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 1:  The three-dimensional harmonic oscillator in Cartesian coordinates.

Consider the three-dimensional Schrödinger equation, where the potential is V = Vx[x] + Vy[y] + Vz[z].

Part (a)  Assume the wave function is separable and of the form X[x]Y[y]Z[z] and derive differential equations for X[x], Y[y], and Z[z].

V = (1/2) (kx x^2 + ky y^2 + kz z^2)

ψ[x,y,z] = X[x] Y[y] Z[z]

SetCoordinates[Cartesian[x,y,z]]

eq1 = schrodinger[V,En] @ ψ[x,y,z]

eq2 = eq1/(X[x] Y[y] Z[z])//ExpandAll

eRule = {En -> Ex + Ey + Ez}

eq3 = eq2/.eRule

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

eq4 = (0 == {select[eq3,Ex,x]
			,select[eq3,Ey,y]
			,select[eq3,Ez,z]}
		)//Thread

eq5 = {xeq,yeq,zeq} =
	(( Solve[eq4,{X''[x],Y''[y],Z''[z]}]//Flatten)
		/.{Rule->Equal}
		//ExpandAll)

Part (b)  Consider the harmonic oscillator potential Vx = (1/2) kx x^2, Vy = (1/2) ky y^2, and Vz = (1/2) kz z^2 .  Solve the Schrödinger equation for the wave function using the one-dimensional harmonic oscillator solutions.

normConst[n_] = 1/Sqrt[Sqrt[π] 2^n Factorial[n]]

X[x_,nx_] = normConst[nx] HermiteH[nx,x] Exp[-x^2/2]

Y[y_,ny_] = normConst[ny] HermiteH[ny,y] Exp[-y^2/2]

Z[z_,nz_] = normConst[nz] HermiteH[nz,z] Exp[-z^2/2]

Part (c)  Plot the two-dimensional wave function X[x]Y[y].

wavePlot1 = Plot3D[X[x,2] Y[y,1]
			,{x,-4,4},{y,-4,4}
			,PlotPoints->35
			,DisplayFunction->Identity];

potPlot =
	Graphics3D[Plot3D[-2+.15(x^2 + y^2)
			,{x,-4,4},{y,-4,4}
			,PlotPoints->35
			,DisplayFunction->Identity]
		];

Show[{potPlot,wavePlot1}
	,PlotRange->{-0.5,0.7}
	,Boxed->False
	,ViewPoint->{1,1,2}
	,Axes->False
	,DisplayFunction->$DisplayFunction];

wavePlot2 = Plot3D[X[x,2] Y[y,3]
			,{x,-4,4},{y,-4,4}
			,PlotPoints->35
			,DisplayFunction->Identity];

Show[{potPlot,wavePlot2}
	,PlotRange->{-0.5,0.7}
	,Boxed->False
	,ViewPoint->{1,1,2}
	,Axes->False
	,DisplayFunction->$DisplayFunction];