Rotation and translation operators do not commute

Now we may think about adding px, Py, Pz, to the above set of operators. The operators H, px, Pv, Pzjfi and Jz do not represent a set of mutual commuting operators. The reason for this is that p,xyJv for which is a consequence of the fact that, in general, rotation and translation operators do not commute as shown in Fig. F.l. 

therefore, impossible to make all the operators H,px,pv, Pz,J and Jz commute in a space fixed coordinate system. What we are able to do, though, is to write the total wave function in the space fixed coordinate i stem as a product of the plane wave exp(/po f cM) depending on the centre-of-mass variables and on the wave function depending on internal coordinates  

The function koAf is the total wave function written in the centre-of-mass coordinate system (a special body-fixed coordinate system, see Appendix I), in which the total angular momentum operators and Jz are now defined. The three operators H,J and Jz commute in any space-fixed or body-fixed coordinate system (including the centre-of-mass coordinate system), and therefore the corresponding physical quantities (energy and angular momentum) have exact values. In this particular coordinate system p = pcM = 0. We may say, therefore, that 

It turns out (as shown by James Maxwell), that //, , p and i are interrelated by the Maxwell equations (c stands for the speed of light) 

The Maxwell equations have an alternative notation, which involves two new quantities the scalar potential 4 and the vector potential A that replace and H  

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Fig. F.l. In general, translation U T) and rotation U a e) operators do not commute. The example shows what happens to a point belonging to the XT plane. (a)If a rx3tationW(o z) by angle a about the r.-axis takes place first, and then a translation t/fT) by a vector T (restricted to the xy plane) is carried out, and (b) shows what happens if the operations are applied in the reverse order. As we can see. the results are different (two points " have different positions in panels a and b) i.e.. the two operators do not commute U(T)U(a z) U(a z)U(T). Expanding W(T) = exp[— (TvPv + TyPy)] andW(a z) = exp(— aJz) in a Taylor series, and, taking into account that Tv Ty, a are arbitrary numbers, leads to the conclusion that [ Jz, P ] Oand J, P ] 0. Note that some translations and rotations do commute e.g., [ Jz, Pzl = [ /, A ] = I A-. PiO = 0, because we see by inspection [as shown in panels (c) and (d)] that any translation by T = (0,0, T ) is independent of any rotation about the r-axis, etc.

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