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Tensor Product of Quantum State Spaces...

Posted: ecterrab 14540 Product: Maple

March 23 2018

4 0

This is about the recent implementation of tensor products of quantum state spaces in the Physics package, in connection with an exchange with the Physics of Information Lab of the University of Waterloo. As usual this development is available to everybody from the Maplesoft R&D Physics webpage. This is the last update for Maple 2017. The updates for Maple 2018, starting with this same material, will begin being distributed through the MapleCloud next week.

Tensor Product of Quantum State Spaces

Basic ideas and design

Suppose and are quantum operators and are, respectively, their eigenkets. The following works since the introduction of the Physics package in Maple

>

(1)

>

(2)

>

(3)

where on the left-hand sides the product operator `*` is used as a sort of inert form (it has all the correct mathematical properties but does not perform the contraction) of the dot product operator `.`, used on the right-hand sides.

Suppose now that and act on different, disjointed, Hilbert spaces.

1) To represent that, a new keyword in Setup , is introduced, to indicate which spaces are disjointed, say as in . We want this notation to pop up at some point as where the indexation indicates all the operators acting on that Hilbert space. The disjointedspaces keyword has for synonyms disjointedhilbertspaces and hilbertspaces. The display is not yet implemented.

NOTE: noncommutative quantum operators acting on disjointed spaces commute between themselves, so after setting - for instance - , automatically, become quantum operators satisfying (see comment (ii) on page 156 of ref. [1])

2) Product of Kets and Bras (KK, BB, KB and BK) where K and B belong to disjointed spaces, are understood as tensor products satisfying, for instance with disjointed spaces and (see footnote on page 154 of ref. [1]),

while of course

	Details

3) All the operators of one disjointed space act transparently over operators, Bras and Kets of the other disjointed spaces, for example

and the same for the Dagger of this equation, that is

And this happens automatically. Hence, when we write the left-hand sides and press enter, they are automatically rewritten (returned) as the right-hand sides.

Note that for the product of an operator times a Bra or a Ket we are not using the notation that expresses the product with the symbol 5.

Regarding the display of Bras and Kets and their tensor products, two enhancements are happening:

•	A new Setup option hideketlabel makes all the labels in Kets and Bras to be hidden when displaying Kets, Bras and Bracket(s), with the indices presented one level up, as if they were a sequence of labels, so that:

is displayed as

This is the notation used more frequently when working in quantum information. This hideketlabel option is already implemented entering

•	Tensor products formed with operators, or Bras and Kets, that belong to disjointed spaces (set as such using Setup ), are displayed with the symbol 5 in between, as in instead of , and instead of .

	Tensor product notation and the hideketlabel option

	The implementation of tensor products using `*` and `.`

	Basic exercising with the new functionality

	Related functionality already in place before these changes

Reference

[1] Cohen-Tannoudji, Diue, Laloe, Quantum Mechanics, Chapter 2, section F.

[2] Griffiths Robert B., Hilbert Space Quantum Mechanics, Quantum Computation and Quantum Information Theory Course, Physics Department, Carnegie Mellon University, 2014.

	See also

Download TensorProductDesign.mw

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

Automatic handling of collision of tenso...

Posted: ecterrab 14540 Product: Maple

March 07 2018

3 0

Automatic handling of collision of tensor indices in products

The design of products of tensorial expressions that have contracted indices got enhanced. The idea: repeated indices in certain subexpressions are actually dummies. So suppose and are tensors, then in , is just a dummy, therefore is a well defined object. The new design automatically maps input like into . This significantly simplifies the manipulation of indices when working with products of tensorial expressions: each tensorial expression being multiplied has its repeated indices automatically transformed into different ones when they would collide with the free or repeated indices of the other expressions being multiplied.

This functionality is available within the Physics update distributed at the Maplesoft R&D Physics webpage (but for what you see under Preview that makes use of a new kernel feature of the Maple version under development).

