Semi-Classical Dynamics Using Hagedorn Wavepackets

Transcription

1 Semi-Classical Dynamics Using Hagedorn Wavepackets Leila Taghizadeh June 6, 2013 Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

2 Outline 1 The Schrödinger Equation in Semi-Classical Scaling 2 Hagedorn Wavepackets 3 A Numerical Integrator for Hagedorn Wavepackets 4 Conclusion Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

3 Section 1: The Schrödinger Equation in the Semi-Classical Scaling Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

4 The Schrödinger Equation... Consider the time-dependent Schrödinger equation in the semi-classical scaling: ıε ψ t = Hψ, (1) where ψ = ψ(x, t) is the wave function depending on the spatial variable x = (x 1,..., x N ) and time t 0. Here, ε is a small positive number representing the scaled Planck constant and ı is the imaginary unit. Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

5 The Schrödinger Equation... The Hamiltonian operator H depends on ε and it is composed of two parts, the kinetic operator T and the potential operator V. Thus we can split H as H = T + V with the following definitions for both operators: T := ε2 2m 2 = j=1 V := V(x), N ε 2 2 2m j x 2, where m j 0 is a particle mass and the real-valued potential V acts as a multiplication operator on ψ. j Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

6 The Schrödinger Equation... Example: In quantum molecular dynamics, equation (1) is a Schrödinger equation for the nuclei on an electronic energy surface. electronic energy surface is a particular eigenvalue of the electronic structure problem which appears as a potential in the time-dependent Schrödinger equation for the nuclei. Here ε 2 is the mass ratio between electrons and nuclei, of magnitude Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

7 The Schrödinger Equation... Numerical approaches to solving equation (1) face two principal difficulties: 1- Highly oscillatory solutions: Typical solutions are wavepackets of width ε, oscillatory with wavelength ε, with the envelope moving at velocity 1. Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

8 The Schrödinger Equation... Numerical approaches to solving equation (1) face two principal difficulties: 1- Highly oscillatory solutions: Typical solutions are wavepackets of width ε, oscillatory with wavelength ε, with the envelope moving at velocity High dimension: For n particles, the spatial dimension in (1) is N = 3n. Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

9 The Schrödinger Equation... Because of the highly oscillatory solution behavior, grid-based methods need very fine resolution for small ε and hence become computationally expensive or infeasible. This also precludes the approximation of the wave function on sparse grids in higher dimensions, because the necessary smoothness requirements for this technique are not met for small ε. The model reduction from full quantum dynamics to Gaussian wavepacket dynamics allows for computationally efficient algorithms, but is often not accurate enough. The Gaussian wavepackets are written as: ) ( ı ψ(x, t) = exp ε (1 2 (x q(t))t C(t)(x q(t))+p(t)(x q(t))+ζ(t)), where C(t) is a complex symmetric matrix and ζ(t) C is a phase and normalization parameter. Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

10 The Schrödinger Equation... Hagedorn constructed parameter-dependent L 2 -orthonormal basis functions and presented Hagedorn s wavepackets. In 2009, Lubich et. al. turned the Hagedorn functions into a computational tool for the numerical solution of (1). The main goal of this talk is to explain Lubich s approach of numerical solution of equation (1) using Hagedorn s wavepackets. We focus on the conceptual and algorithmic aspects of the approach. Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

11 Section 2: Hagedorn Wavepackets Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

12 Hagedorn Parametrization of Gaussian Wavepackets In this section, we review Hagedorn s parametrization of Gaussian wave packets and his parameter-dependent orthonormal basis functions. In Hagedorn s approach, a Gaussian wavepacket is parameterized as: ϕ ε 0 [q, p, Q, P](x) = (πε) N 4(det Q) 1 2 exp [ ı 2ε (x q)t PQ 1 (x q)+ ı ε pt (x q) ], (2) where q R N and p R N represent the position and momentum, respectively, and Q and P are complex N N matrices satisfying the relations Q T P P T Q = 0, (3) which we call them symplecticity relations. Q P P Q = 2 ıi. (4) Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

13 Hagedorn Parametrization of Gaussian Wavepackets Here, Q T denotes the transpose of Q, and Q is the transpose and complex conjugate matrix. Relations (3) and (4) imply that both Q and P are invertible. Lemma Let Q and P be complex N N matrices that satisfy the relations (3) and (4). Then, Q and P are invertible, and C = PQ 1 is complex symmetric with the positive definite imaginary part Im C = (QQ ) 1. (5) Conversely, every complex symmetric matrix C with positive definite imaginary part can be written as C = PQ 1 with matrices Q and P satisfying the relations (3) and (4). Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

