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

Integral Equations in Quantum Mechanics I 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: Bound States in Momentum Space Integro-Differential Equation r r’ r’ N–body interaction reduces to nonlocal V eff ( r ) : d 2 ψ ( r ) − 1 � dr ′ V ( r , r ′ ) ψ ( r ′ ) = E ψ ( r ) + (1) 2 m dr 2 Integro-differential equation Problem: Solve for l = 0 bound-state E i & ψ i 2 / 1

Theory: Momentum-Space Schrödinger Equation Integral Schrödinger Equation Equally Valid Transform Schrödinger Equation to momentum space Replace integro-differential by integral equation: � ∞ k 2 2 µψ ( k ) + 2 dpp 2 V ( k , p ) ψ ( p ) = E ψ ( k ) (1) π 0 V ( k , p ) = p-space representation (TF) of V : � ∞ V ( k , p ) = 1 dr dr ′ sin ( kr ) V ( r , r ′ ) sin ( pr ′ ) (2) kp 0 ψ ( k ) = p-space representation (TF) of ψ : � ∞ ψ ( k ) = dr kr ψ ( r ) sin ( kr ) (3) 0 Will transform into matrix equation (see matrix Chapter) 3 / 1

Algorithm: Integral Equations → Linear Equations Solve on p-Space Grid k k k k 1 2 3 N Integral ≃ weighted sum (see Integration chapter) � ∞ N dpp 2 V ( k , p ) ψ ( p ) ≃ � w j k 2 j V ( k , k j ) ψ ( k j ) (1) 0 j = 1 Integral equation → algebraic equation N k 2 2 µψ ( k ) + 2 � w j k 2 j V ( k , k j ) ψ ( k j ) = E (2) π j = 1 N unknowns ψ ( k j ) Solve on grid, k = k i 1 unknown E → N couple equations Unknown function ψ ( k ) ( N + 1 ) unknowns: 4 / 1

Algorithm: Integral Equations → Linear Equations Solve on p-Space Grid k k k k 1 2 3 N N k 2 2 µψ ( k i ) + 2 � i w j k 2 j V ( k i , k j ) ψ ( k j ) = E ψ ( k i ) , i = 1 , N (1) π j = 1 e.g. N = 2 ⇒ 2 coupled linear equations k 2 2 µψ ( k 1 ) + 2 1 π w 1 k 2 1 V ( k 1 , k 1 ) ψ ( k 1 ) + w 2 k 2 2 V ( k 1 , k 2 ) ψ ( k 2 ) = E ψ ( k 1 ) (2) k 2 2 µψ ( k 2 ) + 2 2 π w 1 k 2 1 V ( k 2 , k 1 ) ψ ( k 1 ) + w 2 k 2 2 V ( k 2 , k 2 ) ψ ( k 2 ) = E ψ ( k 2 ) (3) 5 / 1

Algorithm: Integral Equations → Linear Equations k k k k Solve on p-Space Grid 1 3 2 N N k 2 2 µψ ( k i ) + 2 � i w j k 2 j V ( k i , k j ) ψ ( k j ) = E ψ ( k i ) , i = 1 , N (1) π j = 1 Matrix Schrödinger equation [ H ][ ψ ] = E [ ψ ] ψ ( k ) = N × 1 vector ψ ( k 1 )   k 2   ψ ( k 2 ) 2 µ + 2 1 π V ( k 1 , k 1 ) k 2 π V ( k 1 , k 2 ) k 2 2 π V ( k 1 , k N ) k 2 2   1 w 1 2 w 2 N w N · · ·         ×    ...     k 2    2 µ + 2 N π V ( k N , k N ) k 2   N w N · · · · · · · · ·   ψ ( k N ) ψ ( k 1 )    ψ ( k 2 )      = E (2)   ...         ψ ( k N ) 6 / 1

Eigenvalue Problem Search for Solution; N equations for ( N + 1 ) unknowns? Solution only sometimes, certain E (eigenvalues) Try to solve, multiply both sides by [ H − EI ] inverse: [ H ][ ψ ] = E [ ψ ] (1) [ H − EI ][ ψ ] = [ 0 ] (2) ⇒ [ ψ ] = [ H − EI ] − 1 [ 0 ] (3) ⇒ if inverse ∃ , then only trivial solution ψ ≡ 0 For nontrivial solution inverse can’t ∃ det [ H − EI ] = 0 (bound-state condition) (4) Requisite additional equation for N + 1 unknowns Solve for just eigenvalues, or full e.v. problem 7 / 1

Model: Delta-Shell Potential (Sort of Analytic Solution) 2 Particles Interact When b Apart V ( r ) = λ 2 µδ ( r − b ) (1) � ∞ 1 sin ( k ′ r ′ ) λ V ( k ′ , k ) = 2 µδ ( r − b ) sin ( kr ) dr (2) k ′ k 0 sin ( k ′ b ) sin ( kb ) = λ (too slow decay) (3) 2 µ k ′ k 1 Bound state E = − κ 2 / 2 µ , if e − 2 κ b − 1 = 2 κ (4) λ Only if strong & attractive ( λ < 0) Exercise: Solve transcendental equation ( b = 10, λ =? ) 8 / 1

Bound–State Integral–Equation Code Sample Code Surveys all Parameters Gauss quadrature for pts & wts Two possible libe calls 1. Search on E , det [ H − EI ] = 0 2. Use eigenproblem solver* Both iterative solutions 9 / 1

Your Implementation Modify or Write Eigenvalues, Eigenproblem 2 µ = 1, b = 10, N > 16 1 Set up [ V ( i , j )] and [ H ( i , j ]) for N ≥ 16 2 Observe monotonic relation E ( λ ) 3 True bound state stable with N , others = artifacts 4 Extract best value for E & estimate precision 5 Comparing RHS, LHS [ H ][ ψ ] = E [ ψ ] 6 10 / 1

Exploration: Momentum Space Wave Function* Bound in p Space? Determine ψ ( k ) (analytic ψ ( p ) ∝ [ p 2 − 2 mE ] − 1 ) 1 Is this reasonable, normalizable? 2 Determine ψ ( r ) via transform 3 � ∞ dk ψ ( k ) sin ( kr ) k 2 ψ ( r ) = (1) kr 0 Is this reasonable ψ ( r ) ? 4 Compare to analytic ψ ( r ) , 5  e − κ r − e κ r , for r < b ,  ψ 0 ( r ) ∝ (2) e − κ r , for r > b  11 / 1