>

(1)

>

${B[b], Physics:-Dgamma[a], Physics:-Psigma[a], T[a, b], Physics:-d_[a], Physics:-g_[a, b], Physics:-KroneckerDelta[a, b], Physics:-LeviCivita[a, b, c, d]}$

(2)

This shows the automatic handling of collision of indices

>

(3)

>

(4)

>

Preview only in the upcomming version under development

Consider now the case of three tensors

>

${A[a], B[b], C[a], Physics:-Dgamma[a], Physics:-Psigma[a], T[a, b], Physics:-d_[a], Physics:-g_[a, b], Physics:-KroneckerDelta[a, b], Physics:-LeviCivita[a, b, c, d]}$

(5)

>

(6)

The product above has indeed the index repeated more than once, therefore none of its occurrences got automatically transformed into contravariant in the output, and Check detects the problem interrupting with an error message

>

Error, (in Physics:-Check) wrong use of the summation rule for repeated indices: `a repeated 3 times`, in A[a]*B[a]*C[a]

However, it is now also possible to indicate, using parenthesis, that the product of two of these tensors actually form a subexpression, so that the following two tensorial expressions are well defined, and the colliding dummy is automatically replaced making that explicit

>

(7)

>

(8)

This change in design makes concretely simpler the use of indices in that it eliminates the need for manually replacing dummies. For example, consider the tensorial expression for the angular momentum in terms of the coordinates and momentum vectors, in 3 dimensions

>

(9)

Define respectively representing angular and linear momentum

>

${Physics:-Dgamma[a], L[j], Physics:-Psigma[a], Physics:-d_[a], Physics:-g_[a, b], p[k], Physics:-KroneckerDelta[a, b], Physics:-LeviCivita[a, b, c], Physics:-SpaceTimeVector[a](X)}$

(10)

Introduce the tensorial expression for

>

(11)

The left-hand side has one free index, , while the right-hand side has two dummy indices and

>

(12)

If we want to computewe can now take the square of (11) directly, and the dummy indices on the right-hand side are automatically handled, there is now no need to manually substitute the repeated indices to avoid their collision

>

(13)

The repeated indices on the right-hand side are now

>

(14)

>

Download AutomaticHandlingCollisionOfTensorIndices.mw

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

Minimize the number of tensor components...

Posted: ecterrab 14540 Product: Maple

March 05 2018

5 0

Minimize the number of tensor components according to its symmetries
(and relabel, redefine or count the number of independent tensor components)

The nice development described below is work in collaboration with Pascal Szriftgiser from Laboratoire PhLAM, Université Lille 1, France, used in the Mapleprimes post Magnetic traps in cold-atom physics

A new keyword in Define and Setup : minimizetensorcomponents, allows for automatically minimizing the number of tensor components taking into account the tensor symmetries. For example, if a tensor with two indices in a 4D spacetime is defined as antisymmetric using Define with this new keyword, the number of different tensor components will be exactly 6, and the elements of the diagonal are automatically set equal to 0. After setting this keyword to true with Setup , all subsequent definitions of tensors automatically minimize the number of components while using this keyword with Define makes this minimization only happen with the tensors being defined in the call to Define .

Related to this new functionality, 4 new Library routines were added: MinimizeTensorComponents, NumberOfIndependentTensorComponents, RelabelTensorComponents and RedefineTensorComponents

Example:

>

Define an antisymmetric tensor with two indices

>

${Physics:-Dgamma[mu], F[mu, nu], Physics:-Psigma[mu], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu]}$

(1.1)

Although the system knows that is antisymmetric, you need to use Simplify to apply the (anti)symmetry

>

(1.2)

>

(1.3)

so by default the components of do not automatically reflect the (anti)symmetry; likewise

>

(1.4)

>

(1.5)

and computing the array form of we do not see the elements of the diagonal equal to zero nor the lower-left triangle equal to the upper-right triangle but for a different sign:

>

Matrix(%id = 18446744078270093062)

(1.6)

On the other hand, this new functionality, here called minimizetensorcomponents, makes the symmetries of the tensor be explicitly reflected in its components.

There are three ways to use it. First, one can minimize the number of tensor components of a tensor previously defined. For example

>

Matrix(%id = 18446744078270064630)

(1.7)

After this, both (1.2) and (1.3) are automatically equal to 0 without having to use Simplify

>

(1.8)

>

(1.9)

And the output of TensorArray in (1.6) becomes equal to (1.7).

