1. Quantum Two

3. Angular Momentum and Rotations

4. Angular Momentum and Rotations
The Emergence of Orbital Angular Momentum as The Generator of Rotations

5. In this segment we show how the generator of rotations 𝐽 relates to the usual definition of angular momentum for, e.g., a single spinless particle.
This is most easily done by working in the position representation.
For example, let 𝜓 𝑟 =〈 𝑟 |𝜓〉 be the wave function associated with an arbitrary state |𝜓〉 of a single spinless particle.
Under a rotation R, the state |𝜓〉 is taken onto a new state 𝜓′ = 𝑈 𝐑 |𝜓〉 described by a different wave function 𝜓′ 𝑟 =〈 𝑟 |𝜓′〉. 𝑢
𝜓 𝑟 O

6. In this segment we show how the generator of rotations 𝐽 relates to the usual definition of angular momentum for, e.g., a single spinless particle.
This is most easily done by working in the position representation.
For example, let 𝜓 𝑟 =〈 𝑟 |𝜓〉 be the wave function associated with an arbitrary state |𝜓〉 of a single spinless particle.
Under a rotation R, the state |𝜓〉 is taken onto a new state 𝜓′ = 𝑈 𝐑 |𝜓〉 described by a different wave function 𝜓′ 𝑟 =〈 𝑟 |𝜓′〉. 𝑢
𝜓 𝑟 O

7. In this segment we show how the generator of rotations 𝐽 relates to the usual definition of angular momentum for, e.g., a single spinless particle.
This is most easily done by working in the position representation.
For example, let 𝜓 𝑟 =〈 𝑟 |𝜓〉 be the wave function associated with an arbitrary state |𝜓〉 of a single spinless particle.
Under a rotation R, the state |𝜓〉 is taken onto a new state 𝜓′ = 𝑈 𝐑 |𝜓〉 described by a different wave function 𝜓′ 𝑟 =〈 𝑟 |𝜓′〉. 𝑢
𝜓 𝑟 O

8. In this segment we show how the generator of rotations 𝐽 relates to the usual definition of angular momentum for, e.g., a single spinless particle.
This is most easily done by working in the position representation.
For example, let 𝜓 𝑟 =〈 𝑟 |𝜓〉 be the wave function associated with an arbitrary state |𝜓〉 of a single spinless particle.
Under a rotation R, the state |𝜓〉 is taken onto a new state 𝜓′ = 𝑈 𝐑 |𝜓〉 described by a different wave function 𝜓′ 𝑟 =〈 𝑟 |𝜓′〉. 𝑢
𝜓 𝑟 O

9. In this segment we show how the generator of rotations 𝐽 relates to the usual definition of angular momentum for, e.g., a single spinless particle.
This is most easily done by working in the position representation.
Let 𝜓 𝑟 =〈 𝑟 |𝜓〉 be the wave function associated with an arbitrary state |𝜓〉 of a single spinless particle.
Under a rotation R, the state |𝜓〉 is taken onto a new state 𝜓′ = 𝑈 𝐑 |𝜓〉 described by a different wave function 𝜓′ 𝑟 =〈 𝑟 |𝜓′〉. 𝑢
𝜓′ 𝑟 O

10. In such a rotation, the value of the unrotated wave function 𝜓 at a point 𝑟𝑢
𝜓 𝑟 O

11. In such a rotation, the value of the unrotated wave function 𝜓 at a point 𝑟 must be the same as the value of the rotated wave function 𝜓′ at the rotated point
𝑟 ′ = 𝐴 𝐑 𝑟 .
Thus we can write
𝜓 𝑟 =𝜓′ 𝑟 ′ =𝜓′ 𝐴 𝐑 𝑟
Noting that 𝑟 = 𝐴 𝐑 −1 𝑟 ′ we can also write
𝜓( 𝐴 𝐑 −1 𝑟 ′)=𝜓′ 𝑟 ′
Now drop the primes on 𝑟 ′, and exchange both sides of the equation to find that
𝜓′ 𝑟 =𝜓( 𝐴 𝐑 −1 𝑟 ) 𝑢
𝜓′ 𝑟 ′ O

