Angular Momenta

Abstract

Q36(1) The spherical harmonics Y 1,1 ( θ , φ ), Y 1,0 ( θ , φ ), Y 1, −1( θ , φ ) are given by Eqs. (16.65) to (16.67). In Cartesian coordinates, these functions are given by 1 Y 1 , − 1 ( x , y , z ) = 3 8 π x − i y r , Y 1 , 0 ( x , y , z ) = 3 4 π z r , Y 1 , 1 ( x , y , z ) = − 3 8 π x + i y r . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0667.tif"/> (a) Using the expression in Cartesian coordinates for the quantised orbital angular momentum operator L ^ c z https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1694.tif"/> in Eq. (27.86), verify by explicit calculations that Y 1,1 ( x , y , z ), Y 1,0 ( x , y , z )and Y 1,-1 ( x , y , z ) are the eigenfunctions of L ^ c z https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1695.tif"/> corresponding to eigenvalues ħ , 0 and – ħ . (b) Consider the following coordinate transformations: x → z ′ , y → x ′ , z → y ′ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0668.tif"/> (1) 202 Show that the component of the orbital angular momentum operator along the z ′-direction is the same as that along the x -direction, i.e., show that L ^ c z ′ = L ^ c z https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1696.tif"/> . (2) Show that the simultaneous eigenfunctions of L ^ c 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1697.tif"/> and L ^ c x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1698.tif"/> corresponding to eigenvalues of L ^ c 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1699.tif"/> equal to 2 ħ 2 are given in Cartesian coordinates by X 1 , − 1 ( x , y , z ) = 3 8 π y − i z r , X 1 , 0 ( x , y , z ) = 3 4 π x r , X 1 , 1 ( x , y , z ) = − 3 8 π y + i z r . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0669.tif"/> SQ36(1)(a) In Cartesian coordinates we have 2 : p ^ x Y → 1 , − 1 : = − i ℏ 3 8 π ∂ ∂ x x − i y r = − i ℏ 3 8 π ( 1 r − x − i y r 2 ∂ ∂ x r ) = − i ℏ 3 8 π ( 1 r − x − i y r 2 x r ) = − i ℏ 3 8 π 1 r 3 ( r 2 − x 2 + i x y ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0670.tif"/> p ^ y Y → 1 , − 1 : = − i ℏ 3 8 π ∂ ∂ y x − i y r = − i ℏ 3 8 π ( − i r − x − i y r 2 ∂ ∂ y r ) = − i ℏ 3 8 π ( − i r − x − i y r 2 y r ) = − i ℏ 3 8 π 1 r 3 ( − i r 2 − x y + i y 2 ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0671.tif"/> L ^ z Y → 1 , − 1 : = ( x p ^ y − y p ^ x ) Y 1 , − 1 = − i ℏ 3 8 π 1 r 3 ( ( − i x r 2 − x 2 y + i x y 2 ) − ( y r 2 − y x 2 + i x y 2 ) ) = − i ℏ 3 8 π 1 r 3 ( − i x r 2 − y r 2 ) = − ℏ 3 8 π 1 r { x − i y } = − ℏ Y 1 , − 1 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0672.tif"/> 203 Next we have p ^ x Y → 1 , 0 : = − i ℏ 3 4 π ∂ ∂ x z r = − i ℏ 3 4 π ( − z r 2 ∂ ∂ x r ) = i ℏ 3 4 π z x r 3 . p ^ y Y → 1 , 0 : = − i ℏ 3 4 π ∂ ∂ y z r = − i ℏ 3 4 π ( − z r 2 ∂ ∂ y r ) = i ℏ 3 4 π z y r 3 . L ^ z Y → 1 , 0 : = ( x p ^ y − y p ^ x ) Y 1 , 0 = x ( i ℏ 3 4 π z y r 3 ) − y ( i ℏ 3 4 π z x r 3 ) = 0. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0673.tif"/> Finally we have p ^ x Y → 1 , 1 : = i ℏ 3 8 π ∂ ∂ x x + i y r = − i ℏ 3 8 π ( 1 r − x + i y r 2 ∂ ∂ x r ) = i ℏ 3 8 π ( 1 r − x + i y r 2 x r ) = i ℏ 3 8 π 1 r 3 ( r 2 − x 2 − i x y ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0674.tif"/> p ^ y Y → 1 , 1 : = i ℏ 3 8 π ∂ ∂ y x + i y r = i ℏ 3 8 π ( i r − x + i y r 2 ∂ ∂ y r ) = i ℏ 3 8 π ( i r − x + i y r 2 y r ) = i ℏ 3 8 π 1 r 3 ( i r 2 − x y − i y 2 ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0675.tif"/> L ^ z Y → 1 , 1 : = ( x p ^ y − y p ^ x ) Y 1 , 1 = i ℏ 3 8 π 1 r 3 { ( i x r 2 − x 2 y − i x y 2 ) = − ( y r 2 − y x 2 − i x y 2 ) } = i ℏ 3 8 π 1 r 3 { i x r 2 − y r 2 } = − ℏ 3 8 π 1 r ( x + i y ) = ℏ Y 1 , 1 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0676.tif"/> 204 SQ36(1)(b) (1) Consider a transformation from the original coordinate system ( x , y , z ) to a new coordinate system ( x ′, y ′, z ′): x → z ′ , y → x ′ , z → y ′ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0677.tif"/> We can see that the new z ′-axis coincides with the original x -axis, and that r = x 2 + y 2 + z 2 = x ′ 2 + y ′ 2 + z ′ 2 = r ′ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0678.tif"/> Physically we expect the angular momentum about the x -axis to be the same as the angular momentum about the z′ -axis. We can confirm this mathematically by showing that L ^ x = y p ^ z − z p ^ y = − i ℏ ( y ∂ ∂ z − z ∂ ∂ y ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0679.tif"/> is equal to L ^ z ′ = x ′ p ^ y ′ − y ′ p ^ x ′ = − i ℏ ( x ′ ∂ ∂ y ′ − y ′ ∂ ∂ x ′ ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0680.tif"/> This is obvious since x′ = y , y′ = z . It follows that the eigenfunctions and eigenvalues of L ^ x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1700.tif"/> are the same in terms of the dashed coordinates as that of L ^ z ′ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1701.tif"/> . (2) We know the eigenfunctions Y ℓ , m ℓ ′ ( x ′ , y ′ , z ′ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1702.tif"/> and eigenvalues of L ^ z ′ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1703.tif"/> in the transformed coordinates (x′, y′, z′) as they are of the same form as those of the eigenfunctions and eigenvalues of L ^ z https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1704.tif"/> in the original coordinates, i.e., we have Y 1 , − 1 ′ ( x ′ , y ′ , z ′ ) = 3 8 π x ′ − i y ′ r ′ , Y 1 , 0 ′ ( x ′ , y ′ , z ′ ) = 3 4 π z ′ r ′ Y 1 , 1 ′ ( x ′ , y ′ , z ′ ) = − 3 8 π x ′ + i y ′ r ′ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0681.tif"/> 205 Since L ^ z ′ = L ^ x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1705.tif"/> these are also eigenfunctions of L ^ x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1706.tif"/> Written in terms of the original coordinates these functions become Y 1 , − 1 ′ = 3 8 π y − i z r , Y 1 , 0 ′ = 3 4 π x r , Y 1 , 1 ′ = − 3 8 π y + i z r . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0682.tif"/> We can conclude that the required eigenfunctions L ^ x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1707.tif"/> denoted more conveniently by X 1,1 , X 1,0 , X 1, −1 are X 1 , − 1 = 3 8 π y − i z r , X 1 , 0 = 3 4 π x r , X 1 , 1 = − 3 8 π y + i z r . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0683.tif"/> Q36(2) Suppose L ^ c 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1708.tif"/> and L ^ c z https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1709.tif"/> are measured giving the eigenvalues 2 ħ 2 and – ħ , respectively. A measurement of L ^ c x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1710.tif"/> is then made. What are the possible results of the measurement of L ^ c x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1711.tif"/> ? Find the probability of each of these possible results.