NOTE: in addition, after using minimizetensorcomponents in the definition of a tensor, say F, all the keywords implemented for Physics tensors are available for F:

>

F[mu, nu] = Matrix(%id = 18446744078247910206)

(1.10)

>

(1.11)

>

F[mu, nu] = {(1, 2) = F[1, 2], (1, 3) = F[1, 3], (1, 4) = F[1, 4], (2, 1) = -F[1, 2], (2, 3) = F[2, 3], (2, 4) = F[2, 4], (3, 1) = -F[1, 3], (3, 2) = -F[2, 3], (3, 4) = F[3, 4], (4, 1) = -F[1, 4], (4, 2) = -F[2, 4], (4, 3) = -F[3, 4]}

(1.12)

>

(1.13)

Alternatively, one can define a tensor, specifying that the symmetries should be taken into account to minimize the number of its components passing the keyword minimizetensorcomponents to Define .

Example:

Define a tensor with the symmetries of the Riemann tensor, that is, a tensor of 4 indices that is symmetric with respect to interchanging the positions of the 1st and 2nd pair of indices and antisymmetric with respect to interchanging the position of its 1st and 2nd indices, or 3rd and 4th indices, and define it minimizing the number of tensor components

>

${Physics:-Dgamma[mu], F[mu, nu], Physics:-Psigma[mu], R[mu, nu, alpha, beta], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu]}$

(1.14)

We now have

>

(1.15)

>

(1.16)

•	One can always retrieve the symmetry properties in the abstract notation used by the Define command using the new , its output is ordered, first the symmetric then the antisymmetric properties

>

(1.17)

•	After making the symmetries explicit (and also before that), it is frequently useful to know the number of independent components of a given tensor. For this purpose you can use the new

>

(1.18)

and besides taking into account the symmetries, in the case of the Riemann tensor, after taking into account the first Bianchi identity this number of components is further reduced to 20.

A third way of using the new minimizetensorcomponents functionality is using Setup , so that, automatically, every subsequent definition of tensors with symmetries is performed minimizing the number of its components using the indicated symmetries

Example:

>

(1.19)

So from hereafter you can define tensors taking into account their symmetries explicitly and without having to include the keyword at each definition

>

${C[mu, nu], Physics:-Dgamma[mu], F[mu, nu], Physics:-Psigma[mu], R[mu, nu, alpha, beta], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu]}$

(1.20)

>

C[mu, nu] = Matrix(%id = 18446744078408747598)

(1.21)

•	Two new related functionalities are provided via and , the first one to have the number of tensor components directly reflected in the names of the components, the second one to redefine only one of these components

>

Matrix(%id = 18446744078408729774)

(1.22)

Suppose now we want to make one of these components equal to 1, say

>

C[mu, nu] = Matrix(%id = 18446744078270104390)

(1.23)

This nice development is work in collaboration with Pascal Szriftgiser from Laboratoire PhLAM, UMR CNRS 8523, Université Lille 1, F-59655, France.

>

Download MinimizeTensorComponents.mw

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

Quantum Mechanics using Maple's Physics ...

Posted: ecterrab 14540 Product: Maple

December 15 2017

6 0

It passed through my mind it would be interesting to collect the links to the most relevant Mapleprimes posts about Quantum Mechanics using the Physics package of the last couple of years, to have them all accessible from one place. These posts give an idea of what kind of computation is already doable in quantum mechanics, how close is the worksheet input to what we write with paper and pencil, and how close is the typesetting of the output to what we see in textbooks.

At the end of each page linked below, you will see another link to download the corresponding worksheet, that you can open using Maple (say the current version or the version 1 or 2 years ago).

This other set of three consecutive posts develops one problem split into three parts:

This other link is interesting as a quick and compact entry point to the use of the Physics package:

Mini-Course: Computer Algebra for Physicists

There is an equivalent set of Mapleprimes posts illustrating the Physics package tackling problems in General Relativity, collecting them is for one other time.

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

The hidden SO(4) symmetry of the Hydroge...

