Numerical Calculation of Cross-section

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Theory of Quantum Mechanical Scattering Program for Calculating Cross-section Numerical Calculation of Cross-section Scattering of Hydrogen atoms oﬀ Krypton atoms Vinay Vaibhav Particle Physics-I Presentation Vinay Vaibhav Numerical Calculation of Cross-section

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Theory of Quantum Mechanical Scattering Program for Calculating Cross-section Outline 1 Theory of Quantum Mechanical Scattering Scattering Amplitude Partial Wave Expansion Determination of Phase Shifts 2 Program for Calculating Cross-section Numerov’s Algorithm for the Radial Schrodinger Equation Numerical Calculation of Bessel Functions Calculation of Cross-section Vinay Vaibhav Numerical Calculation of Cross-section

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Theory of Quantum Mechanical Scattering Program for Calculating Cross-section Scattering Amplitude Partial Wave Expansion Determination of Phase Shifts Outline 1 Theory of Quantum Mechanical Scattering Scattering Amplitude Partial Wave Expansion Determination of Phase Shifts 2 Program for Calculating Cross-section Numerov’s Algorithm for the Radial Schrodinger Equation Numerical Calculation of Bessel Functions Calculation of Cross-section Vinay Vaibhav Numerical Calculation of Cross-section

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Theory of Quantum Mechanical Scattering Program for Calculating Cross-section Scattering Amplitude Partial Wave Expansion Determination of Phase Shifts Lippmann-Schwinger Equation Lippmann-Schwinger Equation |ψ± = 1 [E − H0 ± i ] V |ψ± + |φ Position State Wavefunction x|ψ± = − 2m 2 d3x e±ik|x−x | 4π|x − x | V (x ) x|ψ± + x|φ where k = √ 2mE 2 Vinay Vaibhav Numerical Calculation of Cross-section

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Theory of Quantum Mechanical Scattering Program for Calculating Cross-section Scattering Amplitude Partial Wave Expansion Determination of Phase Shifts If we assume FINITE RANGE SCATTERING POTENTIAL then we can write x|ψ+ = − 1 4π 2m 2 eikr r d3x e−ik.x V (x ) x |ψ+ + x|φ where k = kˆ r and ˆ r = x |x| ψ(r) = 1 L3 2 [eik.x + eikr r f (k , k)] where f (k , k) is called Scattering Amplitude, given by f (k , k) = − 1 4π 2m 2 L3 d3x e−ik .x L3 2 V (x ) x |ψ+ = − mL3 2π 2 k |V |ψ+ = − mL3 2π 2 k |T|ψ+ where V |ψ+ = T|φ = T|k Vinay Vaibhav Numerical Calculation of Cross-section

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Theory of Quantum Mechanical Scattering Program for Calculating Cross-section Scattering Amplitude Partial Wave Expansion Determination of Phase Shifts Hence, f (k , k) = − mL3 2π 2 k |T|ψ+ Diﬀerential Scattering Cross-section: dσ dω = |f (k , k)|2 Vinay Vaibhav Numerical Calculation of Cross-section

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Theory of Quantum Mechanical Scattering Program for Calculating Cross-section Scattering Amplitude Partial Wave Expansion Determination of Phase Shifts Free Particle in Spherical Basis Since, free particle hamiltonian commutes with L2 and Lz, so it is possible to consider a simultaneous eigenstate of H0, L2 and Lz as |E, , m x|E, , m = ι 2mk π j (kr)Y m(ˆ r) k|E, , m = √ 2mk δ(E − 2k2 2m )Y m(ˆ k) Now using these basis we can write f (k , k) = − mL3 2π 2 k |T|ψ+ = − 4π k T (E)Y m( ˆ k )Y m∗(ˆ k)δ(E − 2k2 2m ) Vinay Vaibhav Numerical Calculation of Cross-section

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Theory of Quantum Mechanical Scattering Program for Calculating Cross-section Scattering Amplitude Partial Wave Expansion Determination of Phase Shifts Taking k along z axis and using Y m(ˆ k) = 2 + 1 4π δm,0 Y m( ˆ k ) = 2 + 1 4π P (cosθ) f (k , k) = f (θ) = ∞ =0 (2 + 1)f (k)P (cosθ) where f (k) = −T (E) k Vinay Vaibhav Numerical Calculation of Cross-section

