Integral Equations in Quantum Mechanics II I Bound States, II - PowerPoint PPT Presentation

Integral Equations in Quantum Mechanics II I Bound States, II Scattering* Rubin H Landau Sally Haerer, Producer-Director Based on A Survey of Computational Physics by Landau, Páez, & Bordeianu with Support from the National Science Foundation Course: Computational Physics II 1 / 1

Problem: Nonlocal Potential Scattering Integro-Differential Equation m 1 -k k’ -k’ r r’ r’ k m 2 Problem: Scattering from nonlocal potential d 2 ψ ( r ) − 1 � dr ′ V ( r , r ′ ) ψ ( r ′ ) = E ψ ( r ) + (1) 2 m dr 2 Avoid Integro-Differential Equation 2 / 1

Theory: Lippmann–Schwinger Equation Integral Form Schrödinger Equation m 1 -k k’ -k’ k m 2 Better solve scattering amplitudes = observable ( � = ψ ) P = Cauchy principal-value prescription; l = 0, � = 1 � ∞ dp p 2 V ( k ′ , p ) R ( p , k ) R ( k ′ , k ) = V ( k ′ , k ) + 2 π P (1) ( k 2 0 − p 2 ) / 2 µ 0 E = k 2 m 1 m 2 0 2 µ, µ = (2) m 1 + m 2 3 / 1

Math: Computing Singular Integrals How Handle Singularity? Singular integral G has singular integrand g ( k ) : � b G = g ( k ) dk � = ∞ (1) a Computational danger near singularity 4 / 1

Math: Computing Singular Integrals What to Do at Singularity? Give energy k 0 small imaginary part ± i ǫ Cauchy principal-value P = pinch* � + ∞ �� k 0 − ǫ � + ∞ � P f ( k ) dk = lim f ( k ) dk + f ( k ) dk (1) ǫ → 0 k 0 + ǫ −∞ −∞ 5 / 1

Numerical Principal Values Subtract Zero Integral (Hilbert Transform) � + ∞ � + ∞ dk dk P = P = 0 (1) k 2 − k 2 k − k 0 0 −∞ 0 � + ∞ � + ∞ f ( k ) dk [ f ( k ) − f ( k 0 )] dk P = (2) k 2 − k 2 k 2 − k 2 0 0 0 0 NB No RHS P , no k = k 0 singularity 6 / 1

Method: Integral- to Linear- to Matrix-Equations Rewrite Principal-Value as Definite Integral � ∞ dp p 2 V ( k ′ , p ) R ( p , k ) − k 2 R ( k ′ , k ) = V ( k ′ , k ) + 2 0 V ( k ′ , k 0 ) R ( k 0 , k ) (1) π ( k 2 0 − p 2 ) / 2 µ 0 Convert to linear equations; approximate integral: N k 2 j V ( k , k j ) R ( k j , k 0 ) w j R ( k , k 0 ) ≃ V ( k , k 0 ) + 2 � π ( k 2 0 − k 2 j ) / 2 µ j = 1 N − 2 w m π k 2 � 0 V ( k , k 0 ) R ( k 0 , k 0 ) (2) ( k 2 0 − k 2 m ) / 2 µ m = 1 ( N + 1 ) unknown R ( k j , k 0 ) , j = 0 , N 7 / 1

Method: Integral- to Linear- to Matrix-Equations Evaluate for k = N Gauss Points k + Experimental k 0 � k j , j = 1 , N (quadrature points), k = k i = (1) k 0 , i = 0 (experimental point) N N k 2 j V ij R j w j R i = V i + 2 j ) / 2 µ − 2 w m � π k 2 � 0 V i 0 R 0 (2) π ( k 2 0 − k 2 ( k 2 0 − k 2 m ) / 2 µ j = 1 m = 1 Express as matrix equations: w i k 2  + 2 i ) / 2 µ , for i = 1 , N , i  ( k 2 0 − k 2 π  D i = (3) w j k 2 − 2 � N j ) / 2 µ , 0 for i = 0   π j = 1 ( k 2 0 − k 2 R = ( 1 − DV ) − 1 V R = V + DVR ⇒ (4) 8 / 1

Solution via Matrix Inversion, Gaussian Elimination R = V + DVR (1) R = ( 1 − DV ) − 1 V (2) Matrix inversion = direct, not fastest Matrix inversion = standard in mathematical libraries Useful if need [ 1 − DV ] − 1 Else Gaussian elimination 9 / 1

Implementation: Delta-Shell Potential sin 2 δ 0 ∝ l = 0 Cross Section 1 V ( k ′ , k ) = −| λ | sin ( k ′ b ) sin ( kb ) (1) 2 µ k ′ k Check analytic phase shift: Analytic 0 0 2 4 6 kb λ b sin 2 ( kb ) tan δ 0 = (2) kb − λ b sin ( kb ) cos ( kb ) R ( k 0 , k 0 ) = − tan δ (3) 2 µ k 0 Estimate precision by increasing N grid points ( N = 26) 10 / 1