Pascal4QM

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Defining a quantum operator as a tensor ...

Posted: Pascal4QM 478 Product: Maple 2016

August 24 2016

2 2

Hi,

In the following example I introduce some commutation rules that are standard in Quantum Mechanics. A major feature of the Maple Physics Package, is that it is possible to define tensors as Quantum Operators. This is of great interest because powerful tensor simplification rules can then be used in Quantum Mechanics. For an example, see the commutation rules of the components of the angular momentum operator in ?Physics,Examples. Here, I focus on a possible issue: when destroying all quantum operators, the pre-defined commutation rules still apply, which should not be the case. As shown in the post, this is link to the fact that these operators are also tensors.

(1)

First, set a 3D Euclidian space

(2)

Define two rank 1 tensors

${Physics:-Dgamma[a], Physics:-Psigma[a], Physics:-d_[a], Physics:-g_[a, b], p[k], x[k], Physics:-KroneckerDelta[a, b], Physics:-LeviCivita[a, b, c]}$

(3)

Now, further define these tensors as quantum operators and gives the usual commutation rule between position and momentum operators (Quantum Mechanics).

(4)

As expected:

(5)

Now, reset all the Hermitian operators, so that all quantum operators are destroyed. This is useful if, for instance, one needs to compare some the result with the commutative case.

(6)

As expected, there are no quantum operators anymore...

(7)

...so that the following expressions should commute (result should be true)

(8)

Result should be 0

(9)

(10)

Below is just a copy & paste of the above section. The only difference, is that "" has been commented, so that and are not a tensor. In that case, everything behaves as expected (but of course, the interesting feature of tensors is not available).

First, set a 3D Euclidian space

(11)

#Define two rank 1 tensors

Now, further define these tensors as quantum operators and gives the usual commutation rule between position and momentum operators (Quantum Mechanics)