Radial linear momentum operator in quantum mechanics
Masatsugu Sei Suzuki and Itsuko S. Suzuki
Department of Physics, SUNY at Binghamton
(Date: March 17, 2023)

Here we show our Mathematica programs for the radial linear 
momentum in quantum mechanics, in the spherical coordinates. 
 
((Notations))

ex->ux,	ey->uy	ez->uz	er->ur
x->x,		y->y		z->z	
Laplacian->	Lap (spherical coordinate)
grad->		Gra (spherical coordinate)
div->			Diva (spherical coordinate)
L->			L (angular momentum vector operator)
p->			P (linear momentum vector vector)
Lx->			ux·L
Ly->			uy·L
Lz->			uy·L
px->			ux·P
py->			uy·P
pz->			uy·P
Lx+iLy->		LP
Lx-iLy->		LM
pr->			Prq
(L^2)->		    Lsq
(p^2)->		    Psq
Hamiltonian	H[L1]; L1 is an integer.

Clear["Global`"];SetCoordinates[Spherical[r,θ,ϕ]]; ux={Sin[θ] Cos[ϕ],Cos[θ] Cos[ϕ],-Sin[ϕ]};uy={Sin[θ] Sin[ϕ],Cos[θ] Sin[ϕ],Cos[ϕ]};uz={Cos[θ],-Sin[θ],0};ur={1,0,0};x=r ur.ux;y=r ur.uy;z=r ur.uz;Lap:=Laplacian[#,{ r,θ,ϕ},"Spherical"]&;Gra:=Grad[#,{ r,θ,ϕ},"Spherical"]&;Diva:=Div[#,{r,θ, ϕ},"Spherical"]&;Curla:=Curl[#,{r,θ, ϕ},"Spherical"]&;L:=(-I ℏ( Cross[(ur  r),Gra[#]])&)//Simplify;P:=-I ℏ Gra[#]&;Lx:=(ux.L[#]&)//Simplify;Ly:=(uy.L[#]&)//Simplify;Lz:=(uz.L[#]&)//Simplify;px:=(ux.P[#]&)//Simplify;py:=(uy.P[#]&)//Simplify;pz:=(uz.P[#]&)//Simplify;LP:=(Lx[#]+I Ly[#])&//Simplify;LM=(Lx[#]-I Ly[#])&//Simplify;prq:=(((-I ℏ)/2) ur .Gra[#]+ ((-I ℏ)/2) Diva[# ur])&;Lsq:=(Nest[Lx, #,2]+Nest[Ly, #,2]+Nest[Lz, #,2])&;Psq:=(Nest[px, #,2]+Nest[py, #,2]+Nest[pz, #,2])&;H[L1_]:=((1/(2μ)) Nest[prq,#,2]+((ℏ^2)/(2μ (r^2))) L1 (L1+1) #-((Z (e^2))/r)#) &//FullSimplify;

1.  Definition of quantum mechanical radial momentum;  prq

eq11=prq[χ[r]]//Simplify

eq12=Nest[prq, χ[r],2]//Simplify

H[L1][χ[r]]//FullSimplify

2. 
We show that eq1+eq2-eq3=0;  
((Subscript[p, rq])^2)+(1/(r^2))((Lx^2)+(Ly^2)+(Lz^2))=-(ℏ^2)(∇^2)
or	(1/(2m))((Subscript[p, rq])^2)+(1/(2(mr^2)))((Lx^2)+(Ly^2)+(Lz^2)) is equal to (1/(2m))(p^2)

eq21=Nest[prq,χ[r,θ, ϕ],2]//Simplify;

eq22=(1/(r^2))(Lx[Lx[χ[r,θ,ϕ]]]+Ly[Ly[χ[r,θ,ϕ]]]+Lz[Lz[χ[r,θ,ϕ]]])//FullSimplify;

eq23=eq21+eq22//Simplify

eq24=-(ℏ^2) Lap[χ[r,θ, ϕ]]//Simplify

eq23-eq24//FullSimplify

3.  
((Goswami )) Quantum Mechanics second edition
Chapter 12-4
Prove the following equivalence:
(Subscript[p, r]) (r^2) (Subscript[p, r]) =r ((Subscript[p, r])^2)r

g[n_]:=prq[(r^n) prq[χ[r]]]-r   Nest[ prq, r χ[r],n]//FullSimplify

g[1]

g[2]

g[3]

g[4]

