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Angular Momentum Operator Algebra

Michael Fowler

Preliminaries: Translation and Rotation Operators

As a warm up to analyzing how a wave function transforms under rotation, we review the effect of linear translation on a single particle wave function ψ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiYdK3aae WaaeaacaWG4baacaGLOaGaayzkaaaaaa@3A4B@ .  We have already seen an example of this: the coherent states of a simple harmonic oscillator discussed earlier were (at t=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabg2 da9iaaicdaaaa@38B0@  ) identical to the ground state except that they were centered at some point displaced from the origin. In fact, the operator creating such a state from the ground state is a translation operator.

The translation operator T( a ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaabm aabaGaamyyaaGaayjkaiaawMcaaaaa@393F@  is defined at that operator which when it acts on a wave function ket | ψ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa4Haaeqaba GaeqiYdK3aaeWaaeaacaWG4baacaGLOaGaayzkaaaacaGLhWUaayPk Jaaaaa@3CE1@  gives the ket corresponding to that wave function moved over by a, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaacY caaaa@378D@  that is,

T( a )| ψ( x )=| ψ( xa ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaabm aabaGaamyyaaGaayjkaiaawMcaamaaEiaabeqaaiabeI8a5naabmaa baGaamiEaaGaayjkaiaawMcaaaGaay5bSlaawQYiaiabg2da9maaEi aabeqaaiabeI8a5naabmaabaGaamiEaiabgkHiTiaadggaaiaawIca caGLPaaaaiaawEa7caGLQmcacaGGSaaaaa@4A9B@

so, for example, if ψ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiYdK3aae WaaeaacaWG4baacaGLOaGaayzkaaaaaa@3A4B@  is a wave function centered at the origin, T( a ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaabm aabaGaamyyaaGaayjkaiaawMcaaaaa@393F@  moves it to be centered at the point a. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaac6 caaaa@378F@  

We have written the wave function as a ket here to emphasize the parallels between this operation and some later ones, but it is simpler at this point to just work with the wave function as a function, so we will drop the ket bracket for now.  The form of T( a ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaabm aabaGaamyyaaGaayjkaiaawMcaaaaa@393F@  as an operator on a function is made evident by rewriting the Taylor series in operator form:

ψ( xa )=ψ( x )a d dx ψ( x )+ a 2 2! d 2 d x 2 ψ( x ) = e a d dx ψ( x ) =T( a )ψ( x ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaabbeaacqaHip qEdaqadaqaaiaadIhacqGHsislcaWGHbaacaGLOaGaayzkaaGaeyyp a0JaeqiYdK3aaeWaaeaacaWG4baacaGLOaGaayzkaaGaeyOeI0Iaam yyamaalaaabaGaamizaaqaaiaadsgacaWG4baaaiabeI8a5naabmaa baGaamiEaaGaayjkaiaawMcaaiabgUcaRmaalaaabaGaamyyamaaCa aaleqabaGaaGOmaaaaaOqaaiaaikdacaGGHaaaamaalaaabaGaamiz amaaCaaaleqabaGaaGOmaaaaaOqaaiaadsgacaWG4bWaaWbaaSqabe aacaaIYaaaaaaakiabeI8a5naabmaabaGaamiEaaGaayjkaiaawMca aiabgkHiTiablAcilbqaaiabg2da9iaadwgadaahaaWcbeqaaiabgk HiTiaadggadaWcaaqaaiaadsgaaeaacaWGKbGaamiEaaaaaaGccqaH ipqEdaqadaqaaiaadIhaaiaawIcacaGLPaaaaeaacqGH9aqpcaWGub WaaeWaaeaacaWGHbaacaGLOaGaayzkaaGaeqiYdK3aaeWaaeaacaWG 4baacaGLOaGaayzkaaGaaiOlaaaaaa@6E5D@

Now for the quantum connection: the differential operator appearing in the exponential is in quantum mechanics proportional to the momentum operator ( p ^ =id/dx MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiCayaaja Gaeyypa0JaeyOeI0IaamyAaiabl+qiOjaadsgacaGGVaGaamizaiaa dIhaaaa@3E88@  ) so the translation operator

T( a )= e ia p ^ / . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaabm aabaGaamyyaaGaayjkaiaawMcaaiabg2da9iaadwgadaahaaWcbeqa aiabgkHiTiaadMgacaWGHbGabmiCayaajaGaai4laiabl+qiObaaki aac6caaaa@41B9@

An important special case is that of an infinitesimal translation,

T( ε )= e iε p ^ / =1iε p ^ /. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaabm aabaGaeqyTdugacaGLOaGaayzkaaGaeyypa0JaamyzamaaCaaaleqa baGaeyOeI0IaamyAaiabew7aLjqadchagaqcaiaac+cacqWIpecAaa GccqGH9aqpcaaIXaGaeyOeI0IaamyAaiabew7aLjqadchagaqcaiaa c+cacqWIpecAcaGGUaaaaa@4B5F@

The momentum operator p ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiCayaaja aaaa@36FC@  is said to be the generator of the translation.

(A note on possibly confusing notation: Shankar writes (page 281) T( ε )|x=| x+ε. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaabm aabaGaeqyTdugacaGLOaGaayzkaaWaa4HaaeqabaGaamiEaaGaay5b SlaawQYiaiabg2da9maaEiaabeqaaiaadIhacqGHRaWkcqaH1oqzai aawEa7caGLQmcacaGGUaaaaa@4567@  Here |x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa4Haaeqaba GaamiEaaGaay5bSlaawQYiaaaa@398A@  denotes a delta-function type wave function centered at x. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiaac6 caaaa@37A6@  It might be better if he had written T( ε )| x 0 =| x 0 +ε MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaabm aabaGaeqyTdugacaGLOaGaayzkaaWaa4HaaeqabaGaamiEamaaBaaa leaacaaIWaaabeaaaOGaay5bSlaawQYiaiabg2da9maaEiaabeqaai aadIhadaWgaaWcbaGaaGimaaqabaGccqGHRaWkcqaH1oqzaiaawEa7 caGLQmcaaaa@4695@ , then we would see right away that this translates into the wave function transformation T( ε )δ( x x 0 )=δ( x x 0 ε ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaabm aabaGaeqyTdugacaGLOaGaayzkaaGaeqiTdq2aaeWaaeaacaWG4bGa eyOeI0IaamiEamaaBaaaleaacaaIWaaabeaaaOGaayjkaiaawMcaai abg2da9iabes7aKnaabmaabaGaamiEaiabgkHiTiaadIhadaWgaaWc baGaaGimaaqabaGccqGHsislcqaH1oqzaiaawIcacaGLPaaaaaa@4BA4@ , the sign of ε MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdugaaa@379E@  now obviously consistent with our usage above.)