12. In such a rotation, the value of the unrotated wave function 𝜓 at a point 𝑟 must be the same as the value of the rotated wave function 𝜓′ at the rotated point
𝑟 ′ = 𝐴 𝐑 𝑟 .
Thus we can write
𝜓 𝑟 =𝜓′ 𝑟 ′ =𝜓′ 𝐴 𝐑 𝑟
Noting that 𝑟 = 𝐴 𝐑 −1 𝑟 ′ we can also write
𝜓( 𝐴 𝐑 −1 𝑟 ′)=𝜓′ 𝑟 ′
Now drop the primes on 𝑟 ′, and exchange both sides of the equation to find that
𝜓′ 𝑟 =𝜓( 𝐴 𝐑 −1 𝑟 ) 𝑢
𝜓′ 𝑟 ′ O

13. In such a rotation, the value of the unrotated wave function 𝜓 at a point 𝑟 must be the same as the value of the rotated wave function 𝜓′ at the rotated point
𝑟 ′ = 𝐴 𝐑 𝑟 .
Thus we can write
𝜓 𝑟 =𝜓′ 𝑟 ′ =𝜓′ 𝐴 𝐑 𝑟
Noting that 𝑟 = 𝐴 𝐑 −1 𝑟 ′ we can also write
𝜓( 𝐴 𝐑 −1 𝑟 ′)=𝜓′ 𝑟 ′
Now drop the primes on 𝑟 ′, and exchange both sides of the equation to find that
𝜓′ 𝑟 =𝜓( 𝐴 𝐑 −1 𝑟 ) 𝑢
𝜓′ 𝑟 ′ O

14. In such a rotation, the value of the unrotated wave function 𝜓 at a point 𝑟 must be the same as the value of the rotated wave function 𝜓′ at the rotated point
𝑟 ′ = 𝐴 𝐑 𝑟 .
Thus we can write
𝜓 𝑟 =𝜓′ 𝑟 ′ =𝜓′ 𝐴 𝐑 𝑟
Noting that 𝑟 = 𝐴 𝐑 −1 𝑟 ′ we can also write
𝜓( 𝐴 𝐑 −1 𝑟 ′)=𝜓′ 𝑟 ′
Now drop the primes on 𝑟 ′, and exchange both sides of the equation to find that
𝜓′ 𝑟 =𝜓( 𝐴 𝐑 −1 𝑟 ) 𝑢
𝜓′ 𝑟 ′ O

15. In such a rotation, the value of the unrotated wave function 𝜓 at a point 𝑟 must be the same as the value of the rotated wave function 𝜓′ at the rotated point
𝑟 ′ = 𝐴 𝐑 𝑟 .
Thus we can write
𝜓 𝑟 =𝜓′ 𝑟 ′ =𝜓′ 𝐴 𝐑 𝑟
Noting that 𝑟 = 𝐴 𝐑 −1 𝑟 ′ we can also write
𝜓( 𝐴 𝐑 −1 𝑟 ′)=𝜓′ 𝑟 ′
Now drop the primes on 𝑟 ′, and exchange both sides of the equation to find that
𝜓′ 𝑟 =𝜓( 𝐴 𝐑 −1 𝑟 ) 𝑢
𝜓′ 𝑟 ′ O

16. In such a rotation, the value of the unrotated wave function 𝜓 at a point 𝑟 must be the same as the value of the rotated wave function 𝜓′ at the rotated point
𝑟 ′ = 𝐴 𝐑 𝑟 .
Thus we can write
𝜓 𝑟 =𝜓′ 𝑟 ′ =𝜓′ 𝐴 𝐑 𝑟
Noting that 𝑟 = 𝐴 𝐑 −1 𝑟 ′ we can also write
𝜓( 𝐴 𝐑 −1 𝑟 ′)=𝜓′ 𝑟 ′
Now drop the primes on 𝑟 ′, and exchange both sides of the equation to find that
𝜓′ 𝑟 =𝜓( 𝐴 𝐑 −1 𝑟 ) 𝑢
𝜓′ 𝑟 ′ O

17. In such a rotation, the value of the unrotated wave function 𝜓 at a point 𝑟 must be the same as the value of the rotated wave function 𝜓′ at the rotated point
𝑟 ′ = 𝐴 𝐑 𝑟 .
Thus we can write
𝜓 𝑟 =𝜓′ 𝑟 ′ =𝜓′ 𝐴 𝐑 𝑟
Noting that 𝑟 = 𝐴 𝐑 −1 𝑟 ′ we can also write
𝜓( 𝐴 𝐑 −1 𝑟 ′)=𝜓′ 𝑟 ′
Now drop the primes on 𝑟 ′, and exchange both sides of the equation to find that
𝜓′ 𝑟 =𝜓( 𝐴 𝐑 −1 𝑟 ) 𝑢
𝜓′ 𝑟 ′ O

