Differential Operators in Cartesian coordinates (quantum mechanics)
Masatsugu Sei Suzuki  and Itsuko S. Suzuki

Department of Physics, SUNY at Binghamton
Binghamton, New York  U.S.A.
(Date: March 25, 2023)

ex  		->		ux
ey		->		uy
ez		->		uz
r ={x,y,z}	->		r
|r|=(Sqrt[(x^2)+(y^2)+(z^2)])	R
L= r x p	->		L
p		->		p
(p^2)				Psq
Lx		->		Lx
Ly		->		Ly
Lz		->		Lz
(L^2)				Lsq
(∇^2)		->		Lap
Grad	 (∇)	->		Gra
curl (rot)	->		Cur	
Div		->		Diva
pr		->		prq
r·p		->		rP

Clear["Global`"];ux={1,0,0};uy={0,1,0};uz={0,0,1};r={x,y,z};R=(Sqrt[(r.r)]);L:=(-I  ℏ( Cross[r,Grad[#,{x,y,z},"Cartesian"]])&)//Simplify;p=(-I  ℏ Grad[#,{x,y,z},"Cartesian"]&)//Simplify;Lx:=(ux.L[#])&;Ly:=(uy.L[#])&;Lz:=(uz.L[#])&;px:=(ux.p[#])&;py:=(uy.p[#])&;pz:=(uz.p[#])&;Lap:=Laplacian[#,{ x,y,z},"Cartesian"]&;Gra:=Grad[#,{ x,y,z},"Cartesian"]&;Cur:=Curl[#,{x,y, z},"Cartesian"]&;Diva:=Div[#,{x,y, z},"Cartesian"]&;prq:=(-I ℏ)(1/2)( (r/R).Gra[#]+ Diva[(r/R) #])&;Lsq:=(Nest[Lx,#,2]+Nest[Ly,#,2]+Nest[Lz,#,2])&;
Psq:=(Nest[px,#,2]+Nest[py,#,2]+Nest[pz,#,2])&;rPsq:=(-I ℏ)( r.Gra[#])&;

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1.
A. Goswami, Quantum mechanics
second edition (WCB, 1997)
Chapter 11, Problem A - 1
A.1 	Find the commutators [(L^2),x],   [(L^2),y], and  [(L^2),z].

s11=Lsq[x ψ[x,y,z]]-x Lsq[ ψ[x,y,z]]//Simplify

s12=Lsq[y ψ[x,y,z]]-y Lsq[ ψ[x,y,z]]//Simplify

s13=Lsq[z ψ[x,y,z]]-z Lsq[ ψ[x,y,z]]//Simplify

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2.
A. Goswami, Quantum mechanics
second edition (WCB, 1997)
Chapter 11, Problem A - 2

s21=Lx[ψ[x,y,z]]-(y pz[ψ[x,y,z]]-z py[ψ[x,y,z]])//Simplify

s22=Ly[ψ[x,y,z]]-(z px[ψ[x,y,z]]-x pz[ψ[x,y,z]])//Simplify

s23=Lz[ψ[x,y,z]]-(x py[ψ[x,y,z]]-y px[ψ[x,y,z]])//Simplify

s24=Lx[x ψ[x,y,z]]-(x Lx[ψ[x,y,z]])//Simplify

s25=Lx[y ψ[x,y,z]]-(y Lx[ψ[x,y,z]])//Simplify

s26=Lx[z ψ[x,y,z]]-(z Lx[ψ[x,y,z]])//Simplify

s27=Ly[x ψ[x,y,z]]-(x Ly[ψ[x,y,z]])//Simplify

s28=Ly[y ψ[x,y,z]]-(y Ly[ψ[x,y,z]])//Simplify

s29=Ly[z ψ[x,y,z]]-(z Ly[ψ[x,y,z]])//Simplify

s30=Lz[x ψ[x,y,z]]-(x Lz[ψ[x,y,z]])//Simplify

s31=Lz[y ψ[x,y,z]]-(y Lz[ψ[x,y,z]])//Simplify

s32=Lz[z ψ[x,y,z]]-(z Lz[ψ[x,y,z]])//Simplify

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3.
Goswami Quantum mechanics
Problem A-3