14 Hagedorn Parametrization of Gaussian Wavepackets Proof. Multiplying the equation (4) from the left and right with the vectors v C N and v C N, respectively yields (Qv) (Pv) (Pv) (Qv) = 2 ı v 2, which shows that v = 0 is the only vector in the null-space of Q or P. Hence, these matrices are invertible. Because, if there is v 0 that is in the null-space of P and Q, so we have Pv = 0 and Qv = 0, respectively. This is a contradiction! Multiplying the equation (3) from the left with (Q 1 ) T and from the right with Q 1 gives PQ 1 (Q 1 ) T P T = 0 and thus shows that PQ 1 = (PQ 1 ) T. So, C = PQ 1 is complex symmetric. Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

15 Hagedorn Parametrization of Gaussian Wavepackets Furthermore, we have (Im C)(QQ ) = 1 2 ı (PQ 1 (Q 1 ) P )QQ = 1 2 ı (PQ (Q ) 1 (P Q)Q ) = 1 2 ı (PQ (Q ) 1 (Q P 2 ıi)q ) = 1 2 ı PQ 1 2 ı (Q ) 1 Q PQ ı (Q ) 1 (2 ıi)q = 1 2 ı PQ 1 2 ı PQ + I = I. = (Im C) = (QQ ) 1. where we use the equation (4) for substituting P Q = Q P 2 ıi. Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

16 Hagedorn Parametrization of Gaussian Wavepackets Conversely, for a complex symmetric matrix C with positive definite imaginary part we set Q = (Im C) 1/2 and P = CQ. It is easily verified that these matrices satisfy the relations (3) and (4): and Q T P P T Q = [(ImC) 1/2 ] T [C][(ImC) 1/2 ] [(ImC) 1/2 ] T [C] T [(ImC) 1/2 ] = 0 Q P P Q = [(ImC) 1/2 ] [C][(ImC) 1/2 ] [(ImC) 1/2 ] [C] [(ImC) 1/2 = 2 ıim([(imc) 1/2 ] [C][(ImC) 1/2 ]) = 2 ı[(imc) 1/2 ] [ImC][(ImC) 1/2 ] = 2 ı[(imc) 1/2 ] [(ImC) 1/2 ] = 2 ı[(imc) ] 1/2 [(ImC) 1/2 ] = 2 ıi. Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

17 Relationship with Symplectic Matrices Definition A real matrix Y R 2N 2N is symplectic if it satisfies the relation ( ) Y T 0 I JY = J with J =. (6) I 0 Note that the relations (3) and (4) are equivalent to stating that the matrix ( ) Re Q Im Q Y = Re P Im P is symplectic. Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

18 Gaussian Wavepackets and Quadratic Hamiltonians Gaussian wavepackets with appropriately evolving parameters are exact solutions to the time-dependent Schrödinger equation (1) in the case of a quadratic potential V. It turns out that the equations of motion assume a particularly appealing form with Hagedorn s parametrization. Consider the classical equations of motion associated with (1), q = p m, ṗ = V(q). (7) Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

19 Gaussian Wavepackets and Quadratic Hamiltonians Also, consider their linearization along (q(t), p(t)), Q = P m, Ṗ = 2 V(q)Q, (8) and the classical action integral S(t) = t 0 ( p(s) 2 2m ) V(q(s)) ds. (9) Then, there is the following result for Gaussian Wavepackets in a Quadratic Potential due to Hagedorn 1980: Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

20 Gaussian Wavepackets in a Quadratic Potential Theorem Let V be a quadratic potential, and let (q(t), p(t), Q(t), P(t)) for 0 t t be a solution of the classical equations (7) and (8) and S(t) the corresponding action (9). Assume that Q(0) and P(0) satisfy the symplecticity relations. Then, Q(t) and P(t) satisfy the symplecticity relations for all times t, and ψ(x, t) = e ıs(t) ε ϕ ε 0 [q(t), p(t), Q(t), P(t)](x) (10) is a solution of the time-dependent Schrödinger equation (1). Proof. The fact that symplecticity relations are preserved under (8), is a consequence of the next lemma below. A direct, lengthy calculation shows that (10) is a solution of the Schrödinger equation (1). Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

