This is, perhaps, one of the most complicated computations done in this area using the Physics package. To the best of my knowledge never before performed on a computer algebra worksheet. It is exciting to present a computation like this one. At the end the corresponding worksheet is linked so that it can be downloaded and the sections be opened, the computation be reproduced. There is also a link to a pdf with everything open.  Special thanks to Pascal Szriftgiser for bringing this problem. To reproduce the computations below, please update the Physics library with the one distributed from the Maplesoft R&D Physics webpage.

 

Quantum Runge-Lenz Vector and the Hydrogen Atom,

the hidden SO(4) symmetry

 

Pascal Szriftgiser1 and Edgardo S. Cheb-Terrab2 

(1) Laboratoire PhLAM, UMR CNRS 8523, Université Lille 1, F-59655, France

(2) Maplesoft

``

 

Let's consider the Hydrogen atom and its Hamiltonian

H = LinearAlgebra[Norm](`#mover(mi("p"),mo("→"))`)^2/(2*m__e)-kappa/r,

 

where `#mover(mi("p"),mo("→"))`is the electron momentum, m__e its mass, κ a real positive constant, and r the distance of the electron from the proton located at the origin. We assume that the proton's mass is infinite. Classically, from the potential -kappa/r, one can derive a central force `#mover(mi("F"),mo("→"))` = -kappa*`#mover(mi("r"),mo("∧"))`/r^2 that drives the electron's motion. Introducing the angular momentum

 

`#mover(mi("L"),mo("→"))` = `&x`(`#mover(mi("r"),mo("→"))`, `#mover(mi("p"),mo("→"))`),

 

one can further define the Runge-Lenz vector `#mover(mi("Z"),mo("→"))`

 

"Z=1/(`m__e`) (L)*(p)+kappa ( r)/r."

 

It is well known that `#mover(mi("Z"),mo("→"))` is a constant of the motion, i.e. diff(`#mover(mi("Z"),mo("→"))`(t), t) = 0. Switching to Quantum Mechanics, this condition reads

 

%Commutator(H, Z_) = 0.

where, for hermiticity purpose, the expression of `#mover(mi("Z"),mo("→"))` must be symmetrized

 

`#mover(mi("Z"),mo("→"))` = (`&x`(`#mover(mi("L"),mo("→"))`, `#mover(mi("p"),mo("→"))`)-`&x`(`#mover(mi("p"),mo("→"))`, `#mover(mi("L"),mo("→"))`))/(2*m__e)+kappa*`#mover(mi("r"),mo("→"))`/r.

 

Here, departing from the basic commutation rules between position `#mover(mi("r"),mo("→"))`, momentum `#mover(mi("p"),mo("→"))` and angular momentum `#mover(mi("L"),mo("→"))` all in tensor notation, we first derive a useful couple of intermediate identities, then we demonstrate the following commutation rules between the quantum Hamiltonian, angular momentum and Runge-Lenz vector `#mover(mi("Z"),mo("→"))`

 

"[H,L[n]][-]=0   "and   "[H,Z[n]][-]=0",

"[L[m],Z[n]][-]=i `ℏ` `ε`[m,n,o] Z[o]",

"[Z[m],Z[n]][-]=-2 (i `ℏ`)/(`m__e`) H `ε`[m,n,o]  L[o]".

 

Since H commutes with both `#mover(mi("L"),mo("→"))`NULL and `#mover(mi("Z"),mo("→"))`, defining

 

"`M__n`=sqrt(-(`m__e`)/(2 H)) `Z__n`,"

one gets the set of relations (the first one is part of the departure point)

"[L[m],L[n]][-]=i `ℏ` `ε`[m,n,o] L[o],   [L[m],M[n]][-]=i `ℏ` `ε`[m,n,o] M[o],"

"[M[m],M[n]][-]= i `ℏ` `ε`[m,n,o]  L[o]."


This set constitutes the Lie algebra of the SO(4) group (closely related to a Poincaré group in special relativity).

 

I Commutation rules and useful identities

 

Quantum commutation rules basics and the Hamiltonian of the hydrogen atom

   

Identities (I):  `∂__n`(V) = -V^3*X__n,  V^3*X[l]^2 = V  and  `□`(V) = 0

   

Identities (II): the commutation rules between  `#mover(mi("L",mathcolor = "olive"),mo("→"))`, `#mover(mi("p",mathcolor = "olive"),mo("→"))` and the potential V(X)

   

II "[H,L[n]][-]=0" and "[H,Z[n]][-]=0"``

 

"[H,L[n]][-]=0"

   

"[H,Z[n]][-]=0"

 

Setting up the problem and Z[n] is hermitian

   

Algebraic approach

   

Alternative approach using differential operators

   

III "[L[m],Z[n]][-]=i `ℏ` `ε`[m,n,k] Z[k]"

   

IV "[Z[m],Z[n]][-]=-(2 i `ℏ` )/(`m__e`)H `ε`[m,n,o] L[o]"

 

Algebraic approach

   

Alternative approach using differential operators

   


Download HiddenSO4.mw    Download  HiddenSO4.pdf

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft


Please Wait...