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## The hidden SO(4) symmetry of the Hydrogen Atom

Maple

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

,

where is the electron momentum,  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 , one can derive a central force  that drives the electron's motion. Introducing the angular momentum

,

one can further define the Runge-Lenz vector

It is well known that  is a constant of the motion, i.e. . Switching to Quantum Mechanics, this condition reads

where, for hermiticity purpose, the expression of  must be symmetrized

.

Here, departing from the basic commutation rules between position , momentum  and angular momentum  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

and   ,

,

.

Since H commutes with both  and , defining

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

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):  ,    and
 Identities (II): the commutation rules between  ,  and the potential

II  and

 Setting up the problem and  is hermitian
 Algebraic approach
 Alternative approach using differential operators
 III

IV

 Algebraic approach
 Alternative approach using differential operators