10.1: General Principles of Angular Momentum
( \newcommand{\kernel}{\mathrm{null}\,}\)
The three fundamental orbital angular momentum operators, Lx, Ly, and Lz, obey the commutation relations ([e8.6])–([e8.8]), which can be written in the convenient vector form:
L×L=iℏL.
S×S=iℏS.
[Li,Sj]=0,
Let us now consider the electron’s total angular momentum vector
J=L+S.
J×J=(L+S)×(L+S)=L×L+S×S+L×S+S×L=L×L+S×S=iℏL+iℏS=iℏJ.
J×J=iℏJ.
It is thus evident that the three funcdamental total angular momentum operators, Jx, Jy, and Jz, obey analogous commutation relations to the corresponding orbital and spin angular momentum operators. It therefore follows that the total angular momentum has similar properties to the orbital and spin angular momenta. For instance, it is only possible to simultaneously measure the magnitude squared of the total angular momentum vector,
J2=J2x+J2y+J2z,
together with a single Cartesian component. By convention, we shall always choose to measure Jz. A simultaneous eigenstate of Jz and J2 satisfies
Jzψj,mj=mjℏψj,mj,J2ψj,mj=j(j+1)ℏ2ψj,mj,
where the quantum number j can take positive integer, or half-integer, values, and the quantum number mj is restricted to the following range of values:
−j,−j+1,⋯,j−1,j.
Now,
J2=(L+S)⋅(L+S)=L2+S2+2L⋅S,
which can also be written as
We know that the operator L2 commutes with itself, with all of the Cartesian components of L (and, hence, with the raising and lowering operators L±), and with all of the spin angular momentum operators. (See Section [s8.2].) It is therefore clear that
[J2,L2]=0.
[J2,S2]=0.
[J2,Lz]≠0.
Likewise, we can also show that
[J2,Sz]≠0.
Finally, we have
Jz=Lz+Sz,
where [Jz,Lz]=[Jz,Sz]=0.
Recalling that only commuting operators correspond to physical quantities that can be simultaneously measured (see Section [smeas]), it follows, from the previous discussion, that there are two alternative sets of physical variables associated with angular momentum that we can measure simultaneously. The first set correspond to the operators L2, S2, Lz, Sz, and Jz. The second set correspond to the operators L2, S2, J2, and Jz. In other words, we can always measure the magnitude squared of the orbital and spin angular momentum vectors, together with the z-component of the total angular momentum vector. In addition, we can either choose to measure the z-components of the orbital and spin angular momentum vectors, or the magnitude squared of the total angular momentum vector.
Let ψ(1)l,s;m,ms represent a simultaneous eigenstate of L2, S2, Lz, and Sz corresponding to the following eigenvalues:
L2ψ(1)l,s;m,ms=l(l+1)ℏ2ψ(1)l,s;m,ms,S2ψ(1)l,s;m,ms=s(s+1)ℏ2ψ(1)l,s;m,ms,Lzψ(1)l,s;m,ms=mℏψ(1)l,s;m,ms,Szψ(1)l,s;m,ms=msℏψ(1)l,s;m,ms.
It is easily seen that
Jzψ(1)l,s;m,ms=(Lz+Sz)ψ(1)l,s;m,ms=(m+ms)ℏψ(1)l,s;m,ms=mjℏψ(1)l,s;m,ms.
Hence,
In other words, the quantum numbers controlling the z-components of the various angular momentum vectors can simply be added algebraically.
Finally, let ψ(2)l,s;j,mj represent a simultaneous eigenstate of L2, S2, J2, and Jz corresponding to the following eigenvalues:
L2ψ(2)l,s;j,mj=l(l+1)ℏ2ψ(2)l,s;j,mj,S2ψ(2)l,s;j,mj=s(s+1)ℏ2ψ(2)l,s;j,mj,J2ψ(2)l,s;j,mj=j(j+1)ℏ2ψ(2)l,s;j,mj,Jzψ(2)l,s;j,mj=mjℏψ(2)l,s;j,mj.
Contributors and Attributions
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)