21 Gaussian Wavepackets in a Quadratic Potential Lemma Suppose that Q(t), P(t) C N N satisfy the differential equations Q(t) = F(t)P(t) Ṗ(t) = G(t)Q(t) with real symmetric matrices F(t) and G(t). If the symplecticity relations hold at t = 0, then they hold for all t. Proof. We have d dt (QT P P T Q) = Q T P + Q T Ṗ ṖT Q P T Q = (FP) T P + Q T GQ (GQ) T Q P T FP = P T F T P + Q T GQ Q T G T Q P T FP = P T FP + Q T GQ Q T GQ P T FP = 0. Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

22 Gaussian Wavepackets in a Quadratic Potential Thus, We have Q T P P T Q = C = Q T P P T Q = 0. d dt (Q P P Q) = Q P + Q Ṗ Ṗ Q P Q = P FP + Q GQ Q GQ P FP = 0 and consequently, Q P P Q = C, then Q P P Q = 2 ıi. Because (Q P P Q)(0) = 2 ıi. We proved that if the symplecticity relations hold at t = 0, then they hold for all t. We set F = 1 m and G = 2 V(q) to complete the proof. Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

23 Ladder Operators In this section, we introduce the ladder operators to construct the Hagedorn functions. An analogous construction via appropriate parameter-dependent ladder operators yields the Hagedorn functions. Let ε > 0 be the small semi-classical parameter in the Schrödinger equation (1). We let q, p R N position and momentum parameters, and Q, P C N N complex matrices satisfying the symplecticity relations. We denote the position and momentum operators by ˆq = (ˆq j ) N j=1 and ˆp = (ˆp j ) N j=1, respectively. So for ψ, we have (ˆqψ)(x) = xψ(x), (ˆpψ)(x) = ıε ψ(x) (x R N ). Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

24 Ladder Operators Also, we consider the commutator relation 1 ıε [ˆq j,ˆp k ] = δ jk. (11) Hagedorn (1998) introduces the parameter-dependent lowering and raising ladder operators A = (A j ) N j=1 and A = (A j )N j=1 as A = A[q, p, Q, P] = ı 2ε (P T (ˆq q) Q T (ˆp p)) A = A [q, p, Q, P] = ı 2ε (P (ˆq q) Q (ˆp p)). (12) Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

25 Ladder Operators In this section, we state some key properties of the ladder operators as follows. Lemma If Q and P satisfy the symplecticity relations, then we have the commutator identities [A j, A k ] = δ jk. (13) Moreover, A j is adjoint to A j on the Schwartz space S: A k ϕ ψ = ϕ A jψ ϕ,ψ S. (14) Note that the Schwartz space is the function space S = {f L 2 (,+ ) ; sup x α f (β) (x) <, α,β N}. Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

26 Ladder Operators Proof. With Q = (Q jk ) and P = (P jk ), we have (we let q = p = 0 for simplicity) [A j, A ı N k ] = [ (P ljˆq l Q ljˆp l ), 2ε l=1 ı N ( P mkˆq m Q mkˆp m )] 2ε m=1 ( = 1 N )( N ) (P ljˆq l Q ljˆp l ) ( P mkˆq m Q mkˆp m ) 2ε l=1 m=1 ( 1 N )( N ) ( P mkˆq m Q mkˆp m ) (P ljˆq l Q ljˆp l ) 2ε m=1 l=1 Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

27 Ladder Operators = 1 2ε N 1 2ε N m=1 l=1 l=1 m=1 = 1 2ε N l=1 m=1 N (P ljˆq l Q ljˆp l )( P mkˆq m Q mkˆp m ) N ( P mkˆq m Q mkˆp m )(P ljˆq l Q ljˆp l ) N P lj Pmk (ˆq lˆq m ˆq mˆq l )+ P mk Q lj (ˆq mˆp l ˆp lˆq m ) + Q mk P lj (ˆp mˆq l ˆq lˆp m )+Q lj Qmk (ˆp lˆp m ˆp mˆp l ) = 1 N N ( P mk Q lj Q mk P lj )( ıεδ lm ) 2ε l=1 m=1 where we use the commutator relation 1 ıε [ˆq j,ˆp k ] = δ jk. Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

28 Ladder Operators This simplifies to [A j, A k ] = ı 2 N ( P lj Qlk + Q lj Plk ) = ı 2 ( Q P + P Q) k,j. l=1 By the symplecticity relations, this equals ı 2 ( 2 ıi) k,j and δ jk. Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

