7.1: Angular Momenum Operators
( \newcommand{\kernel}{\mathrm{null}\,}\)
In classical mechanics, the vector angular momentum, L, of a particle of position vector r and linear momentum p is defined as L=r×p. It follows that Lx=ypz−zpy,Ly=zpx−xpz,Lz=xpy−ypx. Let us, first of all, consider whether it is possible to use the previous expressions as the definitions of the operators corresponding to the components of angular momentum in quantum mechanics, assuming that the xi and pi (where x1≡x, p1≡px, x2≡y, etc.) correspond to the appropriate quantum mechanical position and momentum operators. The first point to note is that expressions ([e8.1])–([e8.3]) are unambiguous with respect to the order of the terms in multiplicative factors, because the various position and momentum operators appearing in them all commute with one another. [See Equations ([commxp]).] Moreover, given that the xi and the pi are Hermitian operators, it is easily seen that the Li are also Hermitian. This is important, because only Hermitian operators can represent physical variables in quantum mechanics. (See Section [s4.6].) We, thus, conclude that Equations ([e8.1])–([e8.3]) are plausible definitions for the quantum mechanical operators that represent the components of angular momentum.
Let us now derive the commutation relations for the Li. For instance, =[ypz−zpy,zpx−xpz]=ypx[pz,z]+xpy[z,pz]=iℏ(xpy−ypx)=iℏLz, where use has been made of the definitions of the Li [see Equations ([e8.1])–([e8.3])], and commutation relations ([commxx])–([commxp]) for the xi and pi. There are two similar commutation relations: one for Ly and Lz, and one for Lz and Lx. Collecting all of these commutation relations together, we obtain [Lx,Ly]=iℏLz,[Ly,Lz]=iℏLx,[Lz,Lx]=iℏLy.
By analogy with classical mechanics, the operator L2, that represents the magnitude squared of the angular momentum vector, is defined L2=L2x+L2y+L2z. Now, it is easily demonstrated that if A and B are two general operators then [A2,B]=A[A,B]+[A,B]A. Hence, =[L2y,Lx]+[L2z,Lx]=Ly[Ly,Lx]+[Ly,Lx]Ly+Lz[Lz,Lx]+[Lz,Lx]Lz=iℏ(−LyLz−LzLy+LzLy+LyLz)=0, where use has been made of Equations ([e8.6])–([e8.8]). In other words, L2 commutes with Lx. Likewise, it is easily demonstrated that L2 also commutes with Ly, and with Lz. Thus, [L2,Lx]=[L2,Ly]=[L2,Lz]=0.
Recall, from Section [smeas], that in order for two physical quantities to be (exactly) measured simultaneously, the operators that represent them in quantum mechanics must commute with one another. Hence, the commutation relations ([e8.6])–([e8.8]) and ([e8.12]) imply that we can only simultaneously measure the magnitude squared of the angular momentum vector, L2, together with, at most, one of its Cartesian components. By convention, we shall always choose to measure the z-component, Lz.
Finally, it is helpful to define the operators L±=Lx±iLy. Note that L+ and L− are not Hermitian operators, but are the Hermitian conjugates of one another (see Section [s4.6]): that is, (L±)†=L∓, Moreover, it is easily seen that L+L−=(Lx+iLy)(Lx−iLy)=L2x+L2y−i[Lx,Ly]=L2x+L2y+ℏLz=L2−L2z+ℏLz. Likewise, L−L+=L2−L2z−ℏLz, giving [L+,L−]=2ℏLz. We also have [L+,Lz]=[Lx,Lz]+i[Ly,Lz]=−iℏLy−ℏLx=−ℏL+, and, similarly, [L−,Lz]=ℏL−.
Contributors and Attributions
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)