M.S. Rogalski, Stuart B. Palmer
The orbital angular momentum operator has been introduced by Postulate 2 as a vector operator L → ^ = r → × p → ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/inline-eqn8_1_B.tif"/> whose Cartesian components, Eq.(2.18) , correspond to those of the classical observable for a single particle. As a consequence of Postulate 4, the components L ^ x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/inline-eqn8_2_B.tif"/> , L ^ y https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/inline-eqn8_3_B.tif"/> and L ^ z https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/inline-eqn8_4_B.tif"/> do not commute with one another, Eq.(4.47) , and all commute with L ^ 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/inline-eqn8_5_B.tif"/> , and this implies that the eigenstates of L → ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/inline-eqn8_6_B.tif"/> are simultaneous eigenfunctions of L ^ 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/inline-eqn8_7_B.tif"/> and of one component L ^ i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/inline-eqn8_8_B.tif"/> , at a time. Similar properties are exhibited by the angular momentum vector operator of a system of particles, which is defined as the vector sum over the individual particle operators: J → ^ = ∑ p L → ^ ( p ) , J ^ 2 = J ^ x 2 + J ^ y 2 + J ^ z 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn8_108_B.tif"/> such that, because the coordinate and momentum operator for two different particles are commuting operators, we have: 8.1 [ J ^ x , J ^ y ] = [ ∑ p L ^ x ( p ) , ∑ q L ^ y ( q ) ] = ∑ p ∑ q [ L ^ x ( p ) , L ^ y ( q ) ] = ∑ p [ L ^ x ( p ) , L ^ y ( q ) ] = i h ∑ p [ L ^ z ( p ) = i h J ^ z https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn8_1_B.tif"/> and similarly: 8.2 [ J ^ y , J ^ z ] = i h J ^ x , [ J ^ z , J ^ x ] = i h J ^ y https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn8_2_B.tif"/>
CRC Press eBooks•Nov 5, 2020•0 citations