It is important to be clear about whether the system is being translated by a, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaacY caaaa@378D@  as we have done above or whether, alternately, the coordinate axes are being translated by a, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaacY caaaa@378D@  that latter would result in the opposite change in the wave function. Translating the coordinate axes, along with the apparatus and any external fields by a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Iaam yyaaaa@37CA@  relative to the wave function would of course give the same physics as translating the wave function by +a. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4kaSIaam yyaiaac6caaaa@3871@  In fact, these two equivalent operations are analogous to the time development of a wave function being described either by a Schrödinger picture, in which the bras and kets change in time, but not the operators, and the Heisenberg picture in which the operators develop but the bras and kets do not change.  To pursue this analogy a little further, in the “Heisenberg” case

x ^ T 1 ( ε ) x ^ T( ε )= e iε p ^ / x ^ e iε p ^ / = x ^ +iε[ p ^ , x ^ ]/= x ^ +ε MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaaja GaeyOKH4QaamivamaaCaaaleqabaGaeyOeI0IaaGymaaaakmaabmaa baGaeqyTdugacaGLOaGaayzkaaGabmiEayaajaGaamivamaabmaaba GaeqyTdugacaGLOaGaayzkaaGaeyypa0JaamyzamaaCaaaleqabaGa amyAaiabew7aLjqadchagaqcaiaac+cacqWIpecAaaGcceWG4bGbaK aacaWGLbWaaWbaaSqabeaacqGHsislcaWGPbGaeqyTduMabmiCayaa jaGaai4laiabl+qiObaakiabg2da9iqadIhagaqcaiabgUcaRiaadM gacqaH1oqzdaWadaqaaiqadchagaqcaiaacYcaceWG4bGbaKaaaiaa wUfacaGLDbaacaGGVaGaeS4dHGMaeyypa0JabmiEayaajaGaey4kaS IaeqyTdugaaa@64D2@

and p ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiCayaaja aaaa@36FC@  is unchanged since it commutes with the operator.  So there are two possible ways to deal with translations: transform the bras and kets, or transform the operators. We shall almost always leave the operators alone, and transform the bras and kets.

We have established that the momentum operator is the generator of spatial translations (the generalization to three dimensions is trivial).  We know from earlier work that the Hamiltonian is the generator of time translations, by which we mean

ψ( t+a )= e iHa/ ψ( t ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiYdK3aae WaaeaacaWG0bGaey4kaSIaamyyaaGaayjkaiaawMcaaiabg2da9iaa dwgadaahaaWcbeqaaiabgkHiTiaadMgacaWGibGaamyyaiaac+cacq WIpecAaaGccqaHipqEdaqadaqaaiaadshaaiaawIcacaGLPaaacaGG Uaaaaa@48A1@

It is tempting to conclude that the angular momentum must be the operator generating rotations of the system, and, in fact, it is easy to check that this is correct.  Let us consider an infinitesimal rotation δ θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdqMafq iUdeNbaSaaaaa@3964@  about some axis through the origin (the infinitesimal vector being in the direction of the axis).  A wavefunction ψ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiYdK3aae WaaeaaceWGYbGbaSaaaiaawIcacaGLPaaaaaa@3A57@  initially localized at r 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOCayaala WaaSbaaSqaaiaaicdaaeqaaaaa@37E6@  will shift to be localized at r 0 +δ r 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOCayaala WaaSbaaSqaaiaaicdaaeqaaOGaey4kaSIaeqiTdqMabmOCayaalaWa aSbaaSqaaiaaicdaaeqaaaaa@3C66@ , where δ r 0 =δ θ × r 0 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdqMabm OCayaalaWaaSbaaSqaaiaaicdaaeqaaOGaeyypa0JaeqiTdqMafqiU deNbaSaacqGHxdaTceWGYbGbaSaadaWgaaWcbaGaaGimaaqabaGcca GGUaaaaa@42CA@  So, how does a wave function transform under this small rotation?  Just as for the translation case, ψ( r )ψ( r δ r ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiYdK3aae WaaeaaceWGYbGbaSaaaiaawIcacaGLPaaacqGHsgIRcqaHipqEdaqa daqaaiqadkhagaWcaiabgkHiTiabes7aKjqadkhagaWcaaGaayjkai aawMcaaaaa@443F@ .  If you don’t understand the minus sign, reread the discussion on translations and the sign of ε MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdugaaa@379E@ .

Thus

ψ( r )ψ( r ) i δ r . p ^ ψ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiYdK3aae WaaeaaceWGYbGbaSaaaiaawIcacaGLPaaacqGHsgIRcqaHipqEdaqa daqaaiqadkhagaWcaaGaayjkaiaawMcaaiabgkHiTmaalaaabaGaam yAaaqaaiabl+qiObaacqaH0oazceWGYbGbaSaacaGGUaGabmiCayaa lyaajaGaeqiYdK3aaeWaaeaaceWGYbGbaSaaaiaawIcacaGLPaaaaa a@4C8D@

to first order in the infinitesimal quantity, so the rotation operator

R( δ θ )ψ( r )=( 1 i δ θ × r . p ^ )ψ( r ) =( 1 i δ θ . r × p ^ )ψ( r ) =( 1 i δ θ . L ^ )ψ( r ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaabbeaacaWGsb WaaeWaaeaacqaH0oazcuaH4oqCgaWcaaGaayjkaiaawMcaaiabeI8a 5naabmaabaGabmOCayaalaaacaGLOaGaayzkaaGaeyypa0ZaaeWaae aacaaIXaGaeyOeI0YaaSaaaeaacaWGPbaabaGaeS4dHGgaaiabes7a KjqbeI7aXzaalaGaey41aqRabmOCayaalaGaaiOlaiqadchagaWcga qcaaGaayjkaiaawMcaaiabeI8a5naabmaabaGabmOCayaalaaacaGL OaGaayzkaaaabaGaeyypa0ZaaeWaaeaacaaIXaGaeyOeI0YaaSaaae aacaWGPbaabaGaeS4dHGgaaiabes7aKjqbeI7aXzaalaGaaiOlaiqa dkhagaWcaiabgEna0kqadchagaWcgaqcaaGaayjkaiaawMcaaiabeI 8a5naabmaabaGabmOCayaalaaacaGLOaGaayzkaaaabaGaeyypa0Za aeWaaeaacaaIXaGaeyOeI0YaaSaaaeaacaWGPbaabaGaeS4dHGgaai abes7aKjqbeI7aXzaalaGaaiOlaiqadYeagaWcgaqcaaGaayjkaiaa wMcaaiabeI8a5naabmaabaGabmOCayaalaaacaGLOaGaayzkaaGaai Olaaaaaa@76D5@

If we write this as

R( δ θ )ψ( r )= e i δ θ . L ^ ψ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaabm aabaGaeqiTdqMafqiUdeNbaSaaaiaawIcacaGLPaaacqaHipqEdaqa daqaaiqadkhagaWcaaGaayjkaiaawMcaaiabg2da9iaadwgadaahaa WcbeqaaiabgkHiTmaalaaabaGaamyAaaqaaiabl+qiObaacqaH0oaz cuaH4oqCgaWcaiaac6caceWGmbGbaSGbaKaaaaGccqaHipqEdaqada qaaiqadkhagaWcaaGaayjkaiaawMcaaaaa@4ECF@

it is clear that a finite rotation is given by multiplying together a large number of these operators, which just amounts to replacing δ θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdqMafq iUdeNbaSaaaaa@3964@  by θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaS aaaaa@37BF@  in the exponential.  Another way of going from the infinitesimal rotation to a full rotation is to use the identity

lim N ( 1+ Aθ N ) N = e Aθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiaad6eacqGHsgIRcqGHEisPaeqaaOWa aeWaaeaacaaIXaGaey4kaSYaaSaaaeaacaWGbbGaeqiUdehabaGaam OtaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaad6eaaaGccqGH9aqp caWGLbWaaWbaaSqabeaacaWGbbGaeqiUdehaaaaa@4962@

which is clearly valid even if A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaaaa@36BD@  is an operator.

