Conventions in scattering theory IV:
Generating scattering equations
August 30, 2006

Documentation

In this revision, I have adopted the new notation (Greek letters) for conversion constants:
	plane-wave state normalization:		 c has become α
	angular-momentum state normalization:	 q has become ζ
	relationship of the S and T matrices:		 d has become β
	relationship of the S and K matrices:		 e has become γ
Also, these have been placed in the order introduced in the text of the associated paper where written as replacement rules.
Furthermore, this version has an 'in-line' version of the ConjugateFix package so that it is completely self-contained.

BookKeeping

ClearAll["Global`*"]

Off[General::spell]
Off[General::"spell1"]

Needs["Utilities`Notation`"]

Symbolize[\(k\&^\)];

SetOptions[Simplify, 
        Assumptions -> ℓ ∈ Integers];

\(\(\( (*\ 
      this\ cell\ is\ an\ in - 
        line\ version\ of\ the\ ConjugateFix\ package\ written\ by\ Michael\ Morrison\ *) \)\(
\)\( (*unprotect\ system\ functions*) \)\(
\)\(
\)\(\(protected = 
        Unprotect[Re, Im, 
          Conjugate];\)

 (*definitions\ for\ system\ functions\ *) 


    Conjugate[expr_] := expr /. Complex[x_, y_] -> x - I\ y

    Re[expr_] := Expand[\((expr + Conjugate[expr])\)/2]

    Im[expr_] := 
      Expand[\((expr - Conjugate[expr])\)/\((2  I)\)]


    \(\(Protect[
      Evaluate[protected]]\)\(;\)\(\ \ \)\( (*restore\ protection\ of\ system\ symbols*) \)\(
\)\)\)\)\)

Fundamental relations

Mathematics

YlmProdToLegendreP =  (Subscript[Y, ℓ, 𝓂])[(\(k\&^\))'] \(\(\((Y\_\(ℓ, 𝓂\))\)\^*\)[k\&^]\) ->\(\(\(\(\ \)\((2  ℓ + 1)\)\)\/\(4  π\)\) 
            P\_ℓ[k\&^'·k\&^]\);

Scattering matrices

ST  =  1  -  2 π I β T;

TK  =  -(1/( π)) (γ/β) (( K)/( 1  -  I γ K));

SK  =  ((1  +  I γ K)/(1  -  I γ K));

KS  =  -((I (-1 + S))/((1 + 
                    S) γ));

TS  =  ((I (-1 + S))/(2 π β));

KT  =  -((I π T β)/((I + π T β) γ));

({Re[#], Im[#]} & /@  SK) // Simplify

Integral equations

In the following expressions, Tmatel and Vmatel refer to matrix elements of the T operator and of the potential energy with respect to pure momentum states. Vint refers to the integral involving the potential energy in the coordinate-space integral equation for the scattering amplitude f.

Clear[fT, fV, fIE]
fT  =  -(((2  π))^2) (β/(α^2)) (m/(ℏ^2)) Tmatel;
fV  =  -(((2  π))^2) (1/(α^2)) (m/(ℏ^2)) 
          Vmatel;
fIE  =  -(((2  π))^(1/2)) (1/α) (m/(ℏ^2)) Vint;

Partial-wave analysis

Kδ  =  (1/γ) Tan[δ];

Sδ  =  (E^(2 I δ));

Tδ  = -(((E^(I δ)) Sin[δ])/(β π));

fδ  =  -(π/k) β Tδ;

Normalized radial function in partial-wave theory

Symbolize[\(j\&^\_ℓ\)]

uVzero = ζ \(j\&^\_ℓ\);

uBC  =  ζ (E^(I δ)) 
          Sin[kr  -  ℓ (( π)/2) + δ];

The (l^th) term in the expansion of the scattering amplitude in spherical harmonics:

fℓ  =  -(((2  π))^2) ((β )/(α^2)) (m/(ℏ^2)) 
              transformationProduct (ζ^2) (Subscript[T, ℓ]^β)  /. 
            (Subscript[T, ℓ]^β) -> (1/β) 
                (Subscript[T, ℓ]) /.  {ℓ' -> ℓ, 𝓂' -> 𝓂};

σℓ  =   4  π (2  ℓ + 1) 
          Conjugate[fδ]*fδ;

Partial-wave conversion rules

TmatelToPhaseShift  =  
        (Subscript[t, ℓ])[
            k] -> -(1/(β )) (((E^(I (Subscript[δ, ℓ]))) Sin[(Subscript[δ, ℓ])[k]])/π);

TmatelToScatteringAmplitude  =  
        (Subscript[t, ℓ])[k] ->  -(1/β) (k/π) 
            (Subscript[f, ℓ])[k] ;

scatteringAmplitudeToTmatel  =  
        (Subscript[f, ℓ])[k] ->  -β (π/k) 
            (Subscript[t, ℓ])[k] ;

Plane-wave to angular-momentum conversion

Clear[Elmk]
\(Elmk[k_, 
        khat_, ℓ_, 𝓂_]\  := \ \(I\^ℓ\) \(\@\(ℏ\^2\/m\)\) \(k\^\(\(-1\)/
              2\)\) \(Y\_\(ℓ, 𝓂\)\%*\)[
          khat] α\/ζ\)

Clear[kElm]
kElm[k_, 
        khat_, ℓ_, 𝓂_]  :=  (((-I))^ℓ) (Sqrt[((ℏ^2)/m)]) (k^(-1/2)) 
        (Subscript[Y, ℓ, 𝓂])[khat] (α/ζ)

Transformation matrix for the general case (not necessarily rotationally invariant):

\(\(transformationProduct\  = \ 
        Elmk[k, k\&^, ℓ, 𝓂] 
            kElm[k, k\&^', ℓ', 𝓂'] /. \ \((\(-I\))\)\^ℓ'\  -> \ I\^\(-ℓ'\);\)\)

Compare to cases in the literature

auRules  =  {m -> 1, ℏ -> 1};

The lists of replacement rules below effect conventions used in the named sources.

