 generate "annihilation" operators acting on a given space of quantum states - Maple Programming Help

Physics[Annihilation] - generate "annihilation" operators acting on a given space of quantum states

Physics[Creation] - generate "creation" operators acting on a given space of quantum states

 Calling Sequence Annihilation(A, n1, n2, ... notation = ..., phaseconvention = ...) Creation(A, n1, n2, ... notation = ..., phaseconvention = ...)

Parameters

 A - a name, as a label; identifies the space of quantum states on which the operator returned acts n1, n2, ... - positive integers; indicate the position of the quantum numbers on which the operator returned acts notation = ... - (optional) equation of the form notation = explicit or notation = implicit (default); if explicit, the operator returned explicitly shows the space and quantum numbers onto which it acts phaseconvention = ... - (optional) equation of the form phaseconvention = procedure where procedure is a procedure of one argument, telling how to construct the phase factor from the quantum number onto which the operator acts (annihilation default: $\mathrm{expr}$; creation default: $\mathrm{expr}$)

Description

 • The Annihilation and Creation commands return annihilation and creation operators, respectively, which act on the quantum numbers of state vectors, Kets or Bras, of a space of quantum states. The first argument, A, is the label of the basis of quantum states on which the operator returned acts. The second and subsequent arguments are expected to be positive integers indicating the position of the quantum numbers on which the operator returned acts; if not given, it is assumed that there is only one quantum number.
 • The %Annihilation and %Creation commands are the inert forms of Annihilation and Creation. That is, they represent the same mathematical operations while returning the operators unevaluated. To evaluate the inert operations, use the value command.
 • By default, the annihilation and creation operators returned are displayed as ${a}^{-}$ and ${a}^{+}$, respectively (in the underlying computational implementation, ${a}^{-}$ and ${a}^{+}$ are local variables that you can invoke in an input line using the global variables a- and a+). Thus, the default returned operator tells neither the basis of states nor the position of the quantum numbers onto which it acts. You can override this default by using the optional argument notation = explicit. See the Options section for details.
 • The annihilation and creation operators satisfy the commutation relation, ${\left[{a}^{-},{a}^{+}\right]}_{-}=1$, where ${\left[{a}^{-},{a}^{+}\right]}_{-}$ is the Commutator, unless the label of the basis of states on which the operators act is itself an anticommutative variable, in which case the operators satisfy the anticommutation relation, ${\left[{a}^{-},{a}^{+}\right]}_{+}=1$, where ${\left[{a}^{-},{a}^{+}\right]}_{+}$ is the AntiCommutator. When these operators are constructed by calls to Annihilation and Creation, these (anti)commutation relations are automatically set, and so are taken into account by the Commutator and AntiCommutator commands.
 Note: Annihilation and creation operators can be created to operate only on Bras and Kets belonging to discrete spaces of states.

Options

 notation
 • The default value is notation = implicit. You can override this value by using the optional argument notation = explicit. This option is useful, for example, to distinguish among various annihilation (or creation) operators within a single worksheet, and to make ${a}^{-}$ and ${a}^{+}$ (entering the returned indexed annihilation and creation operators) be exports of Physics, not local variables. Alternatively, for the same purpose, you can use the alias command. See the Examples for a demonstration.
 phaseconvention
 • Given a basis of states with label A, and a single quantum number, n, the annihilation operator satisfies the equation ${a}^{-}\left|\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{A}_{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\right⟩=0$ . So the state vector $\left|\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{A}_{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\right⟩$ is defined up to a constant factor. Assuming $\left|\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{A}_{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\right⟩$ is normalized to 1, this factor is of the form ${ⅇ}^{i\mathrm{\theta }}$ with $\mathrm{\theta }$ real. By requiring, additionally, that $\left|\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{A}_{1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\right⟩={c}_{1}{a}^{+}\left|\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{A}_{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\right⟩$, with ${c}_{1}$ real and positive, and taking into account the (anti)commutation relation satisfied by ${a}^{-}$ and ${a}^{+}$, the factor ${c}_{1}$ is equal to 1.
 • In the bosonic case, considering now that $\left|\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{A}_{n}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\right⟩={c}_{n}{a}^{+}\left|\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{A}_{n-1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\right⟩$, by using the same assumptions, the following phase convention arises:

 ${a}^{-}\left|\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{A}_{n}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\right⟩$   = $\sqrt{n}\left|\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{A}_{n-1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\right⟩$ ${a}^{+}\left|\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{A}_{n}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\right⟩$   = $\sqrt{n+1}\left|\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{A}_{n+1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\right⟩$

 • You can override this phase convention with a procedure pc having one argument, by using the optional argument phaseconvention = pc.
 • In the fermionic case, recalling Pauli's exclusion principle, each quantum number can only assume the values 0 or 1, so:

 ${a}^{-}\left|\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{A}_{1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\right⟩$   = $\left|\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{A}_{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\right⟩$ ${a}^{+}\left|\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{A}_{1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\right⟩$   = $0$