29 Ladder Operators To verify (14), we write out A j ϕ ψ = ı d ( P ljˆq l Q ljˆp l )ϕ 2ε ψ l=1 = ϕ ı d (P ljˆq l Q ljˆp l )ψ = ϕ A j ψ, 2ε l=1 where we just use that ˆq l and ˆp l are self-adjoint operators and also we use the property of inner product ax, y = a x, y = x, ay. Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

30 Hagedorn Wavepackets Lemma If Q and P satisfy the symplecticity relations, then the complex Gaussian ϕ 0 = ϕ ε 0 [q, p, Q, P] spans the null-space of A. Proof. If ϕ S is in the null-space of A, then we must have ϕ {x Ax = 0}: 0 = Aϕ = ı ) (P T (ˆq q) Q T (ˆp p) ϕ(x) 2ε = ı 2ε P T (ˆqϕ(x) qϕ(x))+ ı 2ε Q T (ˆpϕ(x) pϕ(x)) = ı 2ε P T (xϕ(x) qϕ(x))+ ı 2ε Q T ( ıε ϕ(x) pϕ(x)) = Q T ( ıε ϕ(x) pϕ(x)) P T (x q)ϕ(x) = Q T ( ıε ϕ(x) pϕ(x)) = P T (x q)ϕ(x). Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

31 Hagedorn Wavepackets So, we have ıε ϕ(x) pϕ(x) = PQ 1 (x q)ϕ(x). Then ϕ must satisfy the linear system of partial differential equations ıε ϕ(x) pϕ(x) = C(x q)ϕ(x) with the complex symmetric matrix C = PQ 1. Multiples of ϕ 0 are the only non-trivial solutions of this equation. Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

32 Hagedorn Functions Hagedorn constructs a complete L 2 -orthonormal set of functions by ϕ k (x) = ϕ ε k [q, p, Q, P](x), ϕ k+ j = 1 kj + 1 A j ϕ k. (15) for multi-indices k = (k 1,..., k N ) with non-negative integers k j. And, j = e j = ( ) denotes the jth unit vector. It then turns out that these functions are orthonormal. Moreover, we have ϕ k j = 1 kj A j ϕ k. (16) Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

33 Hagedorn Functions The above relations imply since, QA + QA = 2 (ˆq q), ε QA + QA ( ) ı = Q ( P(ˆq q) Q(ˆp p)) 2ε = ı 2ε ( Q P(ˆq q) QP(ˆq q) ) = ı (ˆq q)(q P QP) 2ε 2 = (ˆq q). ε ( + Q ı ) (P(ˆq q) Q(ˆp p)) 2ε Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

34 Hagedorn Functions Multiplying (15) and (16) by Q and Q and then adding them, we have Q( k j + 1ϕ k+ j ) = (QA + QA)ϕ k Q( k j ϕ k j ), and recalling QA + QA = relation ( ) Q k j + 1ϕ k+ j (x) = 2 ε (ˆq q), we obtain the recurrence 2 ε (ˆq q)ϕ k(x) Q ( ) k j ϕ k j (x). (17) where j = e j = ( ) denotes the jth unit vector. This relation permits us to compute the functions ϕ k at any given value x. Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

35 Hagedorn Functions The following theorem describes the Hagedorn functions. Theorem The functions ϕ k = ϕ ε k [q, p, Q, P] defined by ϕ ε 0 [q, p, Q, P](x) = (πε) N 4(det Q) 1 2 exp [ ı 2ε (x q)t PQ 1 (x q)+ ı ε pt (x q) ], and ϕ k+ j = 1 kj + 1 A j ϕ k. form a complete L 2 -orthonormal set of functions. Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

36 Evolution of Ladder Operators Under Quadratic Hamiltonians Proof. The orthonormality of the functions ϕ k follows from their property as eigenfunctions of the symmetric operator AA. The completeness is obtained by an extension of the arguments in the proof of completeness of the Hermite functions. Along a solution (q(t), p(t), Q(t), P(t)) of the classical equations of motion, we consider the time-dependent operators A j (t) = A j [q(t), p(t), Q(t), P(t)] and A j (t) = A j [q(t), p(t), Q(t), P(t)]. The next lemma represents the derivatives of these operators. Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

37 Evolution of Ladder Operators Under Quadratic Hamiltonians Lemma In the case of a quadratic potential V : ψ(x) x 2 ψ(x), we have Ȧ j = 1 ıε [A j, H] and Ȧ j = 1 ıε [A j, H]. Proof. With classical equations of motion (7) and (8), we obtain for A(t) = (A j (t)) Ȧ = 1 ıε (QT V(ˆq)+ 1 m PTˆp). The same expression is obtained for 1 ıε [A, H] on using the commutator relations 1 ıε [ˆq j,ˆp k ] = δ jk and the ensuing 1 ıε [ˆq j,ˆp k 2] = δ jk 2ˆp k and 1 ıε [ˆq2 j,ˆp k ] = δ jk 2ˆq j. The result for A is obtained by taking complex conjugates. Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

