solutionofthetime-dependentschrödingerequation usingthecrank ... · 2017-05-10 · peter leitner...

Solution of the Time-dependent Schrödinger Equationusing the Crank-Nicolson algorithm with MPI on a 2-D

regular Cartesian gridSeminar on High Performance Computing 2

Summer term 2017

Peter Leitner & Stefan Hofmeister

Wednesday, May 10, 2017

Peter Leitner & Stefan Hofmeister Crank-Nicolson using MPI Wednesday, May 10, 2017 1 / 13

Time-dependent Schrödinger Equation

Schrödinger Equation in 2-D for an electron in a given potential

i~∂

∂t|ψ(t, x)〉 = H(p, x) |ψ(t, x)〉 , H =

p2x

2m+

p2y

2m+ V (x , y) (1)

Attempt to simply discretize the differential operators,

∂tψ(t, x) ≈ψn+1j ,l − ψ

nj ,l

∆t,

p2

2mψ(t, x) = − ~2

2m∇2ψ(t, x) ≈ − ~2

2m

ψn+1j+1,l − 2ψn+1

j ,l + ψn+1j−1

(∆x)2 + . . .

is – as the von Neumann stability analysis shows – unconditionally stable.However, each solution of Schrödinger’s equation has to fulfill a continuityequation for the probability w(t, x) d3x = ψ∗ψ d3x to find the electronwithin a volume d3x around x ,

∂tw(t, x) +∇ · j = 0, j =~

2im[ψ∗(∇ψ)− (∇ψ∗)ψ] (2)


From the continuity equation follows the conservation of the norm of thewave function:

ddt

∫d3x w(t, x) =

∫d3x ∂tw(t, x) = −

∫d3x ∇ · j (t, x) = −

∫dS · j (t, x)

∫R3

d3x ‖w(0, x)‖2 = 1 SE−→∫R3

d3x ‖w(t, x)‖2 = 1 (3)

The normalization of the wave function ψ(t, x) is however not conserved by the fullyimplicit discretization as this method is not unitary ⇒ a discretization preserving thehermiticity of the Hamiltonian is needed that maintains the normalization of thewavefunction.The temporal evolution of the wave function can be expressed using the unitarytime-evolution operator U(t):

|ψ(t + ∆t, x)〉 ' U(t) |ψ(t, x)〉 , U(t) = exp(−iHt/~) (4)

With the time-evolution operator one can find the explicit Forward-Time Centered-Space(FTCS) scheme ψn+1

j,l = (1− iH∆t)ψnj,l and the implicit Backward-Time scheme

(1 + iH∆t)ψn+1j,l = ψn

j,l . Neither scheme is unitary and thus violates probabilityconservation.


A finite-difference approximation of exp(−iH∆t/~) that is unitary wasfound by Cayley:

exp(−iH∆t/~) =1− iH∆t

2~

1 + iH∆t2~

+O((∆t)2)

(4)⇒(1 +

i2~

H∆t)|ψ(t + ∆t, x)〉 =

(1− i

2~H∆t

)|ψ(t, x)〉 (5)

This is exactly the Crank-Nicolson scheme for the Schrödinger equation.


Discretization

By substituting the 2-D discrete wave function ψ(t; x , y)→ ψnj ,l into

Eq. (5) and considering the Hamiltonian in the form

H = |p = −i~∇| = − ~2

2m∂2

∂x2 −~2

2m∂2

∂y2 + V (x , y)

one obtains the discretized 2-D Schrödinger Equation

ψn+1j,l −

i∆t

2~

(~2

2m

ψj+1,ln + 1− 2ψn+1j,l + ψn+1

j−1,l

(∆x)2+ψn+1j,l+1 − 2ψn+1

j,l + ψn+1j,l−1

(∆y)2− Vj,lψ

n+1j,l

)

= ψnj,l +

i∆t

2~

(~2

2m

ψnj+1,l − 2ψn

j,l + ψnj−1,l

(∆x)2+

~2

2m

ψnj,l+1 − 2ψn

j,l + ψnj,l−1

(∆y)2− Vj,lψj,l

)(6)


Rewrite Eq. (6) in matrix-form. Assign a short-hand notation for the prefactors of the wavefunction at the next time step at various spatial grid points:

ψn+1j,l : bj,l = 1 +

i∆t

~~2

2m

(1

(∆x)2+

1(∆y)2

)+

i∆t

2~Vj,l

ψn+1j−1,l : c = −

i∆t

2~~2

2m1

(∆x)2

ψn+1j+1,l : a = c

ψn+1j,l+1 : d = −

i∆t

2~~2

2m1

(∆x)2

ψn+1j,l−1 : e = d

and for the RHS:

ψnj,l : fj,l = 1−

i∆t

~~2

2m

(1

(∆x)2+

1(∆y)2

)−

i∆t

2~Vj,l

ψnj+1,l : g =

i∆t

2~~2

2m1

(∆x)2

ψnj−1,l : h = g

ψnj,l+1 : k =

i∆t

2~~2

2m1

(∆y)2

ψnj,l−1 : p = k


LHS of the discretized Schrödinger Equation

Map the two indices j and l of the wave function ψnj,l on a single one (j , l) 7→ i :

i = 1 + (j − 1)(L− 1) + l − 1 for j = 0, 1, . . . , J; l = 0, 1, . . . , L

Thereby ψn1,1 → ψn

1, ψn1,2 → ψn

2, . . ., ψn1,J → ψn

J , ψn2,1 → ψn

J+1, . . . For simplicity (and lack ofspace) shown here only for J = L = 4, though later J, L ∼ 103 will be used. It exhibitssub-blocks of shape J × L:

b1,1 d 0 a 0 0 0 0 0e b1,2 d 0 a 0 0 0 00 e b1,3 0 0 a 0 0 0c 0 0 b2,1 d 0 a 0 00 c 0 e b2,2 d 0 a 00 0 c 0 e b2,3 0 0 a0 0 0 c 0 0 b3,1 d 00 0 0 0 c 0 e b3,2 d0 0 0 0 0 c 0 e b3,3