MS95  =   {α -> 
            1, β -> -(1/π), γ -> 
            1, ζ -> 1} ;

Taylor  =  {α -> 1, β -> 1, γ -> 
            1, ζ -> 1};

McCarthyWeigold  =  {α -> 1, β -> 
            1, γ -> 
            1, ζ -> (E^(-I δ))};

Nesbet  =  {α -> (((2  π))^(3/2)), β -> -(1/π), γ -> 
            1, ζ -> (k^(-1/
                    2)) (E^(-I δ))};

Bransden  =  {α -> (((2  π))^(3/2)), β -> -(1/π), γ -> 1, ζ -> 
            1};

Burke  =  {α -> (((2  π))^(3/2)), β -> (I/π), γ -> 
            1, ζ -> (E^(-I δ))};

Sitenko  =  {α -> (((2  π))^(3/2)), β -> 1, γ -> 1, ζ -> 
            4 π (I^ℓ)/k};

RT  =  {α -> (((2  π))^(3/2)), β -> 
            1, γ -> -π, ζ -> 1};

Drukarev  =  {α -> (((2  π))^(3/2)), β -> 2  π/k, ζ -> 1/k};

Joachain  =  {α -> 1, β -> 
            1, γ -> -π, ζ -> ((E^(-I δ))/(k Cos[δ]))};

Taylor

Taylor

{ST, TS} /. Taylor

{TK, KT}  /.  Taylor

{SK, KS}  /.  Taylor

{fT,  fV,  fIE} /.  Taylor

{Kδ, Sδ, Tδ} /. Taylor

{fδ, σℓ}  /.  Taylor

{uVzero, uBC}  /.  Taylor

Joachain

Joachain

{ST, TS} /. Joachain

{TK, KT}  /.  Joachain

{SK, KS}  /.  Joachain

{fT,  fV,  fIE} /.  Joachain

{Kδ, Sδ, Tδ} /. Joachain

{fδ, σℓ}  /.  Joachain

{uVzero, uBC}  /.  Joachain

(uBC  /.  Joachain /. kr - ((π ℓ)/2) -> x // 
            TrigExpand) /. x -> kr - ((π ℓ)/2) // 
      Factor

The above result conforms to the boundary condition (4.86a), which J uses in most of his examples

RT

RT

{ST, TS} /. RT

{TK, KT}  /.  RT

{SK, KS}  /.  RT

{fT,  fV,  fIE} /.  RT

{Kδ, Sδ, Tδ} /. RT

{fδ, σℓ}  /.  RT

{uVzero, uBC}  /.  RT

MS95

MS95

{ST, TS} /. MS95

{TK, KT}  /.  MS95

{SK, KS}  /.  MS95

{fT,  fV,  fIE} /.  MS95

{Kδ, Sδ, Tδ} /. MS95

{fδ, σℓ}  /.  MS95

{uVzero, uBC}  /.  MS95

Nesbet

Nesbet

{ST, TS} /. Nesbet

{TK, KT}  /.  Nesbet

{SK, KS}  /.  Nesbet

{fT,  fV,  fIE} /.  Nesbet

{Kδ, Sδ, Tδ} /. Nesbet

{fδ, σℓ}  /.  Nesbet

{uVzero, uBC}  /.  Nesbet

Burke

Burke

{ST, TS} /. Burke

{TK, KT}  /.  Burke

{SK, KS}  /.  Burke

{fT,  fV,  fIE} /.  Burke

{Kδ, Sδ, Tδ} /. Burke

{fδ, σℓ}  /.  Burke

{uVzero, uBC}  /.  Burke

McCarthyWeigold

McCarthyWeigold

{ST, TS} /. McCarthyWeigold

{TK, KT}  /.  McCarthyWeigold

{SK, KS}  /.  McCarthyWeigold

{fT,  fV,  fIE} /.  McCarthyWeigold

{Kδ, Sδ, Tδ} /. McCarthyWeigold

{fδ, σℓ}  /.  McCarthyWeigold

{uVzero, uBC}  /.  McCarthyWeigold

Sitenko

Sitenko

{ST, TS} /. Sitenko

{fT,  fV,  fIE} /.  Sitenko

{Kδ, Sδ, Tδ} /. Sitenko

{fδ, σℓ}  /.  Sitenko

{uVzero, uBC}  /.  Sitenko

Drukarev

Drukarev

{ST, TS} /. Drukarev

{TK, SK}  /.  Drukarev

{SK, KS}  /.  Drukarev

{fT,  fV,  fIE} /.  Drukarev  /. auRules

{Kδ, Sδ, Tδ} /. Drukarev

{fδ, σℓ}  /.  Drukarev

{uVzero, uBC}  /.  Drukarev

Bransden

Bransden

{ST, TS} /. Bransden

{TK, KT}  /.  Bransden

{SK, KS}  /.  Bransden

{fT,  fV,  fIE} /.  Bransden

{Kδ, Sδ, Tδ} /. Bransden

{fδ, σℓ}  /.  Bransden

{uVzero, uBC}  /.  Bransden