4.1Angular MomentumClassicallytheangularmomentumofanAPisolatedsystemis a constant ofthemotionrsineQuantummechanicallythismeansthatweYexpecttobeabletofind statesofdefiniteangular momentum.In three dimensions, the angular momentum about a point is the magnitude p of themomentum multiplied by theperpendiculardistance of the momentumvectorfromthe point (r sin inthediagram)The angularmomentum describesrotation about an axis perpendicular totheplane containing r and p, so in vector notation it is J= r X p
4.1 Angular Momentum Classically, the angular momentum of an isolated system is a constant of the motion. Quantum mechanically, this means that we expect to be able to find states of definite angular momentum. In three dimensions, the angular momentum about a point is the magnitude p of the momentum multiplied by the perpendicular distance of the momentum vector from the point (r sin θ in the diagram). The angular momentum describes rotation about an axis perpendicular to the plane containing r and p, so in vector notation it is J = r × p
4.2AngularmomentumoperatorsThe angular momentum is the vector product J= r X p. That is,J, = yp. -zPyJ,= zP-XPJ. = xPy - yPxMaking the usual substitutions yields the operatorsaaJ, =-in(vo, -2o,)=-ihayCj, = -in(za, - xa.)FC-tj, =-in(xa, - yo.)y
4.2 Angular momentum operators The angular momentum is the vector product J = r × p. That is, Making the usual substitutions yields the operators x z y y x z z y x J yp zp J zp xp J xp yp ˆ ˆ ˆ x y z J i y z z y J i z x x z J i x y y x ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ x z y y x z z y x J i y z J i z x J i x y
Angular momentum operators don't commuteThe components of the angular momentum operator do not commute with each otherWriting a/ax = a,for brevity, and remembering that a, x = 1 + xax, but a, y = y, anda,o,=a,ox,j,j,w=(-in)(va. -z,)(-in)(zo, -xo.)y--h2 (vo,z0, -yo.xo. -2z0,20+ +z0,xo.)y= -h? (vo, + yza,a, - xya.a. -2a,a + xza,a.)yj,j=(-in)(-o, -xa.)(-in)(vo. -z,)y=-h2 (2o.yo. -xo.yo. -z0,20, +xo,20,)y=-h2(yz0,a. -xya.a,-2a,a, +xa, + xza.a,)ywe find thatJx,j,=jj,-j,j,=-h2(vor-xa,)=inj
Angular momentum operators don’t commute The components of the angular momentum operator do not commute with each other. Writing 𝜕/𝜕𝑥 = 𝜕መ 𝑥 for brevity, and remembering that 𝜕መ 𝑥 𝑥 = 1 + 𝑥𝜕መ 𝑥 , but 𝜕መ 𝑥 𝑦 = 𝑦𝜕መ 𝑥 and 𝜕መ 𝑥𝜕መ 𝑦 = 𝜕መ 𝑦𝜕መ 𝑥 , 2 2 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ x y z y x z z x z z y x y z x z x z z y x y z J J i y z i z x y z y x z z z x y yz xy z xz 2 2 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ y x x z z y x z z z x y z y x z z z x y y z y J J i z x i y z z y x y z z x z yz xy z x xz 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ , x y x y y x x y z J J J J J J y x i J we find that
Commutation relationsforangularmomentumWehavefoundthat7[J,j,]=jj,-j,j,=-ha1[,j,]-injSimilarly,[3.,j,]=inj,[3,j,]-inj,(Note that x, y and z appear in cyclic order in these equations.)The uncertainty principle tells us that, for example,N,N,([.j,J)-.)so in general we cannot find wavefunctions that are simultaneously eigenfunctions of two ormore of Jx , J, and Jz . The only exception is that it is possible to find wavefunctions forwhich Jx, J, and J, are all exactly zero
Commutation relations for angular momentum We have found that Similarly, (Note that x, y and z appear in cyclic order in these equations.) The uncertainty principle tells us that, for example, so in general we cannot find wavefunctions that are simultaneously eigenfunctions of two or more of 𝐽መ 𝑥 , 𝐽መ 𝑦 and 𝐽መ 𝑧 . The only exception is that it is possible to find wavefunctions for which Jx , Jy and Jz are all exactly zero. 2 ˆ ˆ ˆ ˆ ˆ ˆ , ˆ ˆ ˆ , x y x y y x x y z J J J J J J y x x y J J i J ˆ ˆ ˆ , y z x J J i J ˆ ˆ ˆ , z x y J J i J 1 1 ˆ ˆ , 2 2 x y x y z J J J J J
However, Jx, J, and J, all commute with j? = J,2 + j,2 + j2. For example,[.,J3]-J.3?-j2j-JSS-JJJ+JJ-JS[3,j]j,+j[,j]=in(j,j,+j,j)and similarly [z,J,2] = -in(JJx +J,J,), while [Jz,J2] = 0Adding these results together shows that [1 z, J2] = 0Therefore we can find wavefunctions that are eigenfunctions of both f2 and oneonly ofjx , J, and Jz . It is customary to choose Jz
However, 𝐽መ 𝑥 , 𝐽መ 𝑦 and 𝐽መ 𝑧 all commute with 𝑱2 = 𝐽መ 𝑥 2 + 𝐽መ 𝑦 2 + 𝐽መ 𝑧 2 . For example, and similarly 𝐽መ 𝑧 ,𝐽መ 𝑦 2 = −𝑖ℏ 𝐽መ 𝑦 𝐽መ 𝑥 + 𝐽መ 𝑥 𝐽መ 𝑦 , while 𝐽መ 𝑧 ,𝐽መ 𝑧 2 = 0. Adding these results together shows that 𝐽መ 𝑧 , 𝑱2 = 0. Therefore we can find wavefunctions that are eigenfunctions of both 𝑱2 and one only of 𝐽መ 𝑥 , 𝐽መ 𝑦 and 𝐽መ 𝑧 . It is customary to choose 𝐽መ 𝑧 . 2 2 2 ˆ ˆ ˆ ˆ ˆ ˆ , ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ , , ˆ ˆ ˆ ˆ z x z x x z z x x x z x x z x x x z z x x x z x y x x y J J J J J J J J J J J J J J J J J J J J J J J J i J J J J