4.  
Commutation relation of (Subscript[p, rq]) and r;		
[(Subscript[p, r]),r]=-iℏ,		[pr,f(r)]=-iℏ f’(r)

eq41=prq[r χ[r]]-r prq[χ[r]]//FullSimplify

eq42=prq[((1/r)) χ[r]]-(1/r)prq[χ[r] ]//Simplify

eq43=prq[((1/(r^2))) ψ[r,θ,ϕ]]-(1/(r^2))prq[ψ[r,θ,ϕ] ]//Simplify

eq44=prq[((1/(r^n))) ψ[r,θ,ϕ]]-(1/(r^n))prq[ψ[r,θ,ϕ] ]//Simplify

5.  
Goswami Quantum Mechanics second edition
Chapter 12-3
Show that the commutator of the operator  (Subscript[p, r]) with r  is as
	[r, (Subscript[p, r])]=iℏ
Hence show that
	p·r = r·p - 3iℏ

eq51=prq[r ψ[r, θ,ϕ]]-r prq[ ψ[r, θ,ϕ]]//FullSimplify

eq52=px[x ψ[r, θ,ϕ]]+py[y ψ[r, θ,ϕ]]+pz[z ψ[r, θ,ϕ]]//FullSimplify;eq53=x px [ψ[r, θ,ϕ]]+y py[ψ[r, θ,ϕ]]+z pz [ψ[r, θ,ϕ]]//FullSimplify;

eq52

eq53

eq52-eq53//Simplify

6.   
Hamiltonian;
		H[L1]=(1/(2μ)) ((Subscript[p, r])^2)+((ℏ^2)/(2μ (r^2))) L1 (L1+1) -((Z (e^2))/r)
Find the commutation relation
		[H1 , pr],	[H1, [H1 , pr]]

H[L1][ψ[r]]//FullSimplify

S1:=(H[L1][prq[#]]-prq[H[L1][#]])&//FullSimplify;

eq61=S1[ψ[r]]//FullSimplify

eq62=H[L1][S1[ψ[r]]]-S1[H[L1][ψ[r]]]//FullSimplify

6.	P1[n] = (((pr))^n)

P1[n_]:=Nest[prq,#,n]&;

P1[1][f[r]]//Simplify

P1[2][f[r]]//Simplify

P1[3][f[r]]//Simplify

P1[4][f[r]]//Simplify

P1[5][f[r]]//Simplify

7.	Hydrogen atom : ladder operator
J. Binney and D. Skinner, The Physics of Quantum Mechanics (Oxford University 
Press, 2014)

prq is the radial linear momentum
a is the Bohr radius and Z is the atomic number.

The lowering operator: AD

AD[L1_]:=(a/(Sqrt[2]))((I/ℏ)prq[ #]-((L1+1)/r) # +(Z/((L1+1) a)) #)&;

The raising operator: AU

AU[L1_]:=(a/(Sqrt[2]))( -(I/ℏ)prq[ #]-((L1+1)/r) # +(Z/((L1+1) a)) #)&;

The Hamiltonian of hydrogen atom. μ is a reduced mass.

H[L1_]:=((1/(2 μ)) prq[prq[#]]+((L1 (L1+1)(ℏ^2))/(2 μ (r^2))) #- ((Z (e1^2))/( r))#)&;

AU[L] AD[L] (number operator) ;
H[L1]=((ℏ^2)/(μ (a^2)))(AU[L] AD[L] -(((Z^2) )/(2 (((L1+1))^2))))

f1=((ℏ^2)/(μ (a^2)))(AU[L1][AD[L1][χ[r]]]-(((Z^2) χ[r])/(2 (((L1+1))^2))))//Simplify;

f2=H[L1][χ[r]]//Simplify;

f12=(f1-f2)/.{a->((ℏ^2)/(μ (e1^2)))}//Simplify

The commutation relation;
AD[L1] AU[L1] - AU[L1]AD[L] is equal to -(((a^2) μ)/(ℏ^2))(H[L1+1]-H[L1])

f3=(AU[L1][AD[L1][χ[r]]]-  AD[L1][AU[L1][χ[r]]])//Simplify

f4=-(((a^2) μ)/(ℏ^2))(H[L1+1][χ[r]]-H[L1][χ[r]]//Simplify);f34=f3-f4//Simplify

Some comment on various kinds of commutation relations

(a)	AU[L1] AU[L1+1] - AU[L1+1]AU[L1]

AU[L1][AU[L1+1][χ[r]]]-AU[L1+1][AU[L1][χ[r]]]//Simplify

(b)	AD[L1] AD[L1+1] - AD[L1+1]AD[L1]

AD[L1][AD[L1+1][χ[r]]]-AD[L1+1][AD[L1][χ[r]]]//Simplify

(c)	AD[L1] AU[L1+1] - AU[L1+1]AD[L1]

AD[L1][AU[L1+1][χ[r]]]-AU[L1+1][AD[L1][χ[r]]]//Simplify

(d)	AU[L1] AD[L1+1] - AD[L1+1]AU[L1]

AU[L1][AD[L1+1][χ[r]]]-AD[L1+1][AU[L1][χ[r]]]//Simplify

(e)	H[L1+1] - H[L1]

f5=((μ (a^2))/(ℏ^2))(H[L1+1][χ[r]]-H[L1][χ[r]])//Simplify

(f)	AD[L1] H[L1] -  H[L1+1] AD[L1]=0

f6=AD[L1][H[L1][χ[r]]]-H[L1+1][AD[L1][χ[r]]]/.{a->((ℏ^2)/(μ (e1^2)))}//Simplify

(g)	H[L1] AU[L1] - AU[L1] H[L1+1]=0

f7=H[L1][AU[L1][χ[r]]]-AU[L1][H[L1+1][χ[r]]]/.{a->((ℏ^2)/(μ (e1^2)))}//Simplify//Simplify