We have therefore established that the orbital angular momentum operator L ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmitayaaly aajaaaaa@36E9@  is the generator of spatial rotations, by which we mean that if we rotate our apparatus, and the wave function with it, the appropriately transformed wave function is generated by the action of R( θ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaabm aabaGafqiUdeNbaSaaaiaawIcacaGLPaaaaaa@3A1F@  on the original wave function. It is perhaps worth giving an explicit example: suppose we rotate the system, and therefore the wave function, through an infinitesimal angle δ θ z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdqMaeq iUde3aaSbaaSqaaiaadQhaaeqaaaaa@3A7D@  about the z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEaaaa@36F6@  -axis. Denote the rotated wave function by ψ rot ( x,y ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiYdK3aaS baaSqaaiaadkhacaWGVbGaamiDaaqabaGcdaqadaqaaiaadIhacaGG SaGaamyEaaGaayjkaiaawMcaaaaa@3F13@ .  Then

ψ rot ( x,y )=( 1 i ( δ θ z ) L ^ z )ψ( x,y ) =( 1 i ( δ θ z )( i( x d dy y d dx ) ) )ψ( x,y ) =( 1( δ θ z )( x d dy y d dx ) )ψ( x,y ) =ψ( x+( δ θ z )y,y( δ θ z )x ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaabbeaacqaHip qEdaWgaaWcbaGaamOCaiaad+gacaWG0baabeaakmaabmaabaGaamiE aiaacYcacaWG5baacaGLOaGaayzkaaGaeyypa0ZaaeWaaeaacaaIXa GaeyOeI0YaaSaaaeaacaWGPbaabaGaeS4dHGgaamaabmaabaGaeqiT dqMaeqiUde3aaSbaaSqaaiaadQhaaeqaaaGccaGLOaGaayzkaaGabm itayaajaWaaSbaaSqaaiaadQhaaeqaaaGccaGLOaGaayzkaaGaeqiY dK3aaeWaaeaacaWG4bGaaiilaiaadMhaaiaawIcacaGLPaaaaeaacq GH9aqpdaqadaqaaiaaigdacqGHsisldaWcaaqaaiaadMgaaeaacqWI pecAaaWaaeWaaeaacqaH0oazcqaH4oqCdaWgaaWcbaGaamOEaaqaba aakiaawIcacaGLPaaadaqadaqaaiabgkHiTiaadMgacqWIpecAdaqa daqaaiaadIhadaWcaaqaaiaadsgaaeaacaWGKbGaamyEaaaacqGHsi slcaWG5bWaaSaaaeaacaWGKbaabaGaamizaiaadIhaaaaacaGLOaGa ayzkaaaacaGLOaGaayzkaaaacaGLOaGaayzkaaGaeqiYdK3aaeWaae aacaWG4bGaaiilaiaadMhaaiaawIcacaGLPaaaaeaacqGH9aqpdaqa daqaaiaaigdacqGHsisldaqadaqaaiabes7aKjabeI7aXnaaBaaale aacaWG6baabeaaaOGaayjkaiaawMcaamaabmaabaGaamiEamaalaaa baGaamizaaqaaiaadsgacaWG5baaaiabgkHiTiaadMhadaWcaaqaai aadsgaaeaacaWGKbGaamiEaaaaaiaawIcacaGLPaaaaiaawIcacaGL PaaacqaHipqEdaqadaqaaiaadIhacaGGSaGaamyEaaGaayjkaiaawM caaaqaaiabg2da9iabeI8a5naabmaabiqaaq+gcaWG4bGaey4kaSYa aeWaaeaacqaH0oazcqaH4oqCdaWgaaWcbaGaamOEaaqabaaakiaawI cacaGLPaaacaWG5bGaaiilaiaaysW7caWG5bGaeyOeI0YaaeWaaeaa cqaH0oazcqaH4oqCdaWgaaWcbaGaamOEaaqabaaakiaawIcacaGLPa aacaWG4baacaGLOaGaayzkaaGaaiOlaaaaaa@A9AA@

That is to say, the value of the new wave function at ( x,y ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WG4bGaaiilaiaadMhaaiaawIcacaGLPaaaaaa@3A2B@  is the value of the old wave function at the point which was rotated into ( x,y ). MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WG4bGaaiilaiaadMhaaiaawIcacaGLPaaacaGGUaaaaa@3ADD@   

Quantum Generalization of the Rotation Operator

However, it has long been known that in quantum mechanics, orbital angular momentum is not the whole story.  Particles like the electron are found experimentally to have an internal angular momentum, called spin.  In contrast to the spin of an ordinary macroscopic object like a spinning top, the electron’s spin is not just the sum of orbital angular momenta of internal parts, and any attempt to understand it in that way leads to contradictions. 

To take account of this new kind of angular momentum, we generalize the orbital angular momentum L ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmitayaaly aajaaaaa@36E9@  to an operator J ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOsayaaly aajaaaaa@36E7@  which is defined as the generator of rotations on any wave function, including possible spin components, so

R( δ θ )ψ( r )= e i δ θ . J ^ ψ( r ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaabm aabaGaeqiTdqMafqiUdeNbaSaaaiaawIcacaGLPaaacqaHipqEdaqa daqaaiqadkhagaWcaaGaayjkaiaawMcaaiabg2da9iaadwgadaahaa WcbeqaaiabgkHiTmaalaaabaGaamyAaaqaaiabl+qiObaacqaH0oaz cuaH4oqCgaWcaiaac6caceWGkbGbaSGbaKaaaaGccqaHipqEdaqada qaaiqadkhagaWcaaGaayjkaiaawMcaaiaac6caaaa@4F7F@

This is of course identical to the equation we found for L ^ , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmitayaaoy aajaGaaiilaaaa@379C@  but there we derived if from the quantum angular momentum operator including the momentum components written as differentials.  But up to this point ψ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiYdK3aae WaaeaaceWGYbGbaSaaaiaawIcacaGLPaaaaaa@3A57@  has just been a complex valued function of position. From now on, the wave function at a point can have several components, so it is in some vector space, and the rotation operator will operate in this space as well as being a differential operator with respect to position.  For example, the wave function could be a vector at each point, so rotation of the system could rotate this vector as well as moving it to a different r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOCayaala aaaa@3700@ .

To summarize: ψ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiYdK3aae WaaeaaceWGYbGbaSaaaiaawIcacaGLPaaaaaa@3A57@  is in general an n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36EA@  -component function at each point in space, R( δ θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaabm aabaGaeqiTdqMafqiUdeNbaSaaaiaawIcacaGLPaaaaaa@3BC3@  is an n×n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgE na0kaad6gaaaa@39F4@  matrix in the component space, and the above equation is the definition of J ^ . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOsayaaly aajaGaaiOlaaaa@3799@  Starting from this definition, we will find J ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOsayaaly aajaaaaa@36E7@  ’s properties. 

