CRC Press eBooks

Rotations and Angular Momentum

Richard P. Feynman

Published Feb 19, 2018• 0 citations

Abstract

In the last lecture we spoke about an apparatus that produced an object in condition a: https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429492501/e6a0c676-1f8d-4684-97a0-6bf3353f7ce0/content/pg11_1.tif"/>

DOI: 10.1201/9780429492501-3

Related papers

Angular Momentum

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 eBooksNov 5, 20200 citations

Rules of Composition of Angular Momentum

Richard P. Feynman

A spin 1/2 state is characterized by two amplitudes. In general a = a + (1/2) + a – (–1/2) where (1/2) stands for ( 1 0 ) , ( − 1 / 2 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429492501/e6a0c676-1f8d-4684-97a0-6bf3353f7ce0/content/eq43.tif"/> for ( 0 1 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429492501/e6a0c676-1f8d-4684-97a0-6bf3353f7ce0/content/eq44.tif"/> , and a for ( a + a − ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429492501/e6a0c676-1f8d-4684-97a0-6bf3353f7ce0/content/eq45.tif"/> .

CRC Press eBooksFeb 19, 20180 citations

Chapter 11. Angular Momentum and the Position Operators

Dennis Morris

De Gruyter eBooksMay 27, 20160 citations

Spin-1/2 Particles

Richard P. Feynman

Recall the two-component spinor, whose behavior under space rotations of angle θ about unit vector n was described by the operator exp (i θ n · M), where M = (1/2)σ. In Problem 21-3 you were concerned with the behavior of this spinor under Lorentz transformation. As was the case in space rotations it is sufficient to consider infinitesimal transformations. We write the corresponding operator as 1 + i ( v / c ) ⋅ N https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429492501/e6a0c676-1f8d-4684-97a0-6bf3353f7ce0/content/eq413.tif"/>

CRC Press eBooksFeb 19, 20180 citations

Rotations and angular momentum

Bipin R. Desai

Cambridge University Press eBooksDec 3, 20090 citations

Angular Momenta

K. Kong Wan

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.

Aug 27, 20200 citations