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$$\label{eq:TueSep4123411CEST2012}[a,\ud{a}]=1$$ | $$\label{eq:TueSep4123411CEST2012}[a,\ud{a}]=1$$ | ||
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| + | (cf. [[Fermi algebra]]) | ||
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| + | {{I}} have written a [[Mathematica]] piece of code to compute such correlators automatically. I found out in this way formulas already known in the literature, in particular from the work of Blasiak{{cite|blasiak03a}}{{cite|blasiak07a}}. You can '''[[:File:boson-normal-ordering.nb|download it]]''' to play with it (check or compute results useful in your daily quantum algebra). This is detailed in [[Blog:Science/Checking_your_bosonic_quantum_algebra_with_Mathematica|this blog post]]. | ||
== Creation & annihilation == | == Creation & annihilation == | ||
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$$a^n\ud{a}^m=\sum_{k=0}^{\min(n,m)}k!\binom{m}{k}\binom{n}{k}\ud{a}^{m-k}a^{n-k}$$ | $$a^n\ud{a}^m=\sum_{k=0}^{\min(n,m)}k!\binom{m}{k}\binom{n}{k}\ud{a}^{m-k}a^{n-k}$$ | ||
| − | Some useful particular cases | + | Another one, inferred from my Mathematica notebook above, that could certainly be proved by recurrence or directly from combinatorics: |
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| + | $$(a\ud{a})^n=\sum_{k=0}^n | ||
| + | \left\{\begin{matrix} n\\k \end{matrix} \right\} | ||
| + | \ud{a}^ka^k$$ | ||
| + | |||
| + | where $\left\{\begin{matrix} n\\k \end{matrix} \right\}$ are the | ||
| + | Stirling partition numbers (the number of ways to partition a set of | ||
| + | $n$ objects into $k$ non-empty subsets). It is easy to obtain similar results in terms of generalized Stirling and Bell numbers for $(a^k\ud{l})^n$, which are results already provided by Blasiak, e.g.: | ||
| + | |||
| + | $$(\ud{a}a)^n=\sum_{k=1}^nS(n,k)(\ud{a})^ka^k$$ | ||
| + | where $S(n,k)$ are the Stirling numbers of the second kind ([[Mathematica]] <tt>StirlingS2[n,k]</tt>). For instance: | ||
| + | $$(\ud{a}a)^3=\ud{a}^3a^3+3\ud{a}^2a^2+\ud{a}a\,.$$ | ||
| + | |||
| + | === Some useful particular cases === | ||
$$a\ud{a}^n=\ud{a}^na+n\ud{a}^{n-1}$$ | $$a\ud{a}^n=\ud{a}^na+n\ud{a}^{n-1}$$ | ||
$$a^n\ud{a}=\ud{a}a^n+na^{n-1}$$ | $$a^n\ud{a}=\ud{a}a^n+na^{n-1}$$ | ||
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| + | === Some recurrent expressions === | ||
| + | |||
| + | A fairly general result (forall $k, l\in\mathbb{N}$, including $0$): | ||
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| + | $$aa^{\dagger k}a^la^\dagger=a^{\dagger(k+1)}a^{l+1}+(k+l+1)a^{\dagger k}a^l+kla^{\dagger(k-1)}a^{l-1}$$ | ||
| + | |||
| + | I maintain a [[long list of Bose algebra expressions in canonic notations]] to assist me in my computations. You can glance at it to find particular and redundant cases of the above. | ||
== Commutation == | == Commutation == | ||
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*$[\hat n,a^m]=-ma^m$ | *$[\hat n,a^m]=-ma^m$ | ||
*$[a,(\hat n)^k]=\left(\sum_{i=0}^{k-1}\binom{k}{i}\hat n^i\right)a$ | *$[a,(\hat n)^k]=\left(\sum_{i=0}^{k-1}\binom{k}{i}\hat n^i\right)a$ | ||
| + | *$[\ud{a}^ma^n,\hat n]=(n-m)\ud{a}^ma^n$ | ||
Special cases below | Special cases below | ||
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== Particle numbers == | == Particle numbers == | ||
| − | *$a\ud{a}= | + | *$a\ud{a}=\hat n+1$. |
*$\ud{a}^2a^2=\hat n^2-\hat n$. | *$\ud{a}^2a^2=\hat n^2-\hat n$. | ||
| + | |||
| + | == References == | ||
| + | |||
| + | <references/> | ||
Contents |
Here are collected some results related with the operator $a$ which commutation with its adjoint $\ud{a}$ reads:
$$\tag{1}[a,\ud{a}]=1$$
(cf. Fermi algebra)
I have written a Mathematica piece of code to compute such correlators automatically. I found out in this way formulas already known in the literature, in particular from the work of Blasiak[1][2]. You can download it to play with it (check or compute results useful in your daily quantum algebra). This is detailed in this blog post.
