PlanetPhysics/Commutator Algebra

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

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

$$ [q_i,p_j] = i \hbar \delta_{ij} $$

Relations (1) are obvious; in particular the second merely states that operations 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 operators $$p$$:

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

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 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

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

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

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

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

The relations (3) and (4) are particular cases of the 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 wave function gives the same result (see quantum operator concept). 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 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 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

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

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

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

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

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