18. Suppose now that the rotation under consideration is an infinitesimal rotation about the axis 𝑢 through an angle 𝛿𝛼, for which
𝐴 𝐑 𝑟 = 𝑟 +𝛿𝛼( 𝑢 × 𝑟 ) The inverse rotation 𝐴 𝐑 −1 is then given by
𝐴 𝐑 −1 𝑟 = 𝑟 −𝛿𝛼( 𝑢 × 𝑟 ) Thus, under such a rotation, we can write
𝜓′ 𝑟 =𝜓( 𝐴 𝐑 −1 𝑟 )
=𝜓 𝑟 −𝛿𝛼( 𝑢 × 𝑟 ) =𝜓( 𝑟 +𝛿 𝑟 )
where
𝛿 𝑟 =−𝛿𝛼( 𝑢 × 𝑟 ) is an infinitesimal displacement. 𝑢 O

19. Suppose now that the rotation under consideration is an infinitesimal rotation about the axis 𝑢 through an angle 𝛿𝛼, for which
𝐴 𝐑 𝑟 = 𝑟 +𝛿𝛼( 𝑢 × 𝑟 ) The inverse rotation 𝐴 𝐑 −1 is then given by
𝐴 𝐑 −1 𝑟 = 𝑟 −𝛿𝛼( 𝑢 × 𝑟 ) Thus, under such a rotation, we can write
𝜓′ 𝑟 =𝜓( 𝐴 𝐑 −1 𝑟 )
=𝜓 𝑟 −𝛿𝛼( 𝑢 × 𝑟 ) =𝜓( 𝑟 +𝛿 𝑟 )
where
𝛿 𝑟 =−𝛿𝛼( 𝑢 × 𝑟 ) is an infinitesimal displacement. 𝑢 O

20. Suppose now that the rotation under consideration is an infinitesimal rotation about the axis 𝑢 through an angle 𝛿𝛼, for which
𝐴 𝐑 𝑟 = 𝑟 +𝛿𝛼( 𝑢 × 𝑟 ) The inverse rotation 𝐴 𝐑 −1 is then given by
𝐴 𝐑 −1 𝑟 = 𝑟 −𝛿𝛼( 𝑢 × 𝑟 ) Thus, under such a rotation, we can write
𝜓′ 𝑟 =𝜓( 𝐴 𝐑 −1 𝑟 )
=𝜓 𝑟 −𝛿𝛼( 𝑢 × 𝑟 ) =𝜓( 𝑟 +𝛿 𝑟 )
where
𝛿 𝑟 =−𝛿𝛼( 𝑢 × 𝑟 ) is an infinitesimal displacement. 𝑢 O

21. Suppose now that the rotation under consideration is an infinitesimal rotation about the axis 𝑢 through an angle 𝛿𝛼, for which
𝐴 𝐑 𝑟 = 𝑟 +𝛿𝛼( 𝑢 × 𝑟 ) The inverse rotation 𝐴 𝐑 −1 is then given by
𝐴 𝐑 −1 𝑟 = 𝑟 −𝛿𝛼( 𝑢 × 𝑟 ) Thus, under such a rotation, we can write
𝜓′ 𝑟 =𝜓( 𝐴 𝐑 −1 𝑟 )
=𝜓 𝑟 −𝛿𝛼( 𝑢 × 𝑟 ) =𝜓( 𝑟 +𝛿 𝑟 )
where
𝛿 𝑟 =−𝛿𝛼( 𝑢 × 𝑟 ) is an infinitesimal displacement. 𝑢 O

22. Suppose now that the rotation under consideration is an infinitesimal rotation about the axis 𝑢 through an angle 𝛿𝛼, for which
𝐴 𝐑 𝑟 = 𝑟 +𝛿𝛼( 𝑢 × 𝑟 ) The inverse rotation 𝐴 𝐑 −1 is then given by
𝐴 𝐑 −1 𝑟 = 𝑟 −𝛿𝛼( 𝑢 × 𝑟 ) Thus, under such a rotation, we can write
𝜓′ 𝑟 =𝜓( 𝐴 𝐑 −1 𝑟 )
=𝜓 𝑟 −𝛿𝛼( 𝑢 × 𝑟 ) =𝜓( 𝑟 +𝛿 𝑟 )
where
𝛿 𝑟 =−𝛿𝛼( 𝑢 × 𝑟 ) is an infinitesimal displacement. 𝑢 O

