Quantum Commutation Rules Basics
Pascal Szriftgiser^{1} and Edgardo S. ChebTerrab^{2}
(1) Laboratoire PhLAM, UMR CNRS 8523, Université Lille 1, F59655, France
(2) Maplesoft
In Quantum Mechanics, in the coordinates representation, the component of the momentum operator along the x axis is given by the differential operator
The purpose of the exercises below is thus to derive the commutation rules, in the coordinates representation, between an arbitrary function of the coordinates and the related momentum, departing from the differential representation
These two exercises illustrate how to have full control of the computational process by using different elements of the Maple language, including inert representations of abstract vectorial differential operators, Hermitian operators, algebra rules, etc.
These exercises also illustrate a new feature of the Physics package, introduced in Maple 2017, that is getting refined (the computation below requires the Maplesoft updates of the Physics package) which is the ability to perform computations algebraically, using the product operator, but with differential operators, and transform the products into the application of the operators only when we want that, as we do with paper and pencil.



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Start setting the problem:
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all of are Hermitian operators

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all of commute between each other

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tell the system only that the operators are the differentiation variables of the corresponding (differential) operators but do not tell what is the form of the operators

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(1.1) 
Assuming is a smooth function, the idea is to apply the commutator to an arbitrary ket of the Hilbert space , perform the operation explicitly after setting a differential operator representation for , and from there get the commutation rule between and .
Start introducing the commutator, to proceed with full control of the operations we use the inert form %Commutator
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(1.2) 
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(1.3) 
For illustration purposes only (not necessary), expand this commutator
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(1.4) 
Note that , and the ket are operands in the products above and that they do not commute: we indicated that the coordinates are the differentiation variables of . This emulates what we do when computing with these operators with paper and pencil, where we represent the application of a differential operator as a product operation.
This representation can be transformed into the (traditional in computer algebra) application of the differential operator when desired, as follows:
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(1.5) 
Note that, in , the application of is not expanded: at this point nothing is known about , it is not necessarily a linear operator. In the Quantum Mechanics problem at hands, however, it is. So give now the operator an explicit representation as a linear vectorial differential operator (we use the inert form %Nabla, , to be able to proceed with full control one step at a time)
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(1.6) 
The expression (1.5) becomes
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(1.7) 
Activate now the inert operator and simplify taking into account the algebra rules for the coordinate operators
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(1.8) 
To make explicit the gradient in disguise on the righthand side, factor out the arbitrary ket
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(1.9) 
Combine now the expanded gradient into its inert (notexpanded) form
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(1.10) 
Since (1.10) is true for all, this ket can be removed from both sides of the equation. One can do that either taking coefficients (see Coefficients ) or multiplying by the "formal inverse" of this ket, arriving at the (expected) form of the commutation rule between and
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(1.11) 


Tensor notation,


The computation rule for position and momentum, this time in tensor notation, is performed in the same way, just that, additionally, specify that the space indices to be used are lowercase latin letters, and set the relationship between the differential operators and the coordinates directly using tensor notation.
You can also specify that the metric is Euclidean, but that is not necessary: the default metric of the Physics package, a Minkowski spacetime, includes a 3D subspace that is Euclidean, and the default signature, (   +), is not a problem regarding this computation.
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(2.1) 
Define now the tensor
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(2.2) 
Introduce now the Commutator, this time in active form, to show how to reobtain the nonexpanded form at the end by resorting the operands in products
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(2.3) 
Expand first (not necessary) to see how the operator is going to be applied
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(2.4) 
Now expand and directly apply in one ago the differential operator
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(2.5) 
Introducing the explicit differential operator representation for , here again using the inert to keep control of the computations step by step
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(2.6) 
The expanded and applied commutator (2.5) becomes
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(2.7) 
Activate now the inert operators and simplify taking into account Einstein's rule for repeated indices
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(2.8) 
Since the ket is arbitrary, we can take coefficients (or multiply by the formal Inverse of this ket as done in the previous section). For illustration purposes, we use Coefficients and note hwo it automatically expands the commutator
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(2.9) 
One can undo this (frequently undesired) expansion of the commutator by sorting the products on the lefthand side using the commutator between and
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(2.10) 
And that is the result we wanted to compute.
Additionally, to see this rule in matrix form,
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(2.11) 
In the above, we use equation (2.10) multiplied by 1 to avoid a minus sign in all the elements of (2.11), due to having worked with the default signature (   +); this minus sign is not necessary if in the Setup at the beginning one also sets
For display purposes, to see this matrix expressed in terms of the geometrical components of the momentum , redefine the tensor explicitly indicating its Cartesian components
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(2.12) 
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(2.13) 
Finally, in a typical situation, these commutation rules are to be taken into account in further computations, and for that purpose they can be added to the setup via
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(2.14) 
For example, from herein computations are performed taking into account that
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(2.15) 
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