The first point to make is that in contrast to translations, rotations do not commute even for a classical system.  Rotating a book through π/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiWdaNaai 4laiaaikdaaaa@3923@  first about the z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEaaaa@36F6@  -axis then about the x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36F4@  -axis leaves it in a different orientation from that obtained by rotating from the same starting position first π/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiWdaNaai 4laiaaikdaaaa@3923@  about the x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36F4@  -axis then π/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiWdaNaai 4laiaaikdaaaa@3923@  about the z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEaaaa@36F6@  -axis.  Even small rotations do not commute, although the commutator is second order.  Since the R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuaaaa@36CE@  -operators are representations of rotations, they will reflect this commutativity structure, and we can see just how they do that by considering ordinary classical rotations of a real vector in three-dimensional space.

The matrices rotating a vector by θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37AD@  about the x,y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiaacY cacaWG5baaaa@38A2@  and z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEaaaa@36F6@  axes are

R x ( θ )=( 1 0 0 0 cosθ sinθ 0 sinθ cosθ ), R y ( θ )=( cosθ 0 sinθ 0 1 0 sinθ 0 cosθ ), R z ( θ )=( cosθ sinθ 0 sinθ cosθ 0 0 0 1 ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaWG4baabeaakmaabmaabaGaeqiUdehacaGLOaGaayzkaaGa eyypa0ZaaeWaaeaafaqabeWadaaabaGaaGymaaqaaiaaicdaaeaaca aIWaaabaGaaGimaaqaaiGacogacaGGVbGaai4CaiabeI7aXbqaaiab gkHiTiGacohacaGGPbGaaiOBaiabeI7aXbqaaiaaicdaaeaaciGGZb GaaiyAaiaac6gacqaH4oqCaeaaciGGJbGaai4BaiaacohacqaH4oqC aaaacaGLOaGaayzkaaGaaiilaiaaywW7caWGsbWaaSbaaSqaaiaadM haaeqaaOWaaeWaaeaacqaH4oqCaiaawIcacaGLPaaacqGH9aqpdaqa daqaauaabeqadmaaaeaaciGGJbGaai4BaiaacohacqaH4oqCaeaaca aIWaaabaGaci4CaiaacMgacaGGUbGaeqiUdehabaGaaGimaaqaaiaa igdaaeaacaaIWaaabaGaeyOeI0Iaci4CaiaacMgacaGGUbGaeqiUde habaGaaGimaaqaaiGacogacaGGVbGaai4CaiabeI7aXbaaaiaawIca caGLPaaacaGGSaGaaGzbVlaadkfadaWgaaWcbaGaamOEaaqabaGcda qadaqaaiabeI7aXbGaayjkaiaawMcaaiabg2da9maabmaabaqbaeqa bmWaaaqaaiGacogacaGGVbGaai4CaiabeI7aXbqaaiabgkHiTiGaco hacaGGPbGaaiOBaiabeI7aXbqaaiaaicdaaeaaciGGZbGaaiyAaiaa c6gacqaH4oqCaeaaciGGJbGaai4BaiaacohacqaH4oqCaeaacaaIWa aabaGaaGimaaqaaiaaicdaaeaacaaIXaaaaaGaayjkaiaawMcaaiaa c6caaaa@972E@

 

In the limit of rotations about infinitesimal angles (ignoring higher order terms),

R x ( ε )=1+ε( 0 0 0 0 0 1 0 1 0 ), R y ( ε )=1+ε( 0 0 1 0 0 0 1 0 0 ), R z ( ε )=1+ε( 0 1 0 1 0 0 0 0 0 ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaWG4baabeaakmaabmaabaGaeqyTdugacaGLOaGaayzkaaGa eyypa0JaaGymaiabgUcaRiabew7aLnaabmaabaqbaeqabmWaaaqaai aaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGa eyOeI0IaaGymaaqaaiaaicdaaeaacaaIXaaabaGaaGimaaaaaiaawI cacaGLPaaacaGGSaGaaGzbVlaadkfadaWgaaWcbaGaamyEaaqabaGc daqadaqaaiabew7aLbGaayjkaiaawMcaaiabg2da9iaaigdacqGHRa WkcqaH1oqzdaqadaqaauaabeqadmaaaeaacaaIWaaabaGaaGimaaqa aiaaigdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacqGHsislca aIXaaabaGaaGimaaqaaiaaicdaaaaacaGLOaGaayzkaaGaaiilaiaa ywW7caWGsbWaaSbaaSqaaiaadQhaaeqaaOWaaeWaaeaacqaH1oqzai aawIcacaGLPaaacqGH9aqpcaaIXaGaey4kaSIaeqyTdu2aaeWaaeaa faqabeWadaaabaGaaGimaaqaaiabgkHiTiaaigdaaeaacaaIWaaaba GaaGymaaqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaa caaIWaaaaaGaayjkaiaawMcaaiaac6cacaaMf8oaaa@748C@

It is easy to check that

 

[ R x ( ε ), R y ( ε ) ]= ε 2 ( 0 1 0 1 0 0 0 0 0 )= R z ( ε 2 )1. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaaca WGsbWaaSbaaSqaaiaadIhaaeqaaOWaaeWaaeaacqaH1oqzaiaawIca caGLPaaacaGGSaGaamOuamaaBaaaleaacaWG5baabeaakmaabmaaba GaeqyTdugacaGLOaGaayzkaaaacaGLBbGaayzxaaGaeyypa0JaeqyT du2aaWbaaSqabeaacaaIYaaaaOWaaeWaaeaafaqabeWadaaabaGaaG imaaqaaiabgkHiTiaaigdaaeaacaaIWaaabaGaaGymaaqaaiaaicda aeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIWaaaaaGaayjkai aawMcaaiabg2da9iaadkfadaWgaaWcbaGaamOEaaqabaGcdaqadaqa aiabew7aLnaaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaaiabgk HiTiaaigdacaGGUaaaaa@5955@

The rotation operators on quantum mechanical kets must, like all rotations, follow this same pattern, that is, we must have