s31=Lsq[x ψ[x,y,z]]-x Lsq[ ψ[x,y,z]]//Simplify

s32=-I ℏ (Ly[z ψ[x,y,z]]+z Ly[ ψ[x,y,z]])+I ℏ (Lz[y ψ[x,y,z]]+y Lz [ψ[x,y,z]])//Simplify

s32-s31//Simplify

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4.
Goswami Quantum mechanics
Problem A - 4

s41=Lsq[Lsq[x ψ[x,y,z]]]-Lsq[x Lsq[ψ[x,y,z]]]-Lsq[x Lsq[ ψ[x,y,z]]]+x Lsq[ Lsq[ψ[x,y,z]]]//FullSimplify

s42=2 (ℏ^2) (x Lsq[ ψ[x,y,z]]+Lsq[x ψ[x,y,z]])//FullSimplify

s42-s41//Simplify

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5
Show that

s51=-I ℏ (-Ly[z ψ[x,y,z]]+z Ly[ ψ[x,y,z]] +Lz[y ψ[x,y,z]]-y Lz[ ψ[x,y,z]])//FullSimplify

s52=(Lx[Lx[x ψ[x,y,z]]]+Ly[Lx[y ψ[x,y,z]]]+Lz[Lx[z ψ[x,y,z]]])//FullSimplify

s53=(Lx[x Lx[ψ[x,y,z]]]+Ly[y Lx [ψ[x,y,z]]]+Lz[z Lx[ ψ[x,y,z]]])//FullSimplify

s54=(Lx[x Lx[ψ[x,y,z]]]+Lx[y Ly [ψ[x,y,z]]]+Lx[z Lz[ ψ[x,y,z]]])//FullSimplify

s55=(x Lx[ Lx[ψ[x,y,z]]]+y Lx[ Ly [ψ[x,y,z]]]+z Lx[Lz[ ψ[x,y,z]]])//FullSimplify

s51-(s52+s53+s54+s55)//FullSimplify

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6
Show that 
	with

eq61=(1/(R^2))(Lx[Lx[ψ[x,y,z]]]+Ly[Ly[ψ[x,y,z]]]+Lz[Lz[ψ[x,y,z]]])//FullSimplify

eq62=Nest[prq,ψ[x,y,z],2]//FullSimplify

eq13=-(ℏ^2) Lap[ψ[x,y,z]]//FullSimplify

eq61+eq62-eq63//FullSimplify

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7
Linear radial momentum

eq71=prq[ψ[x,y,z]]//Simplify

eq72=prq[R ψ[x,y,z]]-R prq[ψ[x,y,z]]//Simplify

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8.
Show that

		r·p->		rPsq=(-i ℏ)( r.Gra[#])&

rPsq:=(-I ℏ)( r.Gra[#])&

eq80=rPsq[ψ[x,y,z]]//Simplify

eq81=(r.r)Psq[ψ[x,y,z]]+I ℏ rPsq[ψ[x,y,z]]-rPsq[rPsq[ψ[x,y,z]]]//FullSimplify

eq82=Lsq[ψ[x,y,z]]//Simplify

eq82-eq81//Simplify

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9
G. Arfken, H.J. Weber, and F.E. HHarris, Mathematical Methods for Physicists, 
7th editiom (Elsevier, 2013).
Problem 3.5.12
Prove that [(a ·L), (b·L)]=i (a×b)·L

a={ax,ay,az};b={bx, by, bz};a.L[ψ[x,y,z]]

eq91=a.L[b.L[ψ[x,y,z]]]-b.L[a.L[ψ[x,y,z]]]//FullSimplify

eq92=I  ℏ Cross[a,b].L[ψ[x,y,z]]//Simplify

eq91-eq92//Simplify

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10.
Y. Peleg, R. Pnini, and E. Zaarur, Schaum’s Quantum Mechanics  (McGraw-Hill, 1998)
problem 6-2
Commutation relations
[(L^2),Lx], [(L^2),Ly], [(L^2),Lz]
[(Ly^2),Lz]=iℏ (LxLy+LyLx]

eq101=Lsq[Lx[ψ[x,y,z]]]-Lx[Lsq[ψ[x,y,z]]]//Simplify

eq102=Lsq[Ly[ψ[x,y,z]]]-Ly[Lsq[ψ[x,y,z]]]//Simplify

eq103=Lsq[Lz[ψ[x,y,z]]]-Lz[Lsq[ψ[x,y,z]]]//Simplify

eq104=Ly[Ly[Lz[ψ[x,y,z]]]]-Lz[Ly[Ly[ψ[x,y,z]]]]//Simplify

eq105=I ℏ ( Lx[Ly[ψ[x,y,z]]]+ Ly[Lx[ψ[x,y,z]]])//Simplify

eq104-eq105//Simplify

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11
Laplace-Runge-Lenz vector in quantum mechanics

