Start setting the problem:

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

For illustration purposes only (not necessary), expand this commutator

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:

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)

The expression (1.5) becomes

Activate now the inert operator  and simplify taking into account the algebra rules for the coordinate operators

To make explicit the gradient in disguise on the right-hand side, factor out the arbitrary ket

Combine now the expanded gradient into its inert (not-expanded) form

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

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.

Define now the tensor

Introduce now the Commutator, this time in active form, to show how to reobtain the non-expanded form at the end by resorting the operands in products

Expand first (not necessary) to see how the operator  is going to be applied

Now expand and directly apply in one ago the differential operator

Introducing the explicit differential operator representation for , here again using the inert  to keep control of the computations step by step

The expanded and applied commutator (2.5) becomes

Activate now the inert operators  and simplify taking into account Einstein's rule for repeated indices

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

One can undo this (frequently undesired) expansion of the commutator by sorting the products on the left-hand side using the commutator between  and

And that is the result we wanted to compute.

Additionally, to see this rule in matrix form,

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

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

For example, from herein computations are performed taking into account that