( ( 1 i ε J x )( 1 i ε J y )( 1 i ε J y )( 1 i ε J x )+ i ε 2 J z )|ψ=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada qadaqaaiaaigdacqGHsisldaWcaaqaaiaadMgaaeaacqWIpecAaaGa eqyTduMaamOsamaaBaaaleaacaWG4baabeaaaOGaayjkaiaawMcaam aabmaabaGaaGymaiabgkHiTmaalaaabaGaamyAaaqaaiabl+qiObaa cqaH1oqzcaWGkbWaaSbaaSqaaiaadMhaaeqaaaGccaGLOaGaayzkaa GaeyOeI0YaaeWaaeaacaaIXaGaeyOeI0YaaSaaaeaacaWGPbaabaGa eS4dHGgaaiabew7aLjaadQeadaWgaaWcbaGaamyEaaqabaaakiaawI cacaGLPaaadaqadaqaaiaaigdacqGHsisldaWcaaqaaiaadMgaaeaa cqWIpecAaaGaeqyTduMaamOsamaaBaaaleaacaWG4baabeaaaOGaay jkaiaawMcaaiabgUcaRmaalaaabaGaamyAaaqaaiabl+qiObaacqaH 1oqzdaahaaWcbeqaaiaaikdaaaGccaWGkbWaaSbaaSqaaiaadQhaae qaaaGccaGLOaGaayzkaaWaa4HaaeqabaGaeqiYdKhacaGLhWUaayPk JaGaeyypa0JaaGimaaaa@6A3D@

where we have used the definition of the infinitesimal rotation operator on kets, R( δ θ )ψ( r )= e i δ θ . J ^ ψ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaabm aabaGaeqiTdqMafqiUdeNbaSaaaiaawIcacaGLPaaacqaHipqEdaqa daqaaiqadkhagaWcaaGaayjkaiaawMcaaiabg2da9iaadwgadaahaa WcbeqaaiabgkHiTmaalaaabaGaamyAaaqaaiabl+qiObaacqaH0oaz cuaH4oqCgaWcaiaac6caceWGkbGbaSGbaKaaaaGccqaHipqEdaqada qaaiqadkhagaWcaaGaayjkaiaawMcaaaaa@4ECD@ .  The zeroth and first-order terms in ε MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdugaaa@379E@  all cancel, the second-order term gives [ J x , J y ]=i J z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaaca WGkbWaaSbaaSqaaiaadIhaaeqaaOGaaiilaiaadQeadaWgaaWcbaGa amyEaaqabaaakiaawUfacaGLDbaacqGH9aqpcaWGPbGaeS4dHGMaam OsamaaBaaaleaacaWG6baabeaaaaa@41B5@ . The general statement is:

 

[ J i , J j ]=i ε ijk J k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeeG+aaaaaai vzKbWdbmaadmaajaaWbaGaamOsaOWaaSbaaKqaahaacaWGPbaabeaa jaaWcaGGSaGaamOsaOWaaSbaaKqaahaacaWGQbaabeaaaKaaalaawU facaGLDbaacqGH9aqpcaWGPbGaeS4dHGMaeqyTduMcdaWgaaqcbaCa aiaadMgacaWGQbGaam4AaaqabaqcaaSaamOsaOWaaSbaaKqaahaaca WGRbaabeaaaaa@4E85@

This is one of the most important formulas in quantum mechanics.

Consequences of the Commutation Relations

The commutation formula [ J i , J j ]=i ε ijk J k , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaaca WGkbWaaSbaaSqaaiaadMgaaeqaaOGaaiilaiaadQeadaWgaaWcbaGa amOAaaqabaaakiaawUfacaGLDbaacqGH9aqpcaWGPbGaeS4dHGMaeq yTdu2aaSbaaSqaaiaadMgacaWGQbGaam4AaaqabaGccaWGkbWaaSba aSqaaiaadUgaaeqaaOGaaiilaaaa@46EC@  which is, after all, a straightforward extension of the result for ordinary classical rotations, has surprisingly far-reaching consequences: it leads directly to the directional quantization of spin and angular momentum observed in atoms subject to a magnetic field.

It is by now very clear that in quantum mechanical systems such as atoms the total angular momentum, and also the component of angular momentum in a given direction, can only take certain values.  Let us try to construct a basis set of angular momentum states for a given system: a complete set of kets corresponding to all allowed values of the angular momentum.  Now, angular momentum is a vector quantity: it has magnitude and direction.  Let’s begin with the magnitude, the natural parameter is the length squared:

J 2 = J x 2 + J y 2 + J z 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsamaaCa aaleqabaGaaGOmaaaakiabg2da9iaadQeadaqhaaWcbaGaamiEaaqa aiaaikdaaaGccqGHRaWkcaWGkbWaa0baaSqaaiaadMhaaeaacaaIYa aaaOGaey4kaSIaamOsamaaDaaaleaacaWG6baabaGaaGOmaaaaaaa@42B8@ .

Now we must specify direction -- but here we run into a problem. J x , J y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsamaaBa aaleaacaWG4baabeaakiaacYcacaWGkbWaaSbaaSqaaiaadMhaaeqa aaaa@3AA2@  and J z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsamaaBa aaleaacaWG6baabeaaaaa@37F1@  are all mutually non-commuting, so we cannot construct a set of common eigenkets of any two of them, which we would need for a precise specification of direction.  They do all commute with J 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsamaaCa aaleqabaGaaGOmaaaaaaa@37AF@ , since it is spherically symmetric and therefore cannot be affected by any rotation (and, it’s easy to check this commutation explicitly).  

The bottom line, then, is that in attempting to construct eigenkets describing the different possible angular momentum states of a quantum system, the best we can do is to find the common eigenkets of J 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsamaaCa aaleqabaGaaGOmaaaaaaa@37AF@  and one direction, say J z . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsamaaBa aaleaacaWG6baabeaakiaac6caaaa@38AD@  The commutation relations do not allow us to be more precise about direction, analogous to the Uncertainty Principle for position and momentum, which also comes from noncommutativity of the relevant operators. 

We conclude that the appropriate angular momentum basis is the set of common eigenkets of the commuting Hermitian matrices J 2 , J z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsamaaCa aaleqabaGaaGOmaaaakiaacYcacaWGkbWaaSbaaSqaaiaadQhaaeqa aaaa@3A63@  :

J 2 | a,b=a| a,b J z | a,b=b| a,b. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGkb WaaWbaaSqabeaacaaIYaaaaOWaa4HaaeqabaGaamyyaiaacYcacaWG IbaacaGLhWUaayPkJaGaeyypa0JaamyyamaaEiaabeqaaiaadggaca GGSaGaamOyaaGaay5bSlaawQYiaaqaaiaadQeadaWgaaWcbaGaamOE aaqabaGcdaGhcaqabeaacaWGHbGaaiilaiaadkgaaiaawEa7caGLQm cacqGH9aqpcaWGIbWaa4HaaeqabaGaamyyaiaacYcacaWGIbaacaGL hWUaayPkJaGaaiOlaaaaaa@529A@

Our next task is to find the allowed values of a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DD@  and b. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaiaac6 caaaa@3790@  

Ladder Operators

The sets of allowed eigenvalues a,b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaacY cacaWGIbaaaa@3874@  can be found using the “ladder operator” trick previously discussed for the simple harmonic oscillator.  It turns out

J ± = J x ±i J y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsamaaBa aaleaacqGHXcqSaeqaaOGaeyypa0JaamOsamaaBaaaleaacaWG4baa beaakiabgglaXkaadMgacaWGkbWaaSbaaSqaaiaadMhaaeqaaaaa@40C6@

are closely analogous to the simple harmonic oscillator raising and lowering operators a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaaiiiGaaaaaa@37CE@  and a. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaac6 caaaa@378F@  

J + MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsamaaBa aaleaacqGHRaWkaeqaaaaa@37D4@  and J MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsamaaBa aaleaacqGHsislaeqaaaaa@37DF@  have commutation relations with J z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsamaaBa aaleaacaWG6baabeaaaaa@37F1@ :