23. Suppose now that the rotation under consideration is an infinitesimal rotation about the axis 𝑢 through an angle 𝛿𝛼, for which
𝐴 𝐑 𝑟 = 𝑟 +𝛿𝛼( 𝑢 × 𝑟 ) The inverse rotation 𝐴 𝐑 −1 is then given by
𝐴 𝐑 −1 𝑟 = 𝑟 −𝛿𝛼( 𝑢 × 𝑟 ) Thus, under such a rotation, we can write
𝜓′ 𝑟 =𝜓( 𝐴 𝐑 −1 𝑟 )
=𝜓 𝑟 −𝛿𝛼( 𝑢 × 𝑟 ) =𝜓( 𝑟 +𝛿 𝑟 )
where
𝛿 𝑟 =−𝛿𝛼( 𝑢 × 𝑟 ) is an infinitesimal displacement. 𝑢 O

24. Expanding
𝜓 𝑟 +𝛿 𝑟 =𝜓 𝑟 +𝛿 𝑟 ∙ 𝛻 𝜓 𝑟
we deduce that
𝜓 ′ 𝑟 =𝜓 𝑟 +𝛿 𝑟 ∙ 𝛻 𝜓 𝑟 =𝜓 𝑟 −𝛿𝛼 ( 𝑢 × 𝑟 )∙ 𝛻 𝜓 𝑟
Using the easily proven identity
( 𝑢 × 𝑟 )∙ 𝛻 𝜓 𝑟 = 𝑢 ∙( 𝑟 × 𝛻 ) 𝜓 𝑟
yields the relation
𝜓 ′ 𝑟 =𝜓 𝑟 −𝛿𝛼 𝑢 ∙( 𝑟 × 𝛻 ) 𝜓 𝑟 𝑢 O

25. Expanding
𝜓 𝑟 +𝛿 𝑟 =𝜓 𝑟 +𝛿 𝑟 ∙ 𝛻 𝜓 𝑟
we deduce that
𝜓 ′ 𝑟 =𝜓 𝑟 +𝛿 𝑟 ∙ 𝛻 𝜓 𝑟 =𝜓 𝑟 −𝛿𝛼 ( 𝑢 × 𝑟 )∙ 𝛻 𝜓 𝑟
Using the easily proven identity
( 𝑢 × 𝑟 )∙ 𝛻 𝜓 𝑟 = 𝑢 ∙( 𝑟 × 𝛻 ) 𝜓 𝑟
yields the relation
𝜓 ′ 𝑟 =𝜓 𝑟 −𝛿𝛼 𝑢 ∙( 𝑟 × 𝛻 ) 𝜓 𝑟 𝑢 O

26. Expanding
𝜓 𝑟 +𝛿 𝑟 =𝜓 𝑟 +𝛿 𝑟 ∙ 𝛻 𝜓 𝑟
we deduce that
𝜓 ′ 𝑟 =𝜓 𝑟 +𝛿 𝑟 ∙ 𝛻 𝜓 𝑟 =𝜓 𝑟 −𝛿𝛼 ( 𝑢 × 𝑟 )∙ 𝛻 𝜓 𝑟
Using the easily proven identity
( 𝑢 × 𝑟 )∙ 𝛻 𝜓 𝑟 = 𝑢 ∙( 𝑟 × 𝛻 ) 𝜓 𝑟
yields the relation
𝜓 ′ 𝑟 =𝜓 𝑟 −𝛿𝛼 𝑢 ∙( 𝑟 × 𝛻 ) 𝜓 𝑟 𝑢 O

27. Expanding
𝜓 𝑟 +𝛿 𝑟 =𝜓 𝑟 +𝛿 𝑟 ∙ 𝛻 𝜓 𝑟
we deduce that
𝜓 ′ 𝑟 =𝜓 𝑟 +𝛿 𝑟 ∙ 𝛻 𝜓 𝑟 =𝜓 𝑟 −𝛿𝛼 ( 𝑢 × 𝑟 )∙ 𝛻 𝜓 𝑟
Using the easily proven identity
( 𝑢 × 𝑟 )∙ 𝛻 𝜓 𝑟 = 𝑢 ∙( 𝑟 × 𝛻 ) 𝜓 𝑟
yields the relation
𝜓 ′ 𝑟 =𝜓 𝑟 −𝛿𝛼 𝑢 ∙( 𝑟 × 𝛻 ) 𝜓 𝑟 𝑢 O

