PlanetPhysics/Commutator Algebra

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

$[q_i,q_j]=0,[p_i,p_j]=0$

$[q_i,p_j]=i\hslash {\delta }_{ij}$

[[../Bijective/|Relations]] (1) are obvious; in particular the second merely states that [[../Cod/|operations]] of differentiation commute with each other. Relation (2) is a generalization of

$[x,p_x]=\frac{\hslash }{i}[x,\frac{\partial }{\partial x}]=i\hslash \ne 0$

it is readily obtained by using the explicit form of the [[../QuantumOperatorAlgebra4/|operators]] $p$:

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

From the fact that the $q$'s and the $p$'s do not commute in pairs, the precise definition of a dynamical variable $𝒜\equiv A(q_1,\dots ,q_R;p_1,\dots ,p_R)$ requires that one properly specifies the order of the $q^{\prime }s$ and the $p^{\prime }s$ 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 [[../Power/|power]] 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

$[q_i,F(q_1,\dots ,q_R)]=0$

$[p_i,G(p_1,\dots ,p_R)]=0$

$[p_i,F(q_1,\dots ,q_R)]=\frac{\hslash }{i}\frac{\partial F}{\partial q_i}$

$[q_i,G(p_1,\dots ,p_R)]=i\hslash \frac{\partial G}{\partial p_i}$

The relations (3) and (4) are particular cases of the [[../Formula/|theorem]]:

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 [[../CosmologicalConstant2/|wave]] function gives the same result (see [[../Commutator/|quantum operator concept]]). Equation (6) is proved by making an analogous verification in momentum space; let us recall that if $\mathrm{\Phi }(p_1,\dots ,p_R)$ is the wave function of momentum space corresponding to $\mathrm{\Psi }(q_1,\dots ,q_R)$, the function of momentum space coresponding to $q_i\mathrm{\Psi }(q_1,\dots ,q_R)$ is

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

One arrives at the same result using the rules of 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 [[../Commutator/|linear operators]], one has

$[A,B]=-[B,A]$

$[A,B+C]=[A,B]+[A,C]$

$[A,BC]=[A,B]C+B[A,C]$

$[A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0$

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

$[q,p^n]=ni\hslash p^{n-1}$

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

$[p_i,A]=\frac{hbar}{i}\frac{\partial A}{\partial q_i}$

$[q_i,A]=i\hslash \frac{\partial A}{\partial p_i}$

$\partial A/\partial q_i$, $\partial A/\partial p_i$ 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.

References

[1] Messiah, Albert. "[[../QuantumParadox/|Quantum mechanics]]: [[../Volume/|volume]] I." Amsterdam, North-Holland Pub. Co.; New York, Interscience Publishers, 1961-62.

This entry is a derivative of the Public [[../Bijective/|domain]] [[../Work/|work]] [1].