[ J z , J ± ]=± J ± MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaaca WGkbWaaSbaaSqaaiaadQhaaeqaaOGaaiilaiaadQeadaWgaaWcbaGa eyySaelabeaaaOGaay5waiaaw2faaiabg2da9iabgglaXkabl+qiOj aayIW7caWGkbWaaSbaaSqaaiabgglaXcqabaaaaa@4626@

and they of course commute  with J 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsamaaCa aaleqabaGaaGOmaaaaaaa@37AF@ , as do J z , J x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsamaaBa aaleaacaWG6baabeaakiaacYcacaWGkbWaaSbaaSqaaiaadIhaaeqa aaaa@3AA3@  and J y . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsamaaBa aaleaacaWG5baabeaakiaac6caaaa@38AC@   

Therefore, J ± MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsamaaBa aaleaacqGHXcqSaeqaaaaa@38E0@  operating on | a,b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa4Haaeqaba GaamyyaiaacYcacaWGIbaacaGLhWUaayPkJaaaaa@3B0A@  cannot affect the value of a. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaac6 caaaa@378F@  But they do change the value of b: MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaiaacQ daaaa@379C@  

J z J ± | a,b=[ J z , J ± ]| a,b+ J ± J z | a,b =± J ± | a,b+b J ± | a,b =( b± ) J ± | a,b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaabbeaacaWGkb WaaSbaaSqaaiaadQhaaeqaaOGaamOsamaaBaaaleaacqGHXcqSaeqa aOWaa4HaaeqabaGaamyyaiaacYcacaWGIbaacaGLhWUaayPkJaGaey ypa0ZaamWaaeaacaWGkbWaaSbaaSqaaiaadQhaaeqaaOGaaiilaiaa dQeadaWgaaWcbaGaeyySaelabeaaaOGaay5waiaaw2faamaaEiaabe qaaiaadggacaGGSaGaamOyaaGaay5bSlaawQYiaiabgUcaRiaadQea daWgaaWcbaGaeyySaelabeaakiaadQeadaWgaaWcbaGaamOEaaqaba GcdaGhcaqabeaacaWGHbGaaiilaiaadkgaaiaawEa7caGLQmcaaeaa cqGH9aqpcqGHXcqScqWIpecAcaaMi8UaamOsamaaBaaaleaacqGHXc qSaeqaaOWaa4HaaeqabaGaamyyaiaacYcacaWGIbaacaGLhWUaayPk JaGaey4kaSIaamOyaiaadQeadaWgaaWcbaGaeyySaelabeaakmaaEi aabeqaaiaadggacaGGSaGaamOyaaGaay5bSlaawQYiaaqaaiabg2da 9maabmaabaGaamOyaiabgglaXkabl+qiObGaayjkaiaawMcaaiaadQ eadaWgaaWcbaGaeyySaelabeaakmaaEiaabeqaaiaadggacaGGSaGa amOyaaGaay5bSlaawQYiaaaaaa@7EBF@

so if | a,b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa4Haaeqaba GaamyyaiaacYcacaWGIbaacaGLhWUaayPkJaaaaa@3B09@  is an eigenket of J z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsamaaBa aaleaacaWG6baabeaaaaa@37F1@  with eigenvalue b, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaiaacY caaaa@378E@   J ± | a,b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsamaaBa aaleaacqGHXcqSaeqaaOWaa4HaaeqabaGaamyyaiaacYcacaWGIbaa caGLhWUaayPkJaaaaa@3DFC@  is either zero or an eigenket of J z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsamaaBa aaleaacaWG6baabeaaaaa@37F1@  with eigenvalue b± MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaiabgg laXkabl+qiObaa@39F5@ , that is, J ± | a,b= C ± | a,b± MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsamaaBa aaleaacqGHXcqSaeqaaOWaa4HaaeqabaGaamyyaiaacYcacaWGIbaa caGLhWUaayPkJaGaeyypa0Jaam4qamaaBaaaleaacqGHXcqSaeqaaO Waa4HaaeqabaGaamyyaiaacYcacaWGIbGaeyySaeRaeS4dHGgacaGL hWUaayPkJaaaaa@4A18@  where C ± ( a,b ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacqGHXcqSaeqaaOWaaeWaaeaacaWGHbGaaiilaiaadkgaaiaa wIcacaGLPaaaaaa@3CE9@  is a normalization constant, taking the initial | a,b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa4Haaeqaba GaamyyaiaacYcacaWGIbaacaGLhWUaayPkJaaaaa@3B0A@  to be normalized.   Just as with the simple harmonic oscillator, we have to find these normalization constants in order to compute matrix elements.  All the physics is in the matrix elements.

 

The squared norm of J ± | a,b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsamaaBa aaleaacqGHXcqSaeqaaOWaa4HaaeqabaGaamyyaiaacYcacaWGIbaa caGLhWUaayPkJaaaaa@3DFC@  

J ± | a,b 2 = a,b| J ± J ± | a,b= a,b| J J ± | a,b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaauWaaeaaca WGkbWaaSbaaSqaaiabgglaXcqabaGcdaGhcaqabeaacaWGHbGaaiil aiaadkgaaiaawEa7caGLQmcaaiaawMa7caGLkWoadaahaaWcbeqaai aaikdaaaGccqGH9aqpdaGhbaqaaiaadggacaGGSaGaamOyaaqabiaa wMYicaGLhWoacaWGkbWaa0baaSqaaiabgglaXcqaaiaaccciaaGcca WGkbWaaSbaaSqaaiabgglaXcqabaGcdaGhcaqabeaacaWGHbGaaiil aiaadkgaaiaawEa7caGLQmcacqGH9aqpdaGhbaqaaiaadggacaGGSa GaamOyaaqabiaawMYicaGLhWoacaWGkbWaaSbaaSqaaiabloHiTbqa baGccaWGkbWaaSbaaSqaaiabgglaXcqabaGcdaGhcaqabeaacaWGHb GaaiilaiaadkgaaiaawEa7caGLQmcaaaa@6440@

and

J J ± =( J x i J y )( J x ±i J y )= J x 2 + J y 2 ±i[ J x , J y ] = J 2 J z 2 J z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaabbeaacaWGkb WaaSbaaSqaaiabloHiTbqabaGccaWGkbWaaSbaaSqaaiabgglaXcqa baGccqGH9aqpdaqadaqaaiaadQeadaWgaaWcbaGaamiEaaqabaGccq WItisBcaWGPbGaamOsamaaBaaaleaacaWG5baabeaaaOGaayjkaiaa wMcaamaabmaabaGaamOsamaaBaaaleaacaWG4baabeaakiabgglaXk aadMgacaWGkbWaaSbaaSqaaiaadMhaaeqaaaGccaGLOaGaayzkaaGa eyypa0JaamOsamaaDaaaleaacaWG4baabaGaaGOmaaaakiabgUcaRi aadQeadaqhaaWcbaGaamyEaaqaaiaaikdaaaGccqGHXcqScaWGPbWa amWaaeaacaWGkbWaaSbaaSqaaiaadIhaaeqaaOGaaiilaiaadQeada WgaaWcbaGaamyEaaqabaaakiaawUfacaGLDbaaaeaacqGH9aqpcaWG kbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaamOsamaaDaaaleaaca WG6baabaGaaGOmaaaakiabloHiTjabl+qiOjaadQeadaWgaaWcbaGa amOEaaqabaaaaaa@6800@