28. Expanding
𝜓 𝑟 +𝛿 𝑟 =𝜓 𝑟 +𝛿 𝑟 ∙ 𝛻 𝜓 𝑟
we deduce that
𝜓 ′ 𝑟 =𝜓 𝑟 +𝛿 𝑟 ∙ 𝛻 𝜓 𝑟 =𝜓 𝑟 −𝛿𝛼 ( 𝑢 × 𝑟 )∙ 𝛻 𝜓 𝑟
Using the easily proven identity
( 𝑢 × 𝑟 )∙ 𝛻 𝜓 𝑟 = 𝑢 ∙( 𝑟 × 𝛻 ) 𝜓 𝑟
We obtain the relation
𝜓 ′ 𝑟 =𝜓 𝑟 −𝛿𝛼 𝑢 ∙( 𝑟 × 𝛻 ) 𝜓 𝑟 𝑢 O

29. 𝜓 ′ 𝑟 =𝜓 𝑟 −𝛿𝛼 𝑢 ∙( 𝑟 × 𝛻 ) 𝜓 𝑟
= [1−𝛿𝛼 𝑢 ∙( 𝑟 × 𝛻 )] 𝜓 𝑟
= (1−𝑖 𝛿𝛼 𝑢 ∙[( 𝑟 ×(−𝑖 𝛻 )]) 𝜓 𝑟
Introducing Dirac notation, this implies
〈 𝑟 |𝜓′〉=〈 𝑟 | 𝟏−𝑖 𝛿𝛼 𝑢 ∙ 𝑅 × 𝐾 |𝜓〉
Being true for all 〈 𝑟 |, and identifying ℓ = 𝑅 × 𝐾 we find that under this infinitesimal rotation
|𝜓′〉= [𝟏−𝑖 𝛿𝛼 ℓ ∙ 𝑢 ]|𝜓〉= 𝑈 𝑢 𝛿𝛼 |𝜓〉
Comparing this to our general result
𝑈 𝑢 𝛿𝛼 =𝟏−𝑖 𝛿𝛼 𝐽 ∙ 𝑢
We deduce that for a single particle 𝐽 = ℓ = 𝑅 × 𝐾 . 𝑢 O

30. 𝜓 ′ 𝑟 =𝜓 𝑟 −𝛿𝛼 𝑢 ∙( 𝑟 × 𝛻 ) 𝜓 𝑟
= [1−𝛿𝛼 𝑢 ∙( 𝑟 × 𝛻 )] 𝜓 𝑟
= (1−𝑖 𝛿𝛼 𝑢 ∙[( 𝑟 ×(−𝑖 𝛻 )]) 𝜓 𝑟
Introducing Dirac notation, this implies
〈 𝑟 |𝜓′〉=〈 𝑟 | 𝟏−𝑖 𝛿𝛼 𝑢 ∙ 𝑅 × 𝐾 |𝜓〉
Being true for all 〈 𝑟 |, and identifying ℓ = 𝑅 × 𝐾 we find that under this infinitesimal rotation
|𝜓′〉= [𝟏−𝑖 𝛿𝛼 ℓ ∙ 𝑢 ]|𝜓〉= 𝑈 𝑢 𝛿𝛼 |𝜓〉
Comparing this to our general result
𝑈 𝑢 𝛿𝛼 =𝟏−𝑖 𝛿𝛼 𝐽 ∙ 𝑢
We deduce that for a single particle 𝐽 = ℓ = 𝑅 × 𝐾 . 𝑢 O

