Talk:PlanetPhysics/Commutator Algebra

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As long as one deals only with commuting \htmladdnormallink{observables}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} the rules of ordinary algebra may be used without restrition. However, the observables of a given quantum \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/GenericityInOpenSystems.html} do not all \htmladdnormallink{commute}{http://planetphysics.us/encyclopedia/Commutator.html}. More precisely, the observables of a quantum system in $R$ dimensions are \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html} of the \htmladdnormallink{position}{http://planetphysics.us/encyclopedia/Position.html} observables $q_i (i=1,2,\dots,R)$ and the \htmladdnormallink{momentum}{http://planetphysics.us/encyclopedia/Momentum.html} observables $p_i (i=1,2,\dots,R)$, all pairs of which do not commute. The \htmladdnormallink{commutators}{http://planetphysics.us/encyclopedia/Commutator.html} of the $q$'s and the $p$'s play a fundamental role in the theory. One has:

\begin{equation} [q_i,q_j] = 0, \,\,\,\,\,\,\,\, [p_i,p_j]=0 \end{equation}

\begin{equation} [q_i,p_j] = i \hbar \delta_{ij} \end{equation}

\htmladdnormallink{Relations}{http://planetphysics.us/encyclopedia/Bijective.html} (1) are obvious; in particular the second merely states that \htmladdnormallink{operations}{http://planetphysics.us/encyclopedia/Cod.html} of differentiation commute with each other. Relation (2) is a generalization of

$$[x,p_x] = \frac{\hbar}{i}\left[x, \frac{\partial}{\partial x}\right] = i \hbar \ne 0$$

it is readily obtained by using the explicit form of the \htmladdnormallink{operators}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html} $p$:

$$p_i = \frac{\hbar}{i} \frac{\partial}{\partial q_i}$$

From the fact that the $q$'s and the $p$'s do not commute in pairs, the precise definition of a dynamical variable $\mathcal{A} \equiv A(q_1,\dots,q_R;p_1,\dots,p_R)$ requires that one properly specifies the order of the $q's$ and the $p's$ in the explicit expression of the function $A(q_1,\dots,q_R;p_1,\dots,p_R)$. In practice, $A$ is put in the form of a polynomial in $p$ - or possibly in the form of a \htmladdnormallink{power}{http://planetphysics.us/encyclopedia/Power.html} series in $p$ - whose coefficients are functions of $q$. Each term is a product of components $p_i$ and functions of the $q$ arranged in a certain order. The function $A$, considered as an operator, is well defined only when the order in each of its terms is specified.

It is interesting to know he commutators of the $q$'s alone, or of the $p$'s alone, one obtains the relations

\begin{equation} [q_i,F(q_1,\dots,q_R)] = 0 \end{equation}

\begin{equation} [p_i,G(p_1,\dots,p_R)] = 0 \end{equation}

\begin{equation} [p_i,F(q_1,\dots,q_R)] = \frac{\hbar}{i} \frac{\partial F}{\partial q_i} \end{equation}

\begin{equation} [q_i,G(p_1,\dots,p_R)] = i\hbar \frac{\partial G}{\partial p_i} \end{equation}

The relations (3) and (4) are particular cases of the \htmladdnormallink{theorem}{http://planetphysics.us/encyclopedia/Formula.html}:

\emph{If two observables commute, they possess a complete orthonormal set of common eigenfunctions, and conversely.}

To prove equation (5), t suffices to write down the operator $p_i$ explicitly and to verify that the action of each side of the equation on an arbitrary \htmladdnormallink{wave}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html} function gives the same result (see \htmladdnormallink{quantum operator concept}{http://planetphysics.us/encyclopedia/Commutator.html}). Equation (6) is proved by making an analogous verification in momentum space; let us recall that if $\Phi(p_1,\dots,p_R)$ is the wave function of momentum space corresponding to $\Psi(q_1,\dots,q_R)$, the function of momentum space coresponding to $q_i \Psi(q_1,\dots,q_R)$ is

$$ i \hbar \frac{\partial}{\partial p_i} \Phi(p_1,\dots,p_R)$$

One arrives at the same result using the rules of \emph{commutator algebra}. Let us give here the four principal rules. Thse rules are direct consequences of the definition of commutators. If $A$, $B$, and $C$ denote three arbitrary \htmladdnormallink{linear operators}{http://planetphysics.us/encyclopedia/Commutator.html}, one has

\begin{equation} [A,B] = -[B,A] \end{equation}

\begin{equation} [A,B+C] = [A,B] + [A,C] \end{equation}

\begin{equation} [A,BC] = [A,B]C +B[A,C] \end{equation}

\begin{equation} [A,[B,C]] + [B,[C,A]] + [C,[A,B]] = 0 \end{equation}

By repeated application of rule (9), one hs

$$ [A,B^n] = \sum_{s=0}^{n-1} B^s [A,B] B^{n-s-1} $$

In particular, for a one-dimensional system one has

$$ [q,p^n] = n i \hbar p^{n-1} $$

Equation 6 is thus verified when $F$ is an arbitrary power of the $p$; it is thus also verified (rule 8) when $F$ is a polynomial, or else a convergent power series in $p$.

For general functions of the $q$'s and $p$'s, one can also write

\begin{equation} [p_i,A] = \frac{hbar}{i} \frac{\partial A}{\partial q_i} \end{equation}

\begin{equation} [q_i,A] = i \hbar \frac{\partial A}{\partial p_i} \end{equation}

$\partial A/\partial q_i$, $\partial A / \partial p_i$ being defined by partial differentiation of $A$, it being understood that the order of the $p$'s and $q$'s in their explicit expression has been suitably chosen.

\subsection{References}

[1] Messiah, Albert. "\htmladdnormallink{Quantum mechanics}{http://planetphysics.us/encyclopedia/QuantumParadox.html}: \htmladdnormallink{volume}{http://planetphysics.us/encyclopedia/Volume.html} I." Amsterdam, North-Holland Pub. Co.; New York, Interscience Publishers, 1961-62.

This entry is a derivative of the Public \htmladdnormallink{domain}{http://planetphysics.us/encyclopedia/Bijective.html} \htmladdnormallink{work}{http://planetphysics.us/encyclopedia/Work.html} [1].

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