Posted: ecterrab 14540 Product: Maple

December 14 2017

6 0

December 2017: 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.

June 17, 2020: updated taking advantage of new features of Maple 2020.1 and Physics Updates v.705. Submitted to arxiv.org.

January 25, 2021: updated in the arXiv and submitted for publication in Computer Physics Communications.

Quantum Runge-Lenz Vector and the Hydrogen Atom,

the hidden SO(4) symmetry using Computer Algebra

Pascal Szriftgiser¹ and Edgardo S. Cheb-Terrab²

(1) University of Lille, CNRS, UMR 8523 - PhLAM - Physique des Lasers, Atomes et Molécules, F-59000 Lille, France

(2) Maplesoft

Abstract

Pauli first noticed the hidden SO(4) symmetry for the Hydrogen atom in the early stages of quantum mechanics [1]. Departing from that symmetry, one can recover the spectrum of a spinless hydrogen atom and the degeneracy of its states without explicitly solving Schrödinger's equation [2]. In this paper, we derive that SO(4) symmetry and spectrum using a computer algebra system (CAS). While this problem is well known [3, 4], its solution involves several steps of manipulating expressions with tensorial quantum operators, simplifying them by taking into account a combination of commutator rules and Einstein's sum rule for repeated indices. Therefore, it is an excellent model to test the current status of CAS concerning this kind of quantum-and-tensor-algebra computations. Generally speaking, when capable, CAS can significantly help with manipulations that, like non-commutative tensor calculus subject to algebra rules, are tedious, time-consuming and error-prone. The presentation also shows a pattern of computer algebra operations that can be useful for systematically tackling more complicated symbolic problems of this kind.

Introduction

The primary purpose of this work is to derive, step-by-step, the SO(4) symmetry of the Hydrogen atom and its spectrum using a computer algebra system (CAS). To the best of our knowledge, such a derivation using symbolic computation has not been shown before. Part of the goal was also to see whether this computation can be performed entering only the main definition formulas, followed by only simplification commands, and without using previous knowledge of the result. The intricacy of this problem is in the symbolic manipulation and simplification of expressions involving noncommutative quantum tensor operators. The simplifications need to take into account commutator rules, symmetries under permutation of indices of tensorial subexpressions, and use Einstein's sum rule for repeated indices.

We performed the derivation using the Maple 2020 system with the Maplesoft Physics Updates v.705. Generally speaking, the default computational domain of CAS doesn't include tensors, noncommutative operators nor related simplifications. On the other hand, the Maple system is distributed with a Physics package that extends that default domain to include those objects and related operations. Physics includes a Simplify command that takes into account custom algebra rules and the sum rule for repeated indices, and uses tensor-simplification algorithms [5] extended to the noncommutative domain.

A note about notation: when working with a CAS, besides the expectation of achieving a correct result for a complicated symbolic calculation, readability is also an issue. It is desired that one be able to enter the definition formulas and computational steps to be performed (the input, preceded by a prompt >, displayed in black) in a way that resembles as closely as possible their paper and pencil representation, and that the results (the output, computed by Maple, displayed in blue) use textbook mathematical-physics notation. The Physics package implements such dedicated typesetting. In what follows, within text and in the output, noncommutative objects are displayed using a different color, e.g. , vectors and tensor indices are displayed the standard way, as in , and , and commutators are displayed with a minus subscript, e.g. . Although the Maple system allows for providing dedicated typesetting also for the input, we preferred to keep visible the Maple input syntax, allowing for comparison with paper and pencil notation. We collected the names of the commands used and a one line description for them in an Appendix at the end. Maple also implements the concept of inert representations of computations, which are activated only when desired. We use this feature in several places. Inert computations are entered by preceding the command with % and are displayed in grey. Finally, as is usual in CAS, every output has an equation label, which we use throughout the presentation to refer to previous intermediate results.

In Sec.1, we recall the standard formulation of the problem and present the computational goal, which is the derivation of the formulas representing the SO(4) symmetry and related spectrum.

In Sec.2, we set tensorial non-commutative operators representing position and linear and angular momentum, respectively , and , their commutation rules used as departure point, and the form of the quantum Hamiltonian . We also derive a few related identities used in the sections that follow.

