6. Quantum Mechanics

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Null

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 6:  Separation into cylindrical and paraboloidal coordinates

Consider the following problems in cylindrical and paraboloidal coordinates.

Part (a)  Express Schrödinger's equation in cylindrical coordinates {r,θ,z}. Assume a potential of the form V = Vr[r] + Vz[z] and a wave function f[r]Θ[θ]h[z].  Derive equations for f[r], Θ[θ], and h[z].

ψ[r_,θ_,z_] = f[r] Θ[θ] h[z]

SetCoordinates[Cylindrical[r,θ,z]]

eq1 =  (-schrodinger[Vr[r]+Vz[z],En] @ ψ[r,θ,z] //ExpandAll)

eq2 = (eq1 r^2/(ℏ^2/(2 m) ψ[r,θ,z]) //ExpandAll)

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

θEq1 = select[eq2,θ] == -m^2

θEq = Solve[θEq1,{Θ''[θ]},θ][[1,1]]/.{Rule->Equal}

θRule = DSolve[θEq,{Θ},θ][[1]]

eq3 = eq2/r^2 //.θRule//Cancel//ExpandAll

zTerm = (select[eq3,z] == -mz^2//Thread[h[z] #,Equal]&)//ExpandAll

rTerm = (select[eq3,r] == mz^2//Thread[r^2 f[r] #,Equal]&)//ExpandAll

Part (b)   Express Schrödinger's equation in paraboloidal coordinates {u,v,ϕ}. Assume a potential of the form V = (Vu[u] + Vv[v])/(u^2 + v^2),  and a wave function f[u]g[v]Φ[ϕ].  Derive equations for f[u], g[v], and Φ[ϕ].

Clear[ψ];

ψ[u_,v_,ϕ_] = f[u] g[v] Φ[ϕ]

SetCoordinates[Paraboloidal[u,v,ϕ]]

eq4 =  (-schrodinger[V[u,v],En] @ ψ[u,v,ϕ] //ExpandAll)

eq5 = (eq4 (2m u^2 v^2/ℏ^2)/ψ[u,v,ϕ]) //Simplify//Apart//Expand

ϕEq = select[eq5,ϕ] == -mϕ^2//Thread[Φ[ϕ] #,Equal]&//Simplify

ϕRule = DSolve[ϕEq,{Φ},ϕ][[1]]

eq6 = (eq5(u^2+v^2)/(u^2 v^2)/.{u^2+v^2->temp}//Expand)/.{temp->u^2+v^2}//.ϕRule//Simplify

eq7 = (eq6/.{V[u,v]->(Vu[u]+Vv[v])/(u^2+v^2)})//Collect[#,{Vu[u],Vv[v]}]&//Cancel

uEq = (select[eq7,u] == -mu^2//Thread[u^2 f[u] #,Equal]&)//ExpandAll

vEq = (select[eq7,v] == -mv^2//Thread[v^2 g[v] #,Equal]&)//ExpandAll