from which

J ± | a,b 2 = a,b| J 2 J z 2 J z | a,b=a b 2 b, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaauWaaeaaca WGkbWaaSbaaSqaaiabgglaXcqabaGcdaGhcaqabeaacaWGHbGaaiil aiaadkgaaiaawEa7caGLQmcaaiaawMa7caGLkWoadaahaaWcbeqaai aaikdaaaGccqGH9aqpdaGhbaqaaiaadggacaGGSaGaamOyaaqabiaa wMYicaGLhWoacaWGkbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0Iaam OsamaaDaaaleaacaWG6baabaGaaGOmaaaakiabloHiTjabl+qiOjaa dQeadaWgaaWcbaGaamOEaaqabaGcdaGhcaqabeaacaWGHbGaaiilai aadkgaaiaawEa7caGLQmcacqGH9aqpcaWGHbGaeyOeI0IaamOyamaa CaaaleqabaGaaGOmaaaakiabloHiTjabl+qiOjaadkgacaGGSaaaaa@5FB6@

recalling that a,b| a,b=1. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa4Xaaeaaca WGHbGaaiilaiaadkgaaeaacaWGHbGaaiilaiaadkgaaiaawMYicaGL hWUaayPkJaGaeyypa0JaaGymaiaac6caaaa@40DE@

Now a, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaacY caaaa@378D@  being the eigenvalue of a sum of squares of Hermitian operators, is necessarily nonnegative, and b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaaaa@36DE@  is real.  Hence for a given a,b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaacY cacaaMe8UaamOyaaaa@3A01@  is bounded: there must be a b max MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBa aaleaacaqGTbGaaeyyaiaabIhaaeqaaaaa@39D9@  and a (negative or zero) b min . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBa aaleaacaqGTbGaaeyAaiaab6gaaeqaaOGaaiOlaaaa@3A93@  But this must mean that

J + | a, b max 2 =a b max 2 b max =0, J | a, b min 2 =a b min 2 + b min =0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaqbda qaaiaadQeadaWgaaWcbaGaey4kaScabeaakmaaEiaabeqaaiaadgga caGGSaGaamOyamaaBaaaleaaciGGTbGaaiyyaiaacIhaaeqaaaGcca GLhWUaayPkJaaacaGLjWUaayPcSdWaaWbaaSqabeaacaaIYaaaaOGa eyypa0JaamyyaiabgkHiTiaadkgadaqhaaWcbaGaciyBaiaacggaca GG4baabaGaaGOmaaaakiabgkHiTiabl+qiOjaadkgadaWgaaWcbaGa ciyBaiaacggacaGG4baabeaakiabg2da9iaaicdacaGGSaaabaWaau WaaeaacaWGkbWaaSbaaSqaaiabgkHiTaqabaGcdaGhcaqabeaacaWG HbGaaiilaiaadkgadaWgaaWcbaGaciyBaiaacMgacaGGUbaabeaaaO Gaay5bSlaawQYiaaGaayzcSlaawQa7amaaCaaaleqabaGaaGOmaaaa kiabg2da9iaadggacqGHsislcaWGIbWaa0baaSqaaiGac2gacaGGPb GaaiOBaaqaaiaaikdaaaGccqGHRaWkcqWIpecAcaWGIbWaaSbaaSqa aiGac2gacaGGPbGaaiOBaaqabaGccqGH9aqpcaaIWaGaaiOlaaaaaa@7231@

Note that for a given a, b max MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaacY cacaaMe8UaaGjbVlaadkgadaWgaaWcbaGaaeyBaiaabggacaqG4baa beaaaaa@3E89@  and b min MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBa aaleaacaqGTbGaaeyAaiaab6gaaeqaaaaa@39D7@  are determined uniquely -- there cannot be two kets with the same a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DD@  but different b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaaaa@36DE@  annihilated by J + . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsamaaBa aaleaacqGHRaWkaeqaaOGaaiOlaaaa@3890@   It also follows immediately that a= b max ( b max + ) and  b min = b max. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabg2 da9iaadkgadaWgaaWcbaGaciyBaiaacggacaGG4baabeaakmaabmaa baGaamOyamaaBaaaleaaciGGTbGaaiyyaiaacIhaaeqaaOGaey4kaS IaeS4dHGgacaGLOaGaayzkaaGaaeiiaiaabggacaqGUbGaaeizaiaa bccacaWGIbWaaSbaaSqaaiGac2gacaGGPbGaaiOBaaqabaGccqGH9a qpcqGHsislcaWGIbWaaSbaaSqaaiGac2gacaGGHbGaaiiEaiaac6ca aeqaaaaa@51D6@  Furthermore, we know that if we keep operating on | a, b min MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa4Haaeqaba GaamyyaiaacYcacaWGIbWaaSbaaSqaaiGac2gacaGGPbGaaiOBaaqa baaakiaawEa7caGLQmcaaaa@3E12@  with J + , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsamaaBa aaleaacqGHRaWkaeqaaOGaaiilaaaa@388E@  we generate a sequence of kets with J z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsamaaBa aaleaacaWG6baabeaaaaa@37F1@  eigenvalues b min +, b min +2, b min +3, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBa aaleaaciGGTbGaaiyAaiaac6gaaeqaaOGaey4kaSIaeS4dHGMaaiil aiaaywW7caWGIbWaaSbaaSqaaiGac2gacaGGPbGaaiOBaaqabaGccq GHRaWkcaaIYaGaeS4dHGMaaiilaiaaywW7caWGIbWaaSbaaSqaaiGa c2gacaGGPbGaaiOBaaqabaGccqGHRaWkcaaIZaGaeS4dHGMaaiilai ablAcilbaa@4FAC@ .  This series must terminate, and the only possible way for that to happen is for b max MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBa aaleaaciGGTbGaaiyyaiaacIhaaeqaaaaa@39DE@  to be equal to b min +n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBa aaleaaciGGTbGaaiyAaiaac6gaaeqaaOGaey4kaSIaamOBaiabl+qi Obaa@3CE4@  with n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36EA@  an integer, from which it follows that  b max MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBa aaleaacaqGTbGaaeyyaiaabIhaaeqaaaaa@39D9@  is either an integer or half an odd integer times . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeS4dHGMaai Olaaaa@37D2@