J.J. Sakurai and Jim Napolitano,
Modern Quantum Mechanics, 3rd edition
Cambridge University Press (2021)
p.252, 253





The Lenz vector in quantum mechanics is defined as Mx, My, and Mz as follows.H0 is the Hamiltonian for hydrogen-like atom with Z as atomic number.

Mx:=((1/(2m ))(py[Lz[#]]-pz[Ly[#]]-Ly[pz[#]]+Lz[py[#]])-Z ((e^2)/R) x #)&;My:=((1/(2m ))(pz[Lx[#]]-px[Lz[#]]-Lz[px[#]]+Lx[pz[#]])-Z ((e^2)/R) y #)&;Mz:=((1/(2m ))(px[Ly[#]]-py[Lx[#]]-Lx[py[#]]+Ly[px[#]])-Z ((e^2)/R) z #)&;
H0:=((1/(2m))Psq[#]-Z ((e^2)/R) #)&;

Commutation relation	[(Subscript[M, i]),H0]=0		(i = x, y, z)

eq111=Mx[H0[ψ[x,y,z]]]-H0[Mx[ψ[x,y,z]]]//Simplify

eq112=My[H0[ψ[x,y,z]]]-H0[My[ψ[x,y,z]]]//Simplify

eq113=Mz[H0[ψ[x,y,z]]]-H0[Mz[ψ[x,y,z]]]//Simplify

Relation:		 (M^2)=(2/m) H0 ((L^2)+(ℏ^2))+(Z^2)(e^4)

eq112=Mx[Mx[ψ[x,y,z]]]+My[My[ψ[x,y,z]]]+Mz[Mz[ψ[x,y,z]]]//Simplify;

eq113=(2/m) H0[Lsq[ψ[x,y,z]]+(ℏ^2) ψ[x,y,z]]+(Z^2) (e^4) ψ[x,y,z]//Simplify;eq114=eq112-eq113//Simplify

Commutation relation	[(Subscript[M, x]),(Subscript[L, y])]=  iℏ (Subscript[M, z])

eq115=Mx[Ly[ψ[x,y,z]]]-Ly[Mx[ψ[x,y,z]]]-I ℏ Mz[ψ[x,y,z]]//Simplify

eq116=My[Lz[ψ[x,y,z]]]-Lz[My[ψ[x,y,z]]]-I ℏ Mx[ψ[x,y,z]]//Simplify

eq117=Mz[Lx[ψ[x,y,z]]]-Lx[Mz[ψ[x,y,z]]]-I ℏ My[ψ[x,y,z]]//Simplify

Commutation relation	[(Subscript[M, x]),(Subscript[M, y])]= - iℏ(2/m) H0 (Subscript[L, z])

eq118=Mx[My[ψ[x,y,z]]]-My[Mx[ψ[x,y,z]]]+I ((2 ℏ  )/m)H0[Lz[ψ[x,y,z]]]//Simplify

eq119=My[Mz[ψ[x,y,z]]]-Mz[My[ψ[x,y,z]]]+I ((2 ℏ  )/m)H0[Lx[ψ[x,y,z]]]//Simplify

eq120=Mz[Mx[ψ[x,y,z]]]-Mx[Mz[ψ[x,y,z]]]+I ((2 ℏ  )/m)H0[Ly[ψ[x,y,z]]]//Simplify

L·M=M·L=0

eq121=Mx[Lx[ψ[x,y,z]]]+My[Ly[ψ[x,y,z]]]+Mz[Lz[ψ[x,y,z]]]//Simplify

eq122=Lx[Mx[ψ[x,y,z]]]+Ly[My[ψ[x,y,z]]]+Lz[Mz[ψ[x,y,z]]]//Simplify