6. Quantum Mechanics

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Setup

\(Clear["Global`*"]; 
Off[General::spell]; 
Off[General::spell1]; 
    \)

Needs["Calculus`VectorAnalysis`"]

Needs["Graphics`ParametricPlot3D`"]

<< "D:\Mma\Inits\ch6init.m"

Clear["Global`*"];

6.2 Problems

6.2.3  Three Dimensional Schrödinger Equation

Problem 2:  Schrödinger's equation for spherically symmetric potentials.

Consider the three-dimensional Schrödinger equation with a spherically symmetric potential  V = V[r].  Assume the solution separates into the form ψ[r,θ,ϕ] = R[r] Θ[θ] Φ[ϕ] .

Part (a)  Express Schrödinger's equation in spherical coordinates and derive an equation for the angular function Φ[ϕ].  Solve for Φ[ϕ] and show that:
	Φ[ϕ] = c1Exp[-I (Subscript[m, ϕ]) ϕ] + c2Exp[I (Subscript[m, ϕ]) ϕ] .

ψ[r,θ,ϕ] = R[r] Θ[θ] Φ[ϕ]

SetCoordinates[Spherical[r,θ,ϕ]]

eq1 = schrodinger[V[r], En] @ ψ[r,θ,ϕ]

eq2 = (eq1 (r^2 Sin[θ]^2 /ψ[r,θ,ϕ])//ExpandAll)

select[eq_,var_] :=
	Select[eq, (!FreeQ[#,var])&]

ϕEq1 = select[eq2,ϕ] == ℏ^2/(2 m) mϕ^2

select[eq_,var_] :=
	Select[eq, (!FreeQ[#,var])&]

ϕEq2 = Solve[ϕEq1,Φ''[ϕ]][[1,1]]/.{Rule->Equal}

ϕRule = DSolve[ϕEq2,{Φ},ϕ]//Flatten

Part (b)  Eliminate the ϕ variable from Schrödinger's equation using the result from Part (a).  Separate the remaining r and θ variables and derive equarions for R[r] and Θ[θ] .

eq3 = (( eq2 //.ϕRule//Cancel)/Sin[θ]^2 //ExpandAll)

rEq1 = select[eq3,r] == L ℏ^2/m

rEq = (( Solve[rEq1,R''[r]][[1,1]]
		/.{Rule->Equal}
		//ExpandAll
		)/.{L->-1/2 ℓ(ℓ+1)}
		)

θEq = (select[eq3,θ]+L ℏ^2/m == 0
		/.{L->-1/2 ℓ(ℓ+1)}
		)

Part (c)  Let z = Cos[θ] and show that R[r] Θ[θ] equation in Part (b) can be expressed in the canonical form of Legendre's Equation:
	ℓ Θ[z] + (ℓ^2) Θ[z] + (((Subscript[m, ϕ])^2)Θ[z])/(-1 + (z^2)) - 2z Θ'[z] + (1 - (z^2)) Θ''[z] == 0  .

eq4 = vChange[θEq,Θ,θ,z,ArcCos[z]]//PowerExpand//Simplify

θEq1 = ((-2 m Θ[z]/ℏ^2 eq4)//Thread[#,Equal]&)

θEq2 = ((θEq1//.{-1+z^2->zz})//Simplify)//.{zz->-1+z^2}

θEq3 = Collect[θEq2[[1]],{Θ''[z]}]==0

Part (d) Declare ℓ and (Subscript[m, ϕ]) to be integersand solve Legendre's equation.  Verify that the solutions satisfy the equation.  Show that the product of the angular functions P[ ℓ, m, Cos[θ]] and Exp[I m ϕ] are related to spherical harmonics by the equation:
	Y[ℓ, m, θ, ϕ] == C[ℓ, m] P[ℓ, m, Cos[θ]] Exp[I m ϕ]
	where C[ℓ, m]  = Sqrt[ (2ℓ +1) / (4π) * (ℓ - m)! / (ℓ + m)! ]  .

IntegerQ[ℓ]^=True;
IntegerQ[mϕ]^=True;

θRule1 = DSolve[θEq2,{Θ},z]//Flatten

θRule2 = DSolve[θEq2//ExpandAll,{Θ},z]//Flatten

θRule = DSolve[θEq3,{Θ},z]//Flatten

Table[θEq3//.θRule,{ℓ,0,3},{mϕ,0,ℓ}]//Simplify

Coeff[el_,m_] := Sqrt[(2 el + 1)/(4 π)*(el-m)!/(el+m)!]

sphEq :=
	SphericalHarmonicY[ℓ,m,θ,ϕ] == (Coeff[ℓ,m]*LegendreP[ℓ,m,Cos[θ]]
		*Exp[I m ϕ])

Table[sphEq//Simplify//PowerExpand//TrigReduce
	,{ℓ,0,3},{m,0,ℓ}]