38 Hagedorn Functions in a Quadratic Potential We now have all ingredients for the following remarkable result due to Hagedorn, Theorem Let V be a quadratic potential, and let (q(t), p(t), Q(t), P(t)) be a solution of the classical equations of motion (7) and (8), and S(t) the corresponding action (9). Assume that Q(0) and P(0) satisfy the symplecticity relations. Then, for every multi-index k, ψ(x, t) = e ıs(t) ε ϕ ε k [q(t), p(t), Q(t), P(t)](x) is a solution of the time-dependent Schrödinger equation (1). Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

39 Hagedorn Functions in a Quadratic Potential Proof. We know from (10) in the previous theorem, that the statement is correct for k = 0. In view of the construction of the functions ϕ k by ϕ k+ j = 1 kj +1 A j ϕ k, the result follows by induction if we can show that with a solution ψ(., t), also A jψ(., t) is a solution of the Schrödinger equation (1). ψ(., t) is a solution of the Schrödinger equation (1). Thus ıε t ψ = Hψ. Also, we have [A j, H]ψ = A j Hψ HA jψ. So, ıε t (A j ψ) = ıεȧ j ψ + A j Hψ = ( ıεȧ j ψ +[A j, H]ψ)+HA j ψ = HA j ψ. Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

40 Approximation Solutions to the Schrödinger equation We will approximate solutions to the Schrödinger equation (1) in the form ψ(x, t) ψ K (x, t) = e ıs(t) ε with a finite multi-index set K. c k (t)ϕ k (x, t) (18) k K In the following section, we give a fully discrete, explicit, and time-reversible timestepping algorithm to propagate the Gaussian parameters q(t), p(t), Q(t), P(t), the phase S(t), and the coefficients c k (t) of a Hagedorn wavepacket (18). Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

41 Section 3: A Numerical Integrator for Hagedorn Wavepackets Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

42 A Splitting Method for Time Integration We describe an algorithm by Faou, Gradinaru and Lubich (2008) for the approximate solution of the time-dependent Schrödinger equation (1) ıε ψ = Hψ t with the Hamiltonian H = T + V, in the semi-classical regime using Hagedorn wave packets. The method is based on the splitting between the kinetic operator and potential operator T = ε2 2m 2 V = V(x). Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

43 A Splitting Method for Time Integration We consider the free linear Schrödinger equation ıε ψ t and the equation with only a potential = ε2 2m 2 ψ (19) ıε ψ t = V(x)ψ. (20) The potential will be further decomposed into its quadratic part at the current position q and the non-quadratic remainder. So we have H = T + U q(t) + W q(t). Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

44 A Splitting Method for Time Integration We now describe the three main ingredients in the time-stepping algorithm. Starting with a Hagedorn wavepacket (18) as initial data for the Schrödinger equation, we make use of the following: 1- We can solve exactly the free linear Schrödinger equation (19), with the wave function remaining in the Hagedorn wavepacket form (18) with unaltered coefficients c k. Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

45 A Splitting Method for Time Integration We now describe the three main ingredients in the time-stepping algorithm. Starting with a Hagedorn wavepacket (18) as initial data for the Schrödinger equation, we make use of the following: 1- We can solve exactly the free linear Schrödinger equation (19), with the wave function remaining in the Hagedorn wavepacket form (18) with unaltered coefficients c k. 2- For a quadratic potential, we can solve exactly the potential equation (20) with the wave function remaining in the Hagedorn wavepacket form (18) with the same coefficients c k. Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

46 A Splitting Method for Time Integration We now describe the three main ingredients in the time-stepping algorithm. Starting with a Hagedorn wavepacket (18) as initial data for the Schrödinger equation, we make use of the following: 1- We can solve exactly the free linear Schrödinger equation (19), with the wave function remaining in the Hagedorn wavepacket form (18) with unaltered coefficients c k. 2- For a quadratic potential, we can solve exactly the potential equation (20) with the wave function remaining in the Hagedorn wavepacket form (18) with the same coefficients c k. 3- For an arbitrary potential, we can compute the Galerkin approximation of the potential equation (20) on the linear space spanned by the functions ϕ k with fixed parameters q, p, Q, P, letting the coefficients c k in the formulation (18) vary. Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