At this point, we switch to the standard notation.  We have established that the eigenvalues of J z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsamaaBa aaleaacaWG6baabeaaaaa@37F1@  form a finite ladder, spacing MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeS4dHGgaaa@3720@ .  We write them as m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiabl+ qiObaa@3812@ , and j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaaaa@36E6@  is used to denote the maximum value of m, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiaacY caaaa@3799@  so the eigenvalue of J 2 ,a=j( j+1 ) 2 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsamaaCa aaleqabaGaaGOmaaaakiaacYcacaaMe8Uaamyyaiabg2da9iaadQga daqadaqaaiaadQgacqGHRaWkcaaIXaaacaGLOaGaayzkaaGaeS4dHG 2aaWbaaSqabeaacaaIYaaaaOGaaiOlaaaa@43B4@   Both j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaaaa@36E6@  and m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaaaa@36E9@  will be integers or half odd integers, but the spacing of the ladder of  m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaaaa@36E9@  values is always unity. Although we have been writing | a,b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa4Haaeqaba GaamyyaiaacYcacaWGIbaacaGLhWUaayPkJaaaaa@3B0A@  with a=j( j+1 ) 2 ,b=m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabg2 da9iaadQgadaqadaqaaiaadQgacqGHRaWkcaaIXaaacaGLOaGaayzk aaGaeS4dHG2aaWbaaSqabeaacaaIYaaaaOGaaiilaiaaysW7caaMc8 UaamOyaiabg2da9iaad2gacqWIpecAaaa@46D3@  we shall henceforth follow convention and write | j,m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa4Haaeqaba GaamOAaiaacYcacaWGTbaacaGLhWUaayPkJaaaaa@3B1E@ .

Summary

The operators J 2 , J z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOsayaala WaaWbaaSqabeaacaaIYaaaaOGaaiilaiaadQeadaWgaaWcbaGaamOE aaqabaaaaa@3A75@  have a common set of orthonormal eigenkets | j,m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa4Haaeqaba GaamOAaiaacYcacaWGTbaacaGLhWUaayPkJaaaaa@3B1E@ ,

J 2 | j,m=j( j+1 ) 2 | j,m J z | j,m=m| j,m j,m| j,m=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaabbeaaceWGkb GbaSaadaahaaWcbeqaaiaaikdaaaGcdaGhcaqabeaacaWGQbGaaiil aiaad2gaaiaawEa7caGLQmcacqGH9aqpcaWGQbWaaeWaaeaacaWGQb Gaey4kaSIaaGymaaGaayjkaiaawMcaaiabl+qiOnaaCaaaleqabaGa aGOmaaaakmaaEiaabeqaaiaadQgacaGGSaGaamyBaaGaay5bSlaawQ YiaaqaaiaadQeadaWgaaWcbaGaamOEaaqabaGcdaGhcaqabeaacaWG QbGaaiilaiaad2gaaiaawEa7caGLQmcacqGH9aqpcaWGTbGaeS4dHG 2aa4HaaeqabaGaamOAaiaacYcacaWGTbaacaGLhWUaayPkJaaabaWa a4raaeaacaWGQbGaaiilaiaad2gaaeqacaGLPmIaay5bSdWaaaGaae aacaWGQbGaaiilaiaad2gaaiaawQYiaiabg2da9iaaigdaaaaa@641E@

where j,m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiaacY cacaWGTbaaaa@3888@  are integers or half integers. The allowed quantum numbers m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaaaa@36E9@  form a ladder with step spacing unity, the maximum value of m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaaaa@36E9@  is j, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiaacY caaaa@3796@  the minimum value is j. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Iaam OAaiaac6caaaa@3885@    

Normalizing J+ and J-

 It is now straightforward to compute the normalization factors needed to find matrix elements:

J ± | j,m 2 = j,m| J 2 J z 2 J z | j,m=( j( j+1 ) 2 m( m±1 ) 2 ) j,m| j,m, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaauWaaeaaca WGkbWaaSbaaSqaaiabgglaXcqabaGcdaGhcaqabeaacaWGQbGaaiil aiaad2gaaiaawEa7caGLQmcaaiaawMa7caGLkWoadaahaaWcbeqaai aaikdaaaGccqGH9aqpdaGhbaqaaiaadQgacaGGSaGaamyBaaqabiaa wMYicaGLhWoacaWGkbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0Iaam OsamaaDaaaleaacaWG6baabaGaaGOmaaaakiabloHiTjabl+qiOjaa dQeadaWgaaWcbaGaamOEaaqabaGcdaGhcaqabeaacaWGQbGaaiilai aad2gaaiaawEa7caGLQmcacqGH9aqpdaqadaqaaiaadQgadaqadaqa aiaadQgacqGHRaWkcaaIXaaacaGLOaGaayzkaaGaeS4dHG2aaWbaaS qabeaacaaIYaaaaOGaeyOeI0IaamyBamaabmaabaGaamyBaiabggla XkaaigdaaiaawIcacaGLPaaacqWIpecAdaahaaWcbeqaaiaaikdaaa aakiaawIcacaGLPaaadaGhdaqaaiaadQgacaGGSaGaamyBaaqaaiaa dQgacaGGSaGaamyBaaGaayzkJiaawEa7caGLQmcacaGGSaaaaa@7366@

and j,m| j,m=1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa4Xaaeaaca WGQbGaaiilaiaad2gaaeaacaWGQbGaaiilaiaad2gaaiaawMYicaGL hWUaayPkJaGaeyypa0JaaGymaaaa@4054@ , so

J + | j,m= j( j+1 )m( m+1 ) | j,m+1 J | j,m= j( j+1 )m( m1 ) | j,m1. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaqqa6da aaaaGuLrgapeGaamOsamaaBaaaleaacqGHRaWkaeqaaOWaa4Haaeqa baGaamOAaiaacYcacaWGTbaacaGLhWUaayPkJaGaeyypa0ZaaOaaae aacaWGQbWaaeWaaeaacaWGQbGaey4kaSIaaGymaaGaayjkaiaawMca aiabgkHiTiaad2gadaqadaqaaiaad2gacqGHRaWkcaaIXaaacaGLOa GaayzkaaaaleqaaOGaeS4dHG2aa4HaaeqabaGaamOAaiaacYcacaWG TbGaey4kaSIaaGymaaGaay5bSlaawQYiaaqaaiaadQeadaWgaaWcba GaeyOeI0cabeaakmaaEiaabeqaaiaadQgacaGGSaGaamyBaaGaay5b SlaawQYiaiabg2da9maakaaabaGaamOAamaabmaabaGaamOAaiabgU caRiaaigdaaiaawIcacaGLPaaacqGHsislcaWGTbWaaeWaaeaacaWG TbGaeyOeI0IaaGymaaGaayjkaiaawMcaaaWcbeaakiabl+qiOnaaEi aabeqaaiaadQgacaGGSaGaamyBaiabgkHiTiaaigdaaiaawEa7caGL QmcacaGGUaaaaaa@6F3C@

With these formulas, and the base set of normalized eigenkets | j,m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa4Haaeqaba GaamOAaiaacYcacaWGTbaacaGLhWUaayPkJaaaaa@3B1E@ , we are in a position to construct explicit matrix representations of the angular momentum algebra for any integer or half integer value of angular momentum j. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiaac6 caaaa@3798@   

Historical note: the use of m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaaaa@36E9@  to denote the component of angular momentum in one direction came about because a Bohr-type electron in orbit is a current loop, with a magnetic moment parallel to its angular momentum, so the m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaaaa@36E9@  measured the component of magnetic moment in a chosen direction, usually along an external magnetic field, and m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaaaa@36E9@  is often called the magnetic quantum number.

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