Advanced Computational Physics Course Hartmut Ruhl, LMU, Munich. People involved. People involved. Literature. Schrödinger equation.

Size: px

Start display at page:

Download "Advanced Computational Physics Course Hartmut Ruhl, LMU, Munich. People involved. People involved. Literature. Schrödinger equation."

Roxanne Caldwell
5 years ago
Views:

1 June 13, 017

3 ASC, room A 38, phone , hartmut.ruhl@lmu.de Patrick Böhl, ASC, room A05, phone , patrick.boehl@physik.uni-muenchen.de.

4 Useful literature Dennis M. Sullivan, Electromagnetic Simulation Using The FDTD Method, IEEE Press Series on RF and Microwave Technology, Poger D. Pollard and Richard Booton, ISBN: Frederick Ira Moxley III, Fei Zhu, and Weizhong Dai, A Generalized FDTD Method with Absorbing Boundary Condition for Solving a Time-Dependent Linear Equation, American J. of Comp. Math., (01).

5 The generalized FDTD method To derive a generalized FDTD scheme we write Next we expand ψ Re (x, t n) and ψ Re (x, t n 1 ) about t n 0.5 h t ψ Re (x, t) = ( m A + V (x) ) ψ Im (x, t), h (1) h t ψ Im (x, t) = (+ m A V (x) ) ψ Re (x, t). h () ψ Re (x, t n) = ψ Re (x, t n 0.5 ) + t ψ Re (x, t n 1 ) = ψ Re (x, t n 0.5 ) t t ψ Re (x, t n 0.5 ) (3) + 1! ( t ) t ψ Re (x, t n 0.5) + 1 3! ( t ) 3 3 t ψ Re (x, t n 0.5) ±... t ψ Re (x, t n 0.5 ) (4) + 1! ( t ) t ψ Re (x, t n 0.5) 1 3! ( t ) 3 3 t ψ Re (x, t n 0.5) ±...

6 The generalized FDTD method Subtracting ψ Re (x, t n 1 ) from ψ Re (x, t n) yields ψ Re (x, t n) = ψ Re (x, t n 1 ) + ( t p+1 p=0 ) 1 (p + 1)! p+1 ψ t Re (x, t n 0.5 ). (5) With the help of the the time derivatives can be evaluated. We obtain up to order N ψ Re (x, t n) = ψ Re (x, t n 1 ) (6) N + ( t p+1 p+1 p=0 ) ( 1) ha (p + 1)! ( m V h p+1 ) ψ Im (x, t n 0.5 ) +O ( t N+3 ). Expanding ψ Im (x, t n+0.5 ) and ψ Im (x, t n 0.5 ) about ψ Im (x, t n) and applying the yields ψ Im (x, t n+0.5 ) = ψ Im (x, t n 0.5 ) (7) N + ( t p+1 p p=0 ) ( 1) ha (p + 1)! ( m V h p+1 ) ψ Re (x, t n) +O ( t N+3 ).

7 The generalized FDTD method The discrete becomes ψ n N j,re = ψn 1 j,re + ( t p+1 p+1 p=0 ) ( 1) ha (p + 1)! ( ψ n+0.5 j,im = ψn 0.5 j,im N + ( t p=0 p+1 p ) ( 1) (p + 1)! ( ha m V h p+1 ) Next a specific discrete representation of the operator A has to be given. For ψ n 0.5 j,im, (8) m V h p+1 ) ψ n j,re. (9) A ψ n j,re = ψ n j+1,re ψn j,re + ψn j 1,Re x (10) we obtain A ψ n j,re = 1 k x ( 4 sin x ) λn Re ei(jk x), (11) A ψ n+0.5 j,im = 1 k x ( 4 sin x ) λn Im ei(jk x), (1) where the ansatz ψ n j,re = λn Re ei(jk x), ψ n+0.5 j,im = λn Im ei(jk x) (13) has been made.

8 The generalized FDTD method The discrete becomes λ n Re = λn 1 Re λ n Im = λn 1 Im Alternatively, this can be writen as N + p=0 N + p=0 ( 1) p h t (p + 1)! ( m x ( 1) p+1 sin k x h t (p + 1)! ( k x sin m x + V t h ) p+1 λ n 1 Im, (14) + V t h ) p+1 λ n Re. (15) λ n Re = λn 1 Re + α λn 1 Im, (16) λ n Im = λn 1 Im α λn Re, (17) where N ( 1) p α = p=0 (p + 1)! ( h t m x sin k x + V t h ) p+1. (18) We find λ n+1 Re λn Re = λn Re λn 1 Re + α (λn Im + λn 1 Im ), λn Im = λn 1 Im α λn Re. (19)

9 The generalized FDTD method Finally, we find or λ n+1 Re ( α ) λ n Re + λn 1 Re = 0 (0) λ Re ( α ) λ Re + 1 = 0. (1) The solution of this is This leads to λ Re = e iω t e iω t e iω t = α () 1 cos ω t = α sin ω t = α 4. (3) sin ω t N ( 1) p = p=0 (p + 1)! ( h t m x sin k x + V t p+1 h ). (4)

10 The generalized FDTD method For vanishing potential we obtain for the t 4 x order approximation sin ω t = h t k x sin 1 h t m x 3! ( k x sin m x )3. (5) The plots below compare analytical (yellow), t x order FDTD (red), and t 4 x order FDTD (blue) dispersion relations for (a) a Courant factor of 0.99 and (b) a Courant factor 0.6. For the Courant factor 0.6 the second and first orders are almost identical implying that the higher order FDTD scheme is much more stable than the lowest order one (a) (b)