31. 𝜓 ′ 𝑟 =𝜓 𝑟 −𝛿𝛼 𝑢 ∙( 𝑟 × 𝛻 ) 𝜓 𝑟
= [1−𝛿𝛼 𝑢 ∙( 𝑟 × 𝛻 )] 𝜓 𝑟
= (1−𝑖 𝛿𝛼 𝑢 ∙[( 𝑟 ×(−𝑖 𝛻 )]) 𝜓 𝑟
Introducing Dirac notation, this implies
〈 𝑟 |𝜓′〉=〈 𝑟 | 𝟏−𝑖 𝛿𝛼 𝑢 ∙ 𝑅 × 𝐾 |𝜓〉
Being true for all 〈 𝑟 |, and identifying ℓ = 𝑅 × 𝐾 we find that under this infinitesimal rotation
|𝜓′〉= [𝟏−𝑖 𝛿𝛼 ℓ ∙ 𝑢 ]|𝜓〉= 𝑈 𝑢 𝛿𝛼 |𝜓〉
Comparing this to our general result
𝑈 𝑢 𝛿𝛼 =𝟏−𝑖 𝛿𝛼 𝐽 ∙ 𝑢
We deduce that for a single particle 𝐽 = ℓ = 𝑅 × 𝐾 . 𝑢 O

32. 𝜓 ′ 𝑟 =𝜓 𝑟 −𝛿𝛼 𝑢 ∙( 𝑟 × 𝛻 ) 𝜓 𝑟
= [1−𝛿𝛼 𝑢 ∙( 𝑟 × 𝛻 )] 𝜓 𝑟
= (1−𝑖 𝛿𝛼 𝑢 ∙[( 𝑟 ×(−𝑖 𝛻 )]) 𝜓 𝑟
Introducing Dirac notation, this implies
〈 𝑟 |𝜓′〉=〈 𝑟 | 𝟏−𝑖 𝛿𝛼 𝑢 ∙ 𝑅 × 𝐾 |𝜓〉
Being true for all 〈 𝑟 |, and identifying ℓ = 𝑅 × 𝐾 we find that under this infinitesimal rotation
|𝜓′〉= [𝟏−𝑖 𝛿𝛼 ℓ ∙ 𝑢 ]|𝜓〉= 𝑈 𝑢 𝛿𝛼 |𝜓〉
Comparing this to our general result
𝑈 𝑢 𝛿𝛼 =𝟏−𝑖 𝛿𝛼 𝐽 ∙ 𝑢
We deduce that for a single particle 𝐽 = ℓ = 𝑅 × 𝐾 . 𝑢 O

33. 𝜓 ′ 𝑟 =𝜓 𝑟 −𝛿𝛼 𝑢 ∙( 𝑟 × 𝛻 ) 𝜓 𝑟
= [1−𝛿𝛼 𝑢 ∙( 𝑟 × 𝛻 )] 𝜓 𝑟
= (1−𝑖 𝛿𝛼 𝑢 ∙[( 𝑟 ×(−𝑖 𝛻 )]) 𝜓 𝑟
Introducing Dirac notation, this implies
〈 𝑟 |𝜓′〉=〈 𝑟 | 𝟏−𝑖 𝛿𝛼 𝑢 ∙ 𝑅 × 𝐾 |𝜓〉
Being true for all 〈 𝑟 |, and identifying ℓ = 𝑅 × 𝐾 we find that under this infinitesimal rotation
|𝜓′〉= [𝟏−𝑖 𝛿𝛼 ℓ ∙ 𝑢 ]|𝜓〉= 𝑈 𝑢 𝛿𝛼 |𝜓〉
Comparing this to our general result
𝑈 𝑢 𝛿𝛼 =𝟏−𝑖 𝛿𝛼 𝐽 ∙ 𝑢
We deduce that for a single particle 𝐽 = ℓ = 𝑅 × 𝐾 . 𝑢 O

34. 𝜓 ′ 𝑟 =𝜓 𝑟 −𝛿𝛼 𝑢 ∙( 𝑟 × 𝛻 ) 𝜓 𝑟
= [1−𝛿𝛼 𝑢 ∙( 𝑟 × 𝛻 )] 𝜓 𝑟
= (1−𝑖 𝛿𝛼 𝑢 ∙[( 𝑟 ×(−𝑖 𝛻 )]) 𝜓 𝑟
Introducing Dirac notation, this implies
〈 𝑟 |𝜓′〉=〈 𝑟 | 𝟏−𝑖 𝛿𝛼 𝑢 ∙ 𝑅 × 𝐾 |𝜓〉
Being true for all 〈 𝑟 |, and identifying ℓ = 𝑅 × 𝐾 we find that under this infinitesimal rotation
|𝜓′〉= [𝟏−𝑖 𝛿𝛼 ℓ ∙ 𝑢 ]|𝜓〉= 𝑈 𝑢 𝛿𝛼 |𝜓〉
Comparing this to our general result
𝑈 𝑢 𝛿𝛼 =𝟏−𝑖 𝛿𝛼 𝐽 ∙ 𝑢
We deduce that for a single particle 𝐽 = ℓ = 𝑅 × 𝐾 . 𝑢 O

