Produce annihilation and creation operators for a (discrete) basis of state vectors labeled phi regarding the 1st and 2nd quantum numbers of the Ket state vectors of this basis, respectively.
In the above, and are local variables. These operators are applied to the Bras and Kets of this space of states by using the Physics[.] scalar product operator.
You can also use the inert form of Commutator, and evaluate the operation by using the value command when desired.
Construct one more pair of annihilation and creation operators for a different basis, labeled psi. In order to distinguish this new pair from the one acting on the phi basis, use the optional argument notation = explicit.
Note that the indices of and explicitly show the basis of states and position of quantum numbers on which they act. and are are also exports of Physics, not local variables.
The first and second pair of operators constructed only act on state vectors of the basis indicated when the operators were constructed
In order to distinguish between annihilation operators acting on different basis or different quantum numbers, instead of using the option notation = explicit, you can also use an alias.
When the label representing the basis of states is anticommutative, the annihilation and creation operators anticommute. To illustrate the phase convention for fermionic states, consider a case where and act on the third quantum number.
The following is the inert form of , and needs to be evaluated by using the value command.
In the next example, there is one particle associated with the first quantum number, zero associated with the second quantum number, and acts on the third quantum number; so the factor entering the result of the first example is .
A typical situation where the label = ... option is useful is with working with several Hilbert spaces at the same time, so the operators of one space commutes with those of the others, and the same happens with their corresponding annihilation / creation operators. For example, set three quantum operators, , and and tell the system the act on different (disjointed) Hilbert spaces
Set annihilation and creation operators for each of these operators , and . For , we can but we don't need to specify a label, keep the default label
For and we use the label option to indicate the letter to use as root for the display
At this point, you can refer to these operators using and assigned above, and also the default values and
Note you don't need to specify any algebra rule between these annihilation / creation operators: they are all implicit in the fact that they annihilate and create states with quanta that are eigenstates of , and C, and so the operators of any two of them automatically commute
As an applied example, consider in quantum mechanics the Schrieffer-Wolff transformation used to determine an effective (often low-energy) Hamiltonian by decoupling weakly interacting subspaces. The Hamiltonian can be expressed in terms of the annihilation and creation operators set above using Annihilation and Creation:
where and are the coupling constants between modes and , between and and between and . Assuming for simplicity of illustration that the mode is completely decoupled from the modes and , so that
the transformation such that is given by
where
This transformation, assumed to be small, can be expanded using Baker-Cambpbell-Haussdorf formula, which, up to second order is given by (use inert commutators and versions of and to see what is being computed)
Releasing the computation of the expression above using value; we get
Expanding in series and keeping terms up to order 2 in the (small) coupling ,
Rewriting for clarity,
Finally, dropping highly oscillating terms involving tensor products of modes and - they will only weakly contribute to the dynamics - we get
Going to higher orders or keeping track of more modes/systems is tedious with paper and pencil, but straightforward on a worksheet following the approach above.
The display of tensor products in all the lines above includes the tensor product operator . In computations with paper and pencil, however, the display of that operator is frequently omitted. For that purpose, use
The display of is now