itut ing ▁for ▁ ▁and ▁ ▁( and ▁taking ▁care ▁with ▁the ▁analysis ) ▁yield ▁He isen berg ' s ▁familiar ▁uncertainty ▁relation ▁for ▁ ▁and ▁, ▁as ▁usual . ▁ ▁Un certain ty ▁relation ▁for ▁angular ▁momentum ▁operators ▁ ▁For ▁the ▁angular ▁momentum ▁operators ▁, ▁etc ., ▁one ▁has ▁that ▁▁ ▁where ▁ ▁is ▁the ▁Le vi - C iv ita ▁symbol ▁and ▁simply ▁revers es ▁the ▁sign ▁of ▁the ▁answer ▁under ▁pair wise ▁inter change ▁of ▁the ▁indices . ▁An ▁analog ous ▁relation ▁holds ▁for ▁the ▁spin ▁operators . ▁ ▁Here , ▁for ▁ ▁and ▁, ▁in ▁angular ▁momentum ▁multiple ts ▁, ▁one ▁has , ▁for ▁the ▁trans verse ▁components ▁of ▁the ▁▁ ▁Cas imir ▁invariant ▁▁ ▁, ▁the ▁- sym metric ▁relations ▁ ▁, ▁ ▁as ▁well ▁as ▁ ▁. ▁ ▁Con sequently , ▁the ▁above ▁inequality ▁applied ▁to ▁this ▁commut ation ▁relation ▁spec ifies ▁▁ ▁hence ▁▁▁ ▁and ▁therefore ▁ ▁so , ▁then , ▁it ▁yields ▁useful ▁constraints ▁such ▁as ▁a ▁lower ▁bound ▁on ▁the ▁Cas imir ▁invariant : ▁, ▁and ▁hence ▁, ▁among ▁others . ▁ ▁See ▁also ▁ ▁Stone – von ▁Ne umann ▁theorem ▁Can on ical ▁quant ization ▁C CR ▁algebra ▁Lie ▁derivative ▁Mo yal ▁bra cket ▁ ▁References ▁▁ ▁. ▁ ▁. ▁ ▁Category : Quant um ▁mechan ics ▁Category : Math emat ical ▁physics ▁ ▁z h : <0xE5> <0xB0> <0x8D> <0xE6> <0x98> <0x93> <0xE9> <0x97> <0x9C> <0xE4> <0xBF> <0x82> <0x0A> </s> ▁Mik l ós ▁Gel b ai ▁( born ▁ 2 3 ▁December