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Theory of Quantum Mechanical Scattering Program for Calculating Cross-section Scattering Amplitude Partial Wave Expansion Determination of Phase Shifts The Following integral representation of j (kr) j (kr) = 1 2ι eιkrcosθP (cosθ)d(cosθ) allows us to write eιk.r = (2 + 1)ι j (kr)P (cosθ) And using the asymptotic limits of j we can write the wavefunction as ψ(r) = 1 (2π)3 2 [eιkz + f (θ) eikr r ] = 1 (2π)3 2 ∞ =0 (2 + 1) P (cosθ) 2ιk [[1 + 2ιkf (k)] eιkr r − e−ι(kr− π) r ] This is called Partial Wave Expansion Vinay Vaibhav Numerical Calculation of Cross-section

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Theory of Quantum Mechanical Scattering Program for Calculating Cross-section Scattering Amplitude Partial Wave Expansion Determination of Phase Shifts Because of probability conservation and angular momentum conservation, the outgoing ﬂux must equal the incoming ﬂux. |S (k)| = |1 + 2ιkf (k)| = 1 Calling S = 1 + 2ιk = e2ιδ f (θ) = 1 k ∞ =0 (2 + 1)eιδ sinδ P (cosθ) Total Scattering Cross-section σtotal = 4π k2 ∞ =0 (2 + 1)sin2δ Vinay Vaibhav Numerical Calculation of Cross-section

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Theory of Quantum Mechanical Scattering Program for Calculating Cross-section Scattering Amplitude Partial Wave Expansion Determination of Phase Shifts How to Determine Phase Shift δ For the spherically symmetric potential V (r), the solution of Schrodinger equation can be written as ψ(r) = ∞ =0 =− A m u (r) r Y m(θ, φ) where u satisﬁes the radial Schrodinger equation: { 2 2m d2 dr2 + [E − V (r) − 2 ( + 1) 2mr2 ]}u (r) = 0 If potential decays suﬃciently s.t. it is possible to assume V (r) = 0(r > rmax ) then we can approximate u (r) with free particle solution beyond rmax . Vinay Vaibhav Numerical Calculation of Cross-section

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Theory of Quantum Mechanical Scattering Program for Calculating Cross-section Scattering Amplitude Partial Wave Expansion Determination of Phase Shifts Free particle solution can be written either in terms of Bessel function or Spherical Hankel function as x|φ = 1 (2π)3/2 ι (2 + 1)A (r)P (cosθ) where A (r) = c(1)h(1)(kr) + c(2)h(2)(kr) c(1) and c(2) can be determined by comparing it with asymptotic form of parial wave expansion x|ψ+ = 1 (2π)3 2 ∞ =0 (2 + 1) P (cosθ) 2ιk [e2ιδ eιkr r − e−ι(kr− π) r ] Vinay Vaibhav Numerical Calculation of Cross-section

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Theory of Quantum Mechanical Scattering Program for Calculating Cross-section Scattering Amplitude Partial Wave Expansion Determination of Phase Shifts Hence, for r > rmax , u (r) = A (r) = eιδ [j (kr)cosδ − n (kr)sinδ ] Considering the values of u (r) at two diﬀerent points r1 and r2 beyond rmax , we can obtain tanδ = Kj (kr1) − j (kr2) Kn (kr1) − n (kr2) with K = r1u (r2) r2u (r1) σtotal ←− δ ←− u (r) ←− Solve Radial Schrodinger Equation for a given V(r) Vinay Vaibhav Numerical Calculation of Cross-section

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Theory of Quantum Mechanical Scattering Program for Calculating Cross-section Numerov’s Algorithm for the Radial Schrodinger Equation Numerical Calculation of Bessel Functions Calculation of Cross-section Outline 1 Theory of Quantum Mechanical Scattering Scattering Amplitude Partial Wave Expansion Determination of Phase Shifts 2 Program for Calculating Cross-section Numerov’s Algorithm for the Radial Schrodinger Equation Numerical Calculation of Bessel Functions Calculation of Cross-section Vinay Vaibhav Numerical Calculation of Cross-section

