Vector analysis
Angular momentum in the cylindrical coordinates

Clear["Global`"];r1=(Sqrt[(z^2)+(ρ^2)]);ux={Cos[ϕ],-Sin[ϕ],0};uy={ Sin[ϕ],Cos[ϕ],0};uz={0,0,1};r={ρ,0,z};ur=(1/r1){ρ,0,z};Gra:=Grad[#,{ρ,ϕ,z},"Cylindrical"]&;Lap:=Laplacian[#,{ρ,ϕ,z},"Cylindrical"]&;Curla:=Curl[#,{ρ,ϕ,z},"Cylindrical"]&;Diva:=Div[#,{ρ,ϕ,z},"Cylindrical"]&;L:=(-I ℏ Cross[r,Gra[#]])&;Lx:=(ux.L[#]&)//Simplify;Ly:=(uy.L[#]&)//Simplify;Lz:=(uz.L[#]&)//Simplify;LP:=(Lx[#] + I Ly[#])&;LM:=(Lx[#] - I Ly[#])&;prq:=(((-I ℏ)/2) ur .Gra[#]+ ((-I ℏ)/2) Diva[# ur])&;Leq:=(Nest[Lx,#,2]+Nest[Ly,#,2]+Nest[Lz,#,2])&;p:=(-I ℏ  Gra[#])&;px:=(ux.p[#])&;py:=(uy.p[#])&;pz:=(uz.p[#])&;pρ:=({1,0,0}.p[#])&;

Laplacian, grad, div, and rot (curl)

Lap[ψ[ρ,ϕ,z]]//Simplify

Gra[ψ[ρ,ϕ,z]]//Simplify

Diva[{Vρ[ρ,ϕ,z], Vϕ[ρ,ϕ,z],Vz[ρ,ϕ,z]}]