 • In the case of many quantum numbers, for example with ${a}^{-}$ and ${a}^{+}$ acting on the ${n}^{\mathrm{th}}$ quantum number ${q}_{n}$, let $M={\sum }_{m=1}^{n-1}{q}_{m}$ represent the total number of particles occupying states represented by the previous ${q}_{m}$ quantum numbers. Then the phaseconvention is

 ${a}^{-}\left|\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{A}_{{q}_{1},\mathrm{...},1,\mathrm{...}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\right⟩$   = ${\left(-1\right)}^{M}\left|\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{A}_{{q}_{1},\mathrm{...},0,\mathrm{...}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\right⟩$ ${a}^{+}\left|\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{A}_{{q}_{1},\mathrm{...},0,\mathrm{...}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\right⟩$   = ${\left(-1\right)}^{M}\left|\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{A}_{{q}_{1},\mathrm{...},1,\mathrm{...}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\right⟩$

 • The default value for this option in the Annihilation command is phaseconvention = sqrt; the default in the Creation command is phaseconvention = n -> sqrt(n + 1).

Examples

 > $\mathrm{with}\left(\mathrm{Physics}\right):$
 > $\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true}\right)$
 $\left[{\mathrm{mathematicalnotation}}{=}{\mathrm{true}}\right]$ (1)

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.

 > $\mathrm{am}≔\mathrm{Annihilation}\left(\mathrm{\phi },1\right)$
 ${\mathrm{am}}{≔}{{a}}^{{-}}$ (2)
 > $\mathrm{ap}≔\mathrm{Creation}\left(\mathrm{\phi },1\right)$
 ${\mathrm{ap}}{≔}{{a}}^{{+}}$ (3)

In the above, $\mathrm{a-}$ and $\mathrm{a+}$ are local variables. These operators are applied to the Bras and Kets of this space of states by using the Physics[.] scalar product operator.

 > $\mathrm{am}·\mathrm{Ket}\left(\mathrm{\phi },n\right)$
 $\sqrt{{n}}{}{\mathrm{Ket}}{}\left({\mathrm{φ}}{,}{n}{-}{1}\right)$ (4)
 > $\mathrm{Bra}\left(\mathrm{\phi },n\right)·\mathrm{am}$
 $\sqrt{{n}{+}{1}}{}{\mathrm{Bra}}{}\left({\mathrm{φ}}{,}{n}{+}{1}\right)$ (5)
 > $\mathrm{ap}·\mathrm{Ket}\left(\mathrm{\phi },n\right)$
 $\sqrt{{n}{+}{1}}{}{\mathrm{Ket}}{}\left({\mathrm{φ}}{,}{n}{+}{1}\right)$ (6)
 > $\mathrm{Bra}\left(\mathrm{\phi },n\right)·\mathrm{ap}$
 $\sqrt{{n}}{}{\mathrm{Bra}}{}\left({\mathrm{φ}}{,}{n}{-}{1}\right)$ (7)
 > $\mathrm{Commutator}\left(\mathrm{am},\mathrm{ap}\right)$
 ${1}$ (8)

You can also use the inert form of Commutator, and evaluate the operation by using the value command when desired.

 > $\mathrm{c_am_ap}≔\mathrm{%Commutator}\left(\mathrm{am},\mathrm{ap}\right)$
 ${\mathrm{%Commutator}}{}\left({{a}}^{{-}}{,}{{a}}^{{+}}\right)$ (9)
 > $\mathrm{value}\left(\mathrm{c_am_ap}\right)$
 ${1}$ (10)

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.

 > $\mathrm{am2}≔\mathrm{Annihilation}\left(\mathrm{\psi },1,\mathrm{notation}=\mathrm{explicit}\right)$
 ${\mathrm{am2}}{≔}{{{a}}^{{-}}}_{{{\mathrm{\psi }}}_{{1}}}$ (11)
 > $\mathrm{ap2}≔\mathrm{Creation}\left(\mathrm{\psi },2,\mathrm{notation}=\mathrm{explicit}\right)$
 ${\mathrm{ap2}}{≔}{{{a}}^{{+}}}_{{{\mathrm{\psi }}}_{{2}}}$ (12)

Note that the indices of $a-$ and $a+$ explicitly show the basis of states and position of quantum numbers on which they act. $\mathrm{a-}$ and $\mathrm{a+}$ 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

 > $\mathrm{am2}·\mathrm{Ket}\left(\mathrm{\psi },n,m\right)$
 $\sqrt{{n}}{}{\mathrm{Ket}}{}\left({\mathrm{ψ}}{,}{n}{-}{1}{,}{m}\right)$ (13)
 > $\mathrm{ap2}·\mathrm{Ket}\left(\mathrm{\psi },n,m\right)$
 $\sqrt{{m}{+}{1}}{}{\mathrm{Ket}}{}\left({\mathrm{ψ}}{,}{n}{,}{m}{+}{1}\right)$ (14)

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.