In Sec.3, we derive the conservation of both angular momentum and the Runge-Lenz quantum operator, respectively and . Taking advantage of the differentialoperators functionality in the Physics package, we perform the derivation exploring two equivalent approaches; first using only a symbolic tensor representation of the momentum operator, then using an explicit differential operator representation for it in configuration space, . With the first approach, expressions are simplified only using the departing commutation rules and Einstein's sum rule for repeated indices. Using the second approach, the problem is additionally transformed into one where the differentiation operators are applied explicitly to a test function . Presenting both approaches is of potential interest as it offers two partly independent methods for performing the same computation, which is helpful to provide confidence on in the results when unknown, a relevant issue when using computer algebra.

In Sec. 4, we derive and show that the classical relation between angular momentum and the Runge-Lenz vectors, = 0, due to the orbital momentum being perpendicular to the elliptic plane of motion while the Runge-Lenz vector lies in that plane, still holds in quantum mechanics, where the components of these quantum vector operators do not commute but =

In Sec. 5, we derive "[Z[a],Z[b]][-]=-(2 i `&hbar;` `ε`[a,b,c] (H L[c]))/`m__e`" using the two alternative approaches described for Sec.3.

In Sec. 6, we derive the well-known formula for the square of the Runge-Lenz vector, Z[k]^2 = 2*H*(`&hbar;`^2+L[a]^2)/m__e+kappa^2 .

Finally, in Sec. 7, we use the SO(4) algebra derived in the previous sections to obtain the spectrum of the Hydrogen atom. Following the literature, this approach is limited to the bound states for which the energy is negative.

Some concluding remarks are presented at the end, and input syntax details are summarized in an Appendix.

1. The hidden SO(4) symmetry of the Hydrogen atom

Let's consider the Hydrogen atom and its Hamiltonian

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

where is the electron momentum, its mass, κ a real positive constant, the distance of the electron from the proton located at the origin, and is its tensorial representation with components [. We assume that the proton's mass is infinite. The electron and nucleus spin are not taken into account. 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

.

In what follows, departing from the Hamiltonian H, the basic commutation rules between position, momentum and angular momentum in tensor notation, we derive the following commutation rules between the quantum Hamiltonian, angular momentum and Runge-Lenz vector

	=	0
	=	0
	=
	=

Since H commutes with both and , defining

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

these commutation rules can be rewritten as

	=
	=
	=

This set constitutes the Lie algebra of the SO(4) group.

	2. Setting the problem, commutation rules and useful identities

	3. Commutation rules between the Hamiltonian and each of the angular momentum and Runge-Lenz tensors

	4. Commutation rules between the angular momentum and the Runge-Lenz tensors

	5. Commutation rules between the components of the Runge-Lenz tensor

	6. The square of the norm of the Runge-Lenz vector

	7. The atomic hydrogen spectrum

Conclusions

In this presentation, we derived, step-by-step, the SO(4) symmetry of the Hydrogen atom and its spectrum using the symbolic computer algebra Maple system. The derivation was performed without departing from the results, entering only the main definition formulas in eqs. (1), (2) and (5), followed by using a few simplification commands - mainly Simplify, SortProducts and SubstituteTensor - and a handful of Maple basic commands, subs, lhs, rhs and isolate. The computational path that was used to get the results of sections 2 to 7 is not unique. Instead of searching for the shortest path, we prioritized clarity and illustration of the techniques that can be used to crack problems like this one.

This problem is mainly about simplifying expressions using two different techniques. First, expressions with noncommutative operands in products need reduction with respect to the commutator algebra rules that have been set. Second, products of tensorial operators require simplification using the sum rule for repeated indices and the symmetries of tensorial subexpressions. Those techniques, which are part of the Maple Physics simplifier, together with the SortProducts and SubstituteTensor commands for sorting the operands in products to apply tensorial identities, sufficed. The derivations were performed in a reasonably small number of steps.