Curla[{Vρ[ρ,ϕ,z], Vϕ[ρ,ϕ,z],Vz[ρ,ϕ,z]}]//Simplify

Angular momentum in  the cylindrical coordinate
L, Lx, Ly, Lz, Lx+iLy, Lx-iLy

L[ψ[ρ,ϕ,z]]//Simplify

Lx[ψ[ρ,ϕ,z]]//FullSimplify

Ly[ψ[ρ,ϕ,z]]//FullSimplify

Lx[ψ[ρ,ϕ,z]]+I Ly[ψ[ρ,ϕ,z]]//FullSimplify

Lx[ψ[ρ,ϕ,z]]-I Ly[ψ[ρ,ϕ,z]]//FullSimplify

Lz[ψ[ρ,ϕ,z]]//Simplify

Lx[Ly[ψ[ρ,ϕ,z]]]-Ly[Lx[ψ[ρ,ϕ,z]]]-I ℏ Lz[ψ[ρ,ϕ,z]]//Simplify

Ly[Lz[ψ[ρ,ϕ,z]]]-Lz[Ly[ψ[ρ,ϕ,z]]]-I ℏ Lx[ψ[ρ,ϕ,z]]//Simplify

Lz[Lx[ψ[ρ,ϕ,z]]]-Lx[Lz[ψ[ρ,ϕ,z]]]- I ℏ Ly[ψ[ρ,ϕ,z]]//Simplify//Simplify

Leq[Lx[ψ[ρ,ϕ,z]]]-Lx[Leq[ψ[ρ,ϕ,z]]]//Simplify

Leq[Ly[ψ[ρ,ϕ,z]]]-Ly[Leq[ψ[ρ,ϕ,z]]]//Simplify

Leq[Lz[ψ[ρ,ϕ,z]]]-Lz[Leq[ψ[ρ,ϕ,z]]]//Simplify

Lz[LP[ψ[ρ,ϕ,z]]]-LP[Lz[ψ[ρ,ϕ,z]]]-ℏ LP[ψ[ρ,ϕ,z]]//Simplify

Lz[LM[ψ[ρ,ϕ,z]]]-LM[Lz[ψ[ρ,ϕ,z]]]+ℏ LM[ψ[ρ,ϕ,z]]//Simplify

The commutation of the angular momentum in  the cylindrical coordinate

eq5=Lx[Ly[ψ[ρ,ϕ,z]]]- Ly[Lx[ψ[ρ,ϕ,z]]]-I ℏ Lz[ψ[ρ,ϕ,z]]//Simplify

(L^2)	in the cylindrical coordinate

eq61=Lx[Lx[ψ[ρ,ϕ,z]]]+ Ly[Ly[ψ[ρ,ϕ,z]]]+Lz[ Lz[ψ[ρ,ϕ,z]]]//Simplify

eq62=Leq[ψ[ρ,ϕ,z]]//Simplify

eq61-eq62//Simplify

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Schrodionger equation with a speific wavefunction in the cylindrical coordinate

H1=χ[ρ] Exp[I kz z] Exp[I m ϕ];

(((-1)/(2 μ))Lap[H1]+V[ρ] H1) //Simplify

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Radial momentum (quantum)

prq[ ψ[ρ,ϕ,z]]//Simplify

Commutation relation for radial momentum and magnitude of r

prq[r1 ψ[ρ,ϕ,z]]-r1 prq[ψ[ρ,ϕ,z]]//Simplify

Linear momentum

pρ[ψ[ρ,ϕ,z]]//Simplify

px[ψ[ρ,ϕ,z]]//Simplify

py[ψ[ρ,ϕ,z]]//Simplify

pz[ψ[ρ,ϕ,z]]//Simplify

Nest[px,ψ[ρ,ϕ,z],2]+Nest[py,ψ[ρ,ϕ,z],2]+Nest[pz,ψ[ρ,ϕ,z],2]//Simplify

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Goswami
Quantum Mechanics
 p265 Problem 4

prq[(r1^2) prq[ψ[ρ,ϕ,z]]]-r1 prq[ prq[r1 ψ[ρ,ϕ,z]]]//Simplify

pρ[ψ[ρ,ϕ,z]]//Simplify

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Goswami
Quantum Mechanics

 p265 Problem 5

s51=(ρ^2)(Nest[px,ψ[ρ,ϕ,z],2]+Nest[py,ψ[ρ,ϕ,z],2]+Nest[pz,ψ[ρ,ϕ,z],2])//Simplify

s52=ρ pρ[ρ pρ[ψ[ρ,ϕ,z]]]//FullSimplify

s53=Lz[Lz[ψ[ρ,ϕ,z]]]//FullSimplify

s54=(ρ^2) pz[pz[ ψ[ρ,ϕ,z]]]//FullSimplify

s51-(s52+s53+s54)//Simplify

Proof of
[Leq,[Leq,x]]=2 (ℏ^2)(x Leq+Leq x)
using the cylindrical co-ordinates

x1=ux.r

s71=Leq[Leq [x1 ψ[ρ,ϕ,z]]]-Leq[x1 Leq[ψ[ρ,ϕ,z]]]-Leq[x1 Leq[ψ[ρ,ϕ,z]]]+x1 Leq[Leq[ψ[ρ,ϕ,z]]]//FullSimplify

s72=2 (ℏ^2) (x1 Leq[ψ[ρ,ϕ,z]]+Leq[x1 ψ[ρ,ϕ,z]])//FullSimplify

s71-s72//Simplify

(p^2)	in the cylindrical coordinate

t11=px[px[ψ[ρ,ϕ,z]]]+ py[py[ψ[ρ,ϕ,z]]]+pz[ pz[ψ[ρ,ϕ,z]]]//Simplify

(pr^2)	in the cylindrical coordinate

t12=prq[prq[ψ[ρ,ϕ,z]]]//Simplify

t13=(t11-t12)((z^2)+(ρ^2))//Simplify

(L^2) in the cylindrical coordinates

t14=Leq[ψ[ρ,ϕ,z]]//Simplify

t13-t14//Simplify