ψn+11ψn+1

2ψn+1

3ψn+1

4ψn+1

5ψn+1

6ψn+1

7ψn+1

8ψn+1

9

= r


RHS of the discretized Schrödinger Equation

Vectorization of the RHS is not necessary when working with Fortranexclusively but essential when using the numpy module with Python:

r =

f1,1 k 0 g 0 0 0 0 0p f1,2 k 0 g 0 0 0 00 p f1,3 0 0 g 0 0 0h 0 0 f2,1 k 0 g 0 00 h 0 p f2,2 k 0 g 00 0 h 0 p f2,3 0 0 g0 0 0 h 0 0 f3,1 k 00 0 0 0 h 0 p f3,2 k0 0 0 0 0 h 0 p f33

ψn1ψn

2ψn

3ψn

4ψn

5ψn

6ψn

7ψn

8ψn

9


After vectorization, the discretized Schrödinger Equation simplifies to

Aψn+1 = r = Bψn

Pseudo-code:

for n in range(N):

Compute Bψn = r as a matrix-vector productSolve Aψn+1 = r with LU decompositionThis yields ψn+1

Map the rolling index i back to indices j , l : ψn+1i → ψn+1

j ,l

Check that the wave packet remains normalized:∫∆2

d2x ‖ψ(t, x)‖2 ≈J,L∑j ,l=1

∆x∆y∥∥ψn

j ,l

∥∥2 != 1


Setting up the matrices in Python

# Build the main diagonalb_main_diag = [] # coefficient matrix of LHSf_main_diag = [] # coefficient matrix of RHSfor j in range(J-1):

for l in range(J-1):b_main_diag.append(b[j,l]); f_main_diag.append(f[j,l])

e_minor_diag = np.ones((len(b_main_diag)-1,), dtype=complex)*ed_minor_diag = np.ones((len(b_main_diag)-1,), dtype=complex)*da_minor_diag = np.ones((len(b_main_diag)-3,), dtype=complex)*ac_minor_diag = np.ones((len(b_main_diag)-3,), dtype=complex)*ck_minor_diag = np.ones((len(f_main_diag)-1,), dtype=complex)*kp_minor_diag = np.ones((len(f_main_diag)-1,), dtype=complex)*pg_minor_diag = np.ones((len(f_main_diag)-3,), dtype=complex)*gh_minor_diag = np.ones((len(f_main_diag)-3,), dtype=complex)*h

# Build the coefficient matrices for A*psi^(n+1) = B*psi^nA = np.diag(b_main_diag) + np.diag(e_minor_diag,-1) + np.diag(d_minor_diag,1) \

+ np.diag(c_minor_diag,-3) + np.diag(a_minor_diag,3)B = np.diag(f_main_diag) + np.diag(p_minor_diag,-1) + np.diag(k_minor_diag,1) \

+ np.diag(h_minor_diag,-3) + np.diag(g_minor_diag,3)


Setting-up the RHS in Python

for n in range(Nt):if n == 0:

psi_n_mat = psi_0# Rewrite matrix psi_jl as a vector psi_ipsi_n_vec = np.empty(((J-1)**2,), dtype=complex)for j in range(J-1):

for l in range(J-1):i = ind(j,l)psi_n_vec[i] = psi_n_mat[j,l]

else:psi_n_vec = psi_np1_vec

RHS = np.dot(B,psi_n_vec)r = RHS


LU decomposition; finally to be parallelized as Fortran code

# LU decomposition:P, L, U = scipy.linalg.lu(A)LU = L + UnA = np.size(A, 0)y = np.zeros((nA,), dtype=complex)x = np.zeros((nA,), dtype=complex)

r = np.dot(np.linalg.inv(P),r)

for i in range(1, nA+1):s = r[i-1]for j in range(1, i):

s = s - LU[i-1,j-1]*y[j-1]y[i-1] = s

for i in range(nA, 0, -1):s = y[i-1]for j in range(i+1, nA+1):

s = s - LU[i-1,j-1]*x[j-1]x[i-1] = s/LU[i-1,i-1]

psi_np1_vec = x


Parallelizing the code

The following strategies for code parallelization systems are envisaged:Application of Distributed SuperLU (SuperLU_DIST) of the SuperLUlibraryCrank-Nicolson predictor-corrector (CNPC) method

Decomposition of spatial domain Ω in subdomains Ωi which can beassigned to their own processorForward Euler to predict unknown values on the interface Γ betweensubdomains Ωi ⊂ ΩSolving the problem independently on the subdomains Ωi withBackward Euler using the predicted values ψ∗

Γ at tn+1Correction of interface values ψn+1

Γ with Backward Euler using interiorvalues


solutionofthetime-dependentschrödingerequation usingthecrank ... · 2017-05-10 · peter leitner...

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