11 The generalized FDTD method: Comparison For the classical FDTD method we have found i ψ 0,Re sin ( ω t ) = ψ h t 0,Im m x i ψ 0,Im sin ( ω t ) = +ψ h t 0,Re m x sin ( k x ), (6) sin ( k x ) (7) or alternatively i sin ( ω t ) h t m x h t m x sin ( k x sin ( k x ) ) i sin ( ω t ) This yields the lowest order dispersion contribution of the generalized FDTD scheme ( ψ Re ) = 0. (8) ψ Im sin ( ω t ) = ( h t m x ) sin 4 ( k x ). (9)

12 : Dispersion Plot of the discrete dispersion relation of the for (c) x = 0.01µm and t = m e x / h and (d) t = 0.5 m e x / h (c) (d)

13 The generalized FDTD method If we make use of a 4th order discrete representation of the operator A we obtain A ψ n j,re = ψ n j+,re + 16ψn j+1,re 30ψn j,re + 16ψn j 1,Re ψn j,re 1 x (30) A ψ n j,re = 4 k x sin (3 + sin k x 3 x ) λn Re ei(jk x), (31) A ψ n+0.5 j,im = 4 k x sin (3 + sin k x 3 x ) λn Im ei(jk x), (3) where the ansatz ψ n j,re = λn Re ei(jk x), ψ n+0.5 j,im = λn Im ei(jk x) (33) has been made. The discrete becomes ψ n N j,re = ψn 1 j,re + ( t p+1 p+1 p=0 ) ( 1) ha (p + 1)! ( ψ n+0.5 j,im = ψn 0.5 j,im N + ( t p=0 p+1 p ) ( 1) (p + 1)! ( ha m V h p+1 ) ψ n 0.5 j,im, (34) m V h p+1 ) ψ n j,re. (35)

14 The generalized FDTD method For vanishing potential we obtain for the t x 4 order FDTD dispersion sin ω t h t k x = ( sin (3 + sin k x 3m x )) (36) and for the t 4 x 4 order dispersion sin ω t h t k x = ( sin (3 + sin k x 3m x ) (37) 1 h t 3! ( k x sin (3 + sin k x 3m x ))3. The plots below compare analytical (blue), t x 4 order FDTD (red), t x order FDTD (green), t 4 x 4 order FDTD (yellow), and t 4 x order (blue) dispersion relations for (e) a Courant factor of 0.75 and (f) The t 4 x 4 scheme is stable for 0.99 and approximates dispersion better (e) (f)

15 : Exercises Exercise: Write a C-program for the t 4 x and t 4 x 4 order FDTD solvers of the free with periodic boundary conditions in 3D. Assume that x w ψ( x, 0) = e x y w e y z w e z e ik x0 x (38) holds, where w x, w y, and w z determine the widths of the Gaussian wave packet and k x0 is the initial momentum of the wave. Obtain the t 4 x and t 4 x 4 order dispersion relations of the free in 3D. Compare the speed of integration with the t x order scheme.

16 : Dispersion in 3D In 3D we obtain sin ( ω t h t ) = ( m x ) sin 4 kx x ( ) (39) h t + ( m y ) sin 4 ky y ( ) h t + ( m z ) sin 4 kz z ( ) or ω t = arcsin h t ( m x ) sin 4 kx x ( ) (40) h t + ( m y ) sin 4 ky y ( ) 1 h t + ( m z ) sin 4 kz z ( ), where the more general expressions from above have to be inserted for higher order integrators.

17 : Dispersion D Plot of (g) the discrete dispersion relation of the for x = y = 0.01µm and (h) the difference between the analytical and the discrete dispersion relations for 1 t = ( h me x ) + ( h me y ) (41) ky ky (g) kx (h) kx

3 10 5, Gaussian packet widths w x = w y = w z = 0.1. Plot (i) gives the real part

18 Plot of the real part of a Gaussian wave packet propagating to the right with h = m e = 1, box dimensions L x = L y = L z = 1.0, resolution x = y = z = 0.01, t = , Gaussian packet widths w x = w y = w z = 0.1. Plot (i) gives the real part of the wave packet at t = 0 and plot (j) at t = 300 t with the t x order scheme. (i) (j)

3 10 5, Gaussian packet widths w x = w y = w z = 0.1. Plot (k) gives the absolute

19 Plot of the absolute value of a Gaussian wave packet propagating to the right with h = m e = 1, box dimensions L x = L y = L z = 1.0, resolution x = y = z = 0.01, t = , Gaussian packet widths w x = w y = w z = 0.1. Plot (k) gives the absolute value of the wave packet at t = 0 and plot (l) at t = 300 t with the t x order scheme. (k) (l)

20 : Exercises Exercise: Write a C-program for the t x order FDTD solver of the with an attractive and repulsive harmonic potential with periodic boundary conditions in 3D. Assume that x L ψ( x, 0) = e y L x e y z L e z e ik x0 x (4) and x r V ( x) = V 0 e x y r e y z r e z (43) holds, where w x, w y, and w z determine the widths of the Gaussian wave packet and k x0 is the initial momentum of the wave, while r x, r y, and r z represent the widths of the potential.

Superposition of electromagnetic waves

Superposition of electromagnetic waves February 9, So far we have looked at properties of monochromatic plane waves. A more complete picture is found by looking at superpositions of many frequencies. Many