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

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

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:

is displayed as

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.

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).

This shows the automatic handling of collision of indices

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

Define  respectively representing angular and linear momentum

Introduce the tensorial expression for

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

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

The repeated indices on the right-hand side are now

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

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

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

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:

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

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

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:

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

We now have

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:

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

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

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