|
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  dimensions are functions of the position observables
 and the momentum observables
 , all pairs of which do not commute. The commutators of the  's and the  's play a fundamental role in the theory. One has:
![$\displaystyle [q_i,q_j] = 0, \,\,\,\,\,\,\,\, [p_i,p_j]=0$](http://images.physicslibrary.org/cache/objects/834/l2h/img6.png "$\displaystyle [q_i,q_j] = 0, \,\,\,\,\,\,\,\, [p_i,p_j]=0$") |
(1) |
![$\displaystyle [q_i,p_j] = i \hbar \delta_{ij}$](http://images.physicslibrary.org/cache/objects/834/l2h/img7.png "$\displaystyle [q_i,p_j] = i \hbar \delta_{ij}$") |
(2) |
Relations (1) are obvious; in particular the second merely states that operations of differentiation commute with each other. Relation (2) is a generalization of
it is readily obtained by using the explicit form of the operators  :
From the fact that the  's and the  's do not commute in pairs, the precise definition of a dynamical variable
 requires that one properly specifies the order of the  and the  in the explicit expression of the function
 . In practice,  is put in the form of a polynomial in  - or possibly in the form of a power series in  - whose coefficients are functions of  . Each term is a product of components  and functions of the  arranged in a certain order. The function  , 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  's alone, or of the  's alone, one obtains the relations
![$\displaystyle [q_i,F(q_1,\dots,q_R)] = 0$](http://images.physicslibrary.org/cache/objects/834/l2h/img26.png "$\displaystyle [q_i,F(q_1,\dots,q_R)] = 0$") |
(3) |
![$\displaystyle [p_i,G(p_1,\dots,p_R)] = 0$](http://images.physicslibrary.org/cache/objects/834/l2h/img27.png "$\displaystyle [p_i,G(p_1,\dots,p_R)] = 0$") |
(4) |
![$\displaystyle [p_i,F(q_1,\dots,q_R)] = \frac{\hbar}{i} \frac{\partial F}{\partial q_i}$](http://images.physicslibrary.org/cache/objects/834/l2h/img28.png "$\displaystyle [p_i,F(q_1,\dots,q_R)] = \frac{\hbar}{i} \frac{\partial F}{\partial q_i}$") |
(5) |
![$\displaystyle [q_i,G(p_1,\dots,p_R)] = i\hbar \frac{\partial G}{\partial p_i}$](http://images.physicslibrary.org/cache/objects/834/l2h/img29.png "$\displaystyle [q_i,G(p_1,\dots,p_R)] = i\hbar \frac{\partial G}{\partial p_i}$") |
(6) |
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  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
 is the wave function of momentum space corresponding to
 , the function of momentum space coresponding to
 is
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  ,  , and  denote three arbitrary linear operators, one has
![$\displaystyle [A,B] = -[B,A]$](http://images.physicslibrary.org/cache/objects/834/l2h/img38.png "$\displaystyle [A,B] = -[B,A]$") |
(7) |
![$\displaystyle [A,B+C] = [A,B] + [A,C]$](http://images.physicslibrary.org/cache/objects/834/l2h/img39.png "$\displaystyle [A,B+C] = [A,B] + [A,C]$") |
(8) |
![$\displaystyle [A,BC] = [A,B]C +B[A,C]$](http://images.physicslibrary.org/cache/objects/834/l2h/img40.png "$\displaystyle [A,BC] = [A,B]C +B[A,C]$") |
(9) |
![$\displaystyle [A,[B,C]] + [B,[C,A]] + [C,[A,B]] = 0$](http://images.physicslibrary.org/cache/objects/834/l2h/img41.png "$\displaystyle [A,[B,C]] + [B,[C,A]] + [C,[A,B]] = 0$") |
(10) |
By repeated application of rule (9), one hs
In particular, for a one-dimensional system one has
Equation 6 is thus verified when  is an arbitrary power of the  ; it is thus also verified (rule 8) when  is a polynomial, or else a convergent power series in  .
For general functions of the  's and  's, one can also write
![$\displaystyle [p_i,A] = \frac{hbar}{i} \frac{\partial A}{\partial q_i}$](http://images.physicslibrary.org/cache/objects/834/l2h/img50.png "$\displaystyle [p_i,A] = \frac{hbar}{i} \frac{\partial A}{\partial q_i}$") |
(11) |
![$\displaystyle [q_i,A] = i \hbar \frac{\partial A}{\partial p_i}$](http://images.physicslibrary.org/cache/objects/834/l2h/img51.png "$\displaystyle [q_i,A] = i \hbar \frac{\partial A}{\partial p_i}$") |
(12) |
 ,
 being defined by partial differentiation of  , it being understood that the order of the  's and  's in their explicit expression has been suitably chosen.
[1] Messiah, Albert. "Quantum mechanics: volume I." Amsterdam, North-Holland Pub. Co.; New York, Interscience Publishers, 1961-62.
This entry is a derivative of the Public domain work [1].
| 
![$\displaystyle [x,p_x] = \frac{\hbar}{i}\left[x, \frac{\partial}{\partial x}\right] = i \hbar \ne 0$](http://images.physicslibrary.org/cache/objects/834/l2h/img8.png "$\displaystyle [x,p_x] = \frac{\hbar}{i}\left[x, \frac{\partial}{\partial x}\right] = i \hbar \ne 0$")
![$\displaystyle [A,B^n] = \sum_{s=0}^{n-1} B^s [A,B] B^{n-s-1} $](http://images.physicslibrary.org/cache/objects/834/l2h/img42.png "$\displaystyle [A,B^n] = \sum_{s=0}^{n-1} B^s [A,B] B^{n-s-1} $")
![$\displaystyle [q,p^n] = n i \hbar p^{n-1} $](http://images.physicslibrary.org/cache/objects/834/l2h/img43.png "$\displaystyle [q,p^n] = n i \hbar p^{n-1} $")