47 Kinetic Part and Quadratic Potential We will use the following propositions, which are direct consequences of the previous Theorem. Theorem A time-dependent Hagedorn wavepackets (18) solves the free linear Schrödinger equation (19) if q(t) = q(0)+ t m p(0), Q(t) = Q(0)+ t m P(0), (21) S(t) = S(0)+ t 2m p(0) 2, and p(t) = p(0), P(t) = P(0), c k (t) = c k (0). Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

48 Kinetic Part and Quadratic Potential Theorem Let U(x) be a quadratic potential. A time-dependent Hagedorn wavepackets (18) solves the potential equation (20) with V = U if p(t) = p(0) t U(q(0)), P(t) = P(0) t 2 U(q(0))Q(0), (22) S(t) = S(0) tu(q(0)), and q(t) = q(0), Q(t) = Q(0), c k (t) = c k (0). Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

49 The Practical Time-Stepping Algorithm We now give a full algorithmic description. Assume that the stepsize t is given, and let the real N-vectors q n, p n, the complex N N matrices Q n, P n, the real scalar S n, and the complex coefficient vector c n = (c n k ) k K be such that ψ n = e ısn ε ck n ϕε k [qn, p n, Q n, P n ] k K is an approximation to the solution of the Schrödinger equation at time t n = n t. Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

50 The Practical Time-Stepping Algorithm To compute the approximation ψ n+1 at time t n+1 = t n + t we proceed as follows: Half-step of the kinetic part: 1. Compute q n+1/2, Q n+1/2, and S n+1/2, : q n+1/2 = q n + t 2m pn, Q n+1/2 = Q n + t 2m Pn, (23) S n+1/2, = S n + t 4m pn 2. Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

51 The Practical Time-Stepping Algorithm Full step of the potential part: 2. Compute p n+1, P n+1, and S n+1/2,+ : p n+1 = p n t V(q n+1/2 ), P n+1 = P n t 2 V(q n+1/2 )Q n+1/2, (24) S n+1/2,+ = S n+1/2, tv(q n+1/2 ). Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

52 The Practical Time-Stepping Algorithm Half-step of the kinetic part: 3. Compute q n+1, Q n+1, and S n+1 : q n+1 = q n+1/2 + t 2m pn+1, Q n+1 = Q n+1/2 + t 2m Pn+1, (25) S n+1 = S n+1/2,+ + t 4m pn+1 2. Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

53 Numerical Results Example: Let s consider the time-dependent Schrödinger equation on R 2 and the modified Henon-Heiles potential V(x) = 1 2 (x x 2 2 )2 + 2x 1 x x 3 1. As initial value we take the normalized Gaussian wavepacket in Hagedorn s parametrization with the identity matrix Q 0 = I and P 0 = ıi, localized around q 0 = ( 0.1, 0) T and with p 0 = (0, 0.1) T. We apply the numerical method with the stepsize t = 0.1 and coefficients c k for k = (k 1, k 2 ) such that (1+k 1 )(1+k 2 ) 8 and 20 basis functions. Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

54 Comparison with the Fourier method Error vs. time in a 2D example (Henon-Heiles) Hagedorn with 20 basis functions at the left, Fourier with 2 20 basis functions at the right. Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

55 Error Behaviour Maximum error vs. number of basis functions at t = 1, t = 5. Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

56 Conclusion Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

57 Conclusion We considered the approximation of multi-particle quantum dynamics in the semiclassical regime by Hagedorn wavepackets, which are products of complex Gaussians with polynomials that form an orthonormal L 2 basis and preserve their type under propagation in Schrödinger equations with quadratic potentials. We built a time-reversible, fully explicit time-stepping algorithm to approximate the solution of the Hagedorn wavepacket dynamics. The algorithm is based on a splitting of the Hamiltonian operator into the kinetic and potential parts, as well as on a splitting of the potential into its local quadratic approximation and the remainder. The algorithm preserves the symplecticity relations and the L 2 norm of the wavepackets. Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

58 References C. Lubich, From quantum to classical molecular dynamics: reduced models and numerical analysis, EMS, Zürich, E. Faou, V. Gradinaru, C. Lubich, Computing semi-classical quantum dynamics with Hagedorn wavepackets, SIAM J. Sci. Comp. 31 (2009), G. A. Hagedorn, Raising and lowering operators for semi-classical wavepackets, Ann. Physics 269 (1998), Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56

59 Thank you for your attention... Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, / 56