The basic rules are:
$$ \begin{align*} a\ket{n}&=\sqrt{n}\ket{n-1}\,,&\bra{n}\,&a=\bra{n+1}\sqrt{n+1}\,,\\ \ud{a}\ket{n}&=\sqrt{n+1}\ket{n+1}\,,&\bra{n}\,&\ud{a}=\bra{n-1}\sqrt{n}\,, \end{align*} $$
of which a general expression can be drawn:
$$a^{\dagger i}a^j a^{\dagger k}\ket{n}={(n+k)!\over(n+k-j)!}\sqrt{(n+i+k-j)!\over n!}\ket{n+i+k-j}\,.$$
Some particular cases:
$$ \begin{align*} \kern-1cm{(\mathrm{for}~i\le n+j)}\kern1cm a^i{\ud{a}}^j\ket{n}&={(n+j)!\over\sqrt{n!}\sqrt{(n+j-i)!}}\ket{n+j-i}\,,\\ \kern-1cm{(\mathrm{for}~i\le n)}\kern1cm a^{\dagger j}a^i\ket{n}&={\sqrt{n!}\sqrt{(n+j-i)!}\over(n-i)!}\ket{n+j-i}\,. \end{align*} $$
A general result, based on Wick theorem:
$$a^n\ud{a}^m=\sum_{k=0}^{\min(n,m)}k!\binom{m}{k}\binom{n}{k}\ud{a}^{m-k}a^{n-k}$$
Another one, inferred from my Mathematica notebook above, that could certainly be proved by recurrence or directly from combinatorics:
$$(a\ud{a})^n=\sum_{k=0}^n \left\{\begin{matrix} n\\k \end{matrix} \right\} \ud{a}^ka^k$$
where $\left\{\begin{matrix} n\\k \end{matrix} \right\}$ are the Stirling partition numbers (the number of ways to partition a set of $n$ objects into $k$ non-empty subsets). It is easy to obtain similar results in terms of generalized Stirling and Bell numbers for $(a^k\ud{l})^n$, which are results already provided by Blasiak, e.g.:
$$(\ud{a}a)^n=\sum_{k=1}^nS(n,k)(\ud{a})^ka^k$$ where $S(n,k)$ are the Stirling numbers of the second kind (Mathematica StirlingS2[n,k]). For instance: $$(\ud{a}a)^3=\ud{a}^3a^3+3\ud{a}^2a^2+\ud{a}a\,.$$
$$a\ud{a}^n=\ud{a}^na+n\ud{a}^{n-1}$$ $$a^n\ud{a}=\ud{a}a^n+na^{n-1}$$
A fairly general result (forall $k, l\in\mathbb{N}$, including $0$):
$$aa^{\dagger k}a^la^\dagger=a^{\dagger(k+1)}a^{l+1}+(k+l+1)a^{\dagger k}a^l+kla^{\dagger(k-1)}a^{l-1}$$
I maintain a long list of Bose algebra expressions in canonic notations to assist me in my computations. You can glance at it to find particular and redundant cases of the above.
A compilation of useful results derived from Eq.~(1):
We define $\hat n\equiv\ud{a}a$.
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Special cases below
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