the energy defined as E := ψ ĤĴm() ψ ψ Ĵm() ψ have the lower bound and what is its relation to the ground state energy E 0. For the ground state E 0 the energy is equal to E 0 for any E 0 ĤĴm() E 0 E 0 Ĵm() E 0 = E 0 E 0Ĵm() E 0 E 0 Ĵm() E 0 = E 0. (.3)..3 Method C We define the averaged energy {H} as 6 {H} := R N dx Ĥψ(x) R N dx ψ(x) 5 One may imagine wave function N cos (0x)e x. When the wave function is poorly sampled, the samples looks like samples of function N e x. However the rapid changes in wave functions are more common for excited states. 6 We can notice that the definition can be expressed as averaged local energy weighed by wave function R N dx Ĥψ(x) ψ(x) ψ(x) R N dx ψ(x).. 5

30 For separable Hamiltonian from equation (.) we can write [ dx ] R {H} = N N + V (x) ψ(x) = dx ψ(x) dx V (x)ψ(x) R N R + N dx ψ(x) dx ψ(x) dx ψ(x) R N R N R N = ψ(x) dσ dx V (x)ψ(x) dx V (x)ψ(x) (R N ) R + N R = N =: {V }, dx ψ(x) dx ψ(x) dx ψ(x) R N R N R N where in step we used Stokes theorem 7, in step we used fact, that the wave function ψ(x) vanishes to zero at boundary of space (R N ) therefore the gradient ψ(x) vanishes to zero too. We defined averaged potential energy {V }, which is identical to {H}. The averaged energy {H} can be evaluated on samples as with estimate of error ˆ V {H} { H} = n V (x i ), n i= ˆ V {H} = n(n ) n i= [ V (x i ) { H}]. In Appendix A.7 we showed that in the limit + the Method B becomes the Method C in precise form and also in terms of estimators. Also in Appendix A.8 we showed that the averaged energy {H} is non-variational energy and the infimum is min {V (x)}. x RN For the ground state E 0 the averaged energy {H} is equal to E 0..4 Method D {H} = R N dx Ĥψ 0(x) R N dx ψ 0 (x) = R N dx E 0 ψ 0 (x) R N dx ψ 0 (x) = E 0. This method (exponential decay method) uses change in number of samples in one iteration and does not need to handle the individual samples. We start with the operator Ĝ i () as the approximant of the operator e Ĥ. We can write e Ĥ ψ = Ĝi() ψ + O( n ). dx x e Ĥ ψ = dx x Ĝi() ψ + O( n ). R N R N When the state ψ is near the ground state E 0 (or any eigenvector), then we can approximate Ĥ ψ E ψ. dx x e Ĥ ψ e E R N dx x ψ R N dx x Ĝi() ψ R N 7 In this special case it is also called divergence theorem or Gauss s theorem. 6

31 We can express the energy as E ln dx x ψ R N. dx x Ĝi() ψ R N We denote the number of samples in the j-th iteration as n j. Then the estimator of energy Ẽ can be expressed as Ẽ = ( ) ln nj. n j+ However this method cannot be used in our case. In each iteration we chose optimal, where the energy has minimum. At this point the operator Ĝi() already fails to approximate the exponential operator, because we do not observe exponential decay. Second disadvantage is dependence on the evolution of wave function. Also for small parameter the error is huge, for larger the relation is inaccurate.. Implementation of the Operators Ĝi() The approximants Ĝi() consist of two types of operators in sense of action on wave function: multiplication with non-negative real-valued function (operators e i+ Ĉ i ) and convolution (operator e ˆT )... Multiplication Let the {x i } n i= be initial set of samples of wave function ψ(x) and we need samples of wave function x ˆF ψ = F (x)ψ(x), where F ( ) is non-negative real-valued function. Implementation: Each sample x i is duplicated F (x i ) times and we add extra duplicate with probability (F (x i ) F (x i ) ). The above mentioned procedure can be used to implement action of operators e i+ Ĉ i... Convolution Let the {x i } n i= be initial set of samples of wave function ψ(x) and we need samples of wave function x e ˆT ψ (A.8) = dy e x y (π) N ψ(y). (.4) / R N The samples can be understood as sum of Dirac delta distributions (equation (.4)). Then the relation (.4) becomes x e ˆT ψ n x x i n (π) N e. / i= Implementation: For each sample x i we generate random vector a from N-dimensional Gaussian distribution ρ G (a; ) ρ G (a; σ) = a (πσ ) N e σ, / and we add vector a to vector x i to make new sample x i x i = x i + a. 7