 > $\mathrm{alias}\left(\mathrm{Am1}=\mathrm{Annihilation}\left(\mathrm{\psi },1\right)\right):$
 > $\mathrm{alias}\left(\mathrm{Am2}=\mathrm{Annihilation}\left(\mathrm{\psi },2\right)\right):$
 > $\mathrm{Am1}$
 ${\mathrm{Am1}}$ (15)
 > $\mathrm{Am1}·\mathrm{Ket}\left(\mathrm{\psi },n,m\right)$
 $\sqrt{{n}}{}{\mathrm{Ket}}{}\left({\mathrm{ψ}}{,}{n}{-}{1}{,}{m}\right)$ (16)
 > $\mathrm{Am2}$
 ${\mathrm{Am2}}$ (17)
 > $\mathrm{Am2}·\mathrm{Ket}\left(\mathrm{\psi },n,m\right)$
 $\sqrt{{m}}{}{\mathrm{Ket}}{}\left({\mathrm{ψ}}{,}{n}{,}{m}{-}{1}\right)$ (18)

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 $a-$ and $a+$ act on the third quantum number.

 > $\mathrm{Physics}:-\mathrm{Setup}\left(\mathrm{anticommutativeprefix}=\mathrm{\theta }\right)$
 $\left[{\mathrm{anticommutativeprefix}}{=}\left\{{\mathrm{\theta }}\right\}\right]$ (19)
 > $\mathrm{am}≔\mathrm{Annihilation}\left(\mathrm{\theta },3,\mathrm{notation}=\mathrm{explicit}\right)$
 ${\mathrm{am}}{≔}{{{a}}^{{-}}}_{{{\mathrm{\theta }}}_{{3}}}$ (20)
 > $\mathrm{ap}≔\mathrm{Creation}\left(\mathrm{\theta },3,\mathrm{notation}=\mathrm{explicit}\right)$
 ${\mathrm{ap}}{≔}{{{a}}^{{+}}}_{{{\mathrm{\theta }}}_{{3}}}$ (21)

The following is the inert form of $\mathrm{AntiCommutator}\left(\mathrm{am},\mathrm{ap}\right)$, and needs to be evaluated by using the value command.

 > $\mathrm{a_am_ap}≔\mathrm{%AntiCommutator}\left(\mathrm{am},\mathrm{ap}\right)$
 ${\mathrm{%AntiCommutator}}{}\left({{{a}}^{{-}}}_{{{\mathrm{θ}}}_{{3}}}{,}{{{a}}^{{+}}}_{{{\mathrm{θ}}}_{{3}}}\right)$ (22)
 > $\mathrm{value}\left(\mathrm{a_am_ap}\right)$
 ${1}$ (23)

In the next example, there is one particle associated with the first quantum number, zero associated with the second quantum number, and $a-$ acts on the third quantum number; so the factor entering the result of the first example is ${\left(-1\right)}^{1}=-1$.

 > $\mathrm{am_K}≔\mathrm{%.}\left(\mathrm{am},\mathrm{Ket}\left(\mathrm{θ},1,0,1\right)\right)$
 ${\mathrm{am_K}}{≔}{\mathrm{%.}}{}\left({{{a}}^{{-}}}_{{{\mathrm{θ}}}_{{3}}}{,}{\mathrm{Ket}}{}\left({\mathrm{θ}}{,}{1}{,}{0}{,}{1}\right)\right)$ (24)
 > $\mathrm{value}\left(\mathrm{am_K}\right)$
 ${-}{\mathrm{Ket}}{}\left({\mathrm{θ}}{,}{1}{,}{0}{,}{0}\right)$ (25)
 > $\mathrm{am_K}≔\mathrm{%.}\left(\mathrm{am},\mathrm{Ket}\left(\mathrm{θ},1,1,1\right)\right)$
 ${\mathrm{am_K}}{≔}{\mathrm{%.}}{}\left({{{a}}^{{-}}}_{{{\mathrm{θ}}}_{{3}}}{,}{\mathrm{Ket}}{}\left({\mathrm{θ}}{,}{1}{,}{1}{,}{1}\right)\right)$ (26)
 > $\mathrm{value}\left(\mathrm{am_K}\right)$
 ${\mathrm{Ket}}{}\left({\mathrm{θ}}{,}{1}{,}{1}{,}{0}\right)$ (27)
 > $\mathrm{ap}·\mathrm{Ket}\left(\mathrm{\theta },1,0,1\right)$
 ${0}$ (28)
 > $\mathrm{ap}·\mathrm{Ket}\left(\mathrm{\theta },1,0,0\right)$
 ${-}{\mathrm{Ket}}{}\left({\mathrm{θ}}{,}{1}{,}{0}{,}{1}\right)$ (29)
 > 

References

 Cohen-Tannoudji, C.; Diu, B.; and Laloe, F. Quantum Mechanics. Chapter II. Paris, France: Hermann, 1977.