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Theory of Quantum Mechanical Scattering Program for Calculating Cross-section Numerov’s Algorithm for the Radial Schrodinger Equation Numerical Calculation of Bessel Functions Calculation of Cross-section The radial SE can be written as 2 2m d2 dr2 u(r) = F( , r, E)u(r) where, F( , r, E) = V (r) + 2 ( + 1) 2mr2 − E This is an eﬃcient method to solve diﬀerential equations of the form (with an error of order h6) d2 dx2 u(r) = F(r)u(r) u(r) is expanded (taylor series) about r=h and r=-h then added, which gives u(r + h) + u(r − h) = 2u(r) + h2u(2)(r) + h4u(4)(r) + O(h6) Vinay Vaibhav Numerical Calculation of Cross-section

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Theory of Quantum Mechanical Scattering Program for Calculating Cross-section Numerov’s Algorithm for the Radial Schrodinger Equation Numerical Calculation of Bessel Functions Calculation of Cross-section u2(r) = u(r + h) + u(r − h) − 2u(r)h2 − h2 12 u4(4) + O(h6) Now if we operate (1 + h2 12 d2 dr2 ) over original equation and eliminate fourth order derivative term then with slight rearrangement we get following, which has error of order h6 u(r+h) = 2(1 + 5 12 h2F(r))u(r) − (1 − 1 12 h2F(r − h))u(r − h) (1 − 1 12 h2F(r + h)) +O(h6) Vinay Vaibhav Numerical Calculation of Cross-section

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Theory of Quantum Mechanical Scattering Program for Calculating Cross-section Numerov’s Algorithm for the Radial Schrodinger Equation Numerical Calculation of Bessel Functions Calculation of Cross-section Lennard-Jones Interaction Here we are considering H-Kr interaction. The two atom interaction is generally modelled by Lennard-Jones potential, which has following form: V (r) = [( σ r )12 − ( σ r )6] For H-Kr, = 5.9meV and σ = 3.57Ao 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 r −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 U(r) Lennard-Jones Potential Vinay Vaibhav Numerical Calculation of Cross-section

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Theory of Quantum Mechanical Scattering Program for Calculating Cross-section Numerov’s Algorithm for the Radial Schrodinger Equation Numerical Calculation of Bessel Functions Calculation of Cross-section Because of numerical inconsistency at r=0 and high value of potential near origin we start the integration from r = rmin = 0.5σ For r < rmin, the term 1 r12 dominates the other term and SE reduces to d2 dr2 u(r) = 2m 2 σ12r12u(r) ⇒ u(r) exp(− 2m σ12 25 2 r−5) The value of u(r) at ﬁrst two points can be calculated using above expression The maximum value of r (rmax ) is taken to be Vinay Vaibhav Numerical Calculation of Cross-section

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Theory of Quantum Mechanical Scattering Program for Calculating Cross-section Numerov’s Algorithm for the Radial Schrodinger Equation Numerical Calculation of Bessel Functions Calculation of Cross-section Spherical Bessel Functions:Iterative Method For = 0, 1 , Spherical Bessel Functions are given by: j0(x) = sinx x ; n0(x) = −cosx x j1(x) = sinx x2 − cosx x ; n1(x) = −cosx x2 − sinx x For higher , the follwing recursion relation can be used: s +1(x) + s −1(x) = 2 + 1 x s (x) where s is either j or n Vinay Vaibhav Numerical Calculation of Cross-section

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Theory of Quantum Mechanical Scattering Program for Calculating Cross-section Numerov’s Algorithm for the Radial Schrodinger Equation Numerical Calculation of Bessel Functions Calculation of Cross-section After determining the valus of u (r) at different points r1 and r2 beyond rmax , we can find phase shift using following expression (as already discussed) tanδ = Kj (kr1) − j (kr2) Kn (kr1) − n (kr2) with K = r1u (r2) r2u (r1) This phase shift can be used to evaluate total cross-section by summing the following series σtotal = 4π k2 ∞ =0 (2 + 1)sin2δ Further, total cross-section can be calculated for different values of energy(0.1-3.5 meV) and can be plotted. Vinay Vaibhav Numerical Calculation of Cross-section

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Theory of Quantum Mechanical Scattering Program for Calculating Cross-section Numerov’s Algorithm for the Radial Schrodinger Equation Numerical Calculation of Bessel Functions Calculation of Cross-section 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Energy [meV] 0 100 200 300 400 500 600 700 Total Cross-section Vinay Vaibhav Numerical Calculation of Cross-section