Two different computational strategies - with and without differential operators - were used in sections 3 and 5, showing an approach for verifying results, a relevant issue in general when performing complicated algebraic manipulations. The Maple Physics ability to handle differential operators as noncommutative operands in products (as frequently done in paper and pencil computations) facilitates readability and ease in entering the computations. The complexity of those operations is then handled by one Physics:-Library command, ApplyProductsOfDifferentialOperators (see eqs. (47) and (83)).

Besides the Maple Physics ability to handle noncommutative tensor operators and simplify such operators using commutator algebra rules, it is interesting to note: a) the ability of the system to factorize expressions involving products of noncommutative operands (see eqs. (90) and (108)) and b) the extension of the algorithms for simplifying tensorial expressions [5] to the noncommutativity domain, used throughout this presentation.

It is also worth mentioning how equation labels can reduce the whole computation to entering the main definitions, followed by applying a few commands to equation labels. That approach helps to reduce the chance of typographical errors to a very strict minimum. Likewise, the fact that commands and equations distribute over each other allows cumbersome manipulations to be performed in simple ways, as done, for instance, in eqs. (8), (9) and (13).

Finally, it was significantly helpful for us to have the typesetting of results using standard mathematical physics notation, as shown in the presentation above.

Appendix

In this presentation, the input lines are preceded by a prompt > and the commands used are of three kinds: some basic Maple manipulation commands, the main Physics package commands to set things and simplify expressions, and two commands of the Physics:-Library to perform specialized, convenient, operations in expressions.

The basic Maple commands used

•	interface is used once at the beginning to set the letter used to represent the imaginary unit (default is I but we used i).

•	isolate is used in several places to isolate a variable in an expression, for example isolating x in results in

•	lhs and rhs respectively get the left-hand side and right-hand side of an equation

•	subs substitutes the left-hand side of an equation by the righ-hand side in a given target, for example results in

•	@ is used to compose commands. So is the same as . This command is useful to express an abstract combo of manipulations, for example as in (108) ≡ .

The Physics commands used

•	Setup is used to set algebra rules as well as the dimension of space, type of metric, and conventions as the kind of letter used to represent indices.

•	Commutator computes the commutator between two objects using the algebra rules set using Setup. If no rules are known to the system, it outputs a representation for the commutator that the system understands.

•	CompactDisplay is used to avoid redundant display of the functionality of a function.

•	d_[n] represents the tensorial differential operator.

•	Define is used to define tensors, with or without specifying its components.

•	Dagger computes the Hermitian transpose of an expression.

•	Normal, Expand, Factor respectively normalizes, expands and factorizes expressions that involve products of noncommutative operands.

•	Simplify performs simplification of tensorial expressions involving products of noncommutative factors taking into account Einstein's sum rule for repeated indices, symmetries of the indices of tensorial subexpressions and custom commutator algebra rules.

•	SortProducts uses the commutation rules set using Setup to sort the non-commutative operands of a product in an indicated ordering.

The Physics:-Library commands used

•	Library:-ApplyProductsOfDifferentialOperators applies the differential operators found in a product to the product operands that appear to its right. For example, applying this command to results in

•	equalizes the repeated indices in the terms of a sum, so for instance applying this command to results in

References

[1] W. Pauli, "On the hydrogen spectrum from the standpoint of the new quantum mechanics,” Z. Phys. 36, 336–363 (1926)

[2] S. Weinberg, "Lectures on Quantum Mechanics, second edition, Cambridge University Press," 2015.

[3] Veronika Gáliková, Samuel Kováčik, and Peter Prešnajder, "Laplace-Runge-Lenz vector in quantum mechanics in noncommutative space", J. Math. Phys. 54, 122106 (2013)

[4] Castro, P.G., Kullock, R. "Physics of the hydrogen atom". Theor. Math. Phys. 185, 1678–1684 (2015).

[5] L. R. U. Manssur, R. Portugal, and B. F. Svaiter, "Group-Theoretic Approach for Symbolic Tensor Manipulation," International Journal of Modern Physics C, Vol. 13, No. 07, pp. 859-879 (2002).

Download Hidden_SO4_symmetry_of_the_hydrogen_atom.mw

Download Hidden_SO4_symmetry_submitted_to_CPC.pdf (all sections open)

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

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