35. 𝜓 ′ 𝑟 =𝜓 𝑟 −𝛿𝛼 𝑢 ∙( 𝑟 × 𝛻 ) 𝜓 𝑟
= [1−𝛿𝛼 𝑢 ∙( 𝑟 × 𝛻 )] 𝜓 𝑟
= (1−𝑖 𝛿𝛼 𝑢 ∙[( 𝑟 ×(−𝑖 𝛻 )]) 𝜓 𝑟
Introducing Dirac notation, this implies
〈 𝑟 |𝜓′〉=〈 𝑟 | 𝟏−𝑖 𝛿𝛼 𝑢 ∙ 𝑅 × 𝐾 |𝜓〉
Being true for all 〈 𝑟 |, and identifying ℓ = 𝑅 × 𝐾 we find that under this infinitesimal rotation
|𝜓′〉= [𝟏−𝑖 𝛿𝛼 ℓ ∙ 𝑢 ]|𝜓〉= 𝑈 𝑢 𝛿𝛼 |𝜓〉
Comparing this to our general result
𝑈 𝑢 𝛿𝛼 =𝟏−𝑖 𝛿𝛼 𝐽 ∙ 𝑢
We deduce that for a single particle 𝐽 = ℓ = 𝑅 × 𝐾 . 𝑢 O

36. 𝜓 ′ 𝑟 =𝜓 𝑟 −𝛿𝛼 𝑢 ∙( 𝑟 × 𝛻 ) 𝜓 𝑟
= [1−𝛿𝛼 𝑢 ∙( 𝑟 × 𝛻 )] 𝜓 𝑟
= (1−𝑖 𝛿𝛼 𝑢 ∙[( 𝑟 ×(−𝑖 𝛻 )]) 𝜓 𝑟
Introducing Dirac notation, this implies
〈 𝑟 |𝜓′〉=〈 𝑟 | 𝟏−𝑖 𝛿𝛼 𝑢 ∙ 𝑅 × 𝐾 |𝜓〉
Being true for all 〈 𝑟 |, and identifying ℓ = 𝑅 × 𝐾 we find that under this infinitesimal rotation
|𝜓′〉= [𝟏−𝑖 𝛿𝛼 ℓ ∙ 𝑢 ]|𝜓〉= 𝑈 𝑢 𝛿𝛼 |𝜓〉
Comparing this to our general result
𝑈 𝑢 𝛿𝛼 =𝟏−𝑖 𝛿𝛼 𝐽 ∙ 𝑢
we deduce that for a single particle 𝐽 = ℓ = 𝑅 × 𝐾 . 𝑢 O

37. So for a single spinless particle
𝐽 = ℓ
Generator of Rotations = Total Angular Momentum
Note that this allows us to write the rotation operators for a spinless particle in the explicit form
So what happens with a collection of spinless particles?
Recall: the state of a collection of 𝑁 such particles is an element of the direct product space
𝑆=𝑆(1)⊗…⊗𝑆(𝑁)
formed the individual single particle spaces.

38. So for a single spinless particle
𝐽 = ℓ
Generator of Rotations = Total Angular Momentum
Note that this allows us to write the rotation operators for a spinless particle in the explicit form
So what happens with a collection of spinless particles?
Recall: the state of a collection of 𝑁 such particles is an element of the direct product space
𝑆=𝑆(1)⊗…⊗𝑆(𝑁)
formed the individual single particle spaces.

39. So for a single spinless particle
𝐽 = ℓ
Generator of Rotations = Total Angular Momentum
Note that this allows us to write the rotation operators for a spinless particle in the explicit form
So what happens with a collection of spinless particles?
Recall: the state of a collection of 𝑁 such particles is an element of the direct product space
𝑆=𝑆(1)⊗…⊗𝑆(𝑁)
formed the individual single particle spaces.

40. So for a single spinless particle
𝐽 = ℓ
Generator of Rotations = Total Angular Momentum
Note that this allows us to write the rotation operators for a spinless particle in the explicit form
So what happens with a collection of spinless particles?
Recall: the state of a collection of 𝑁 such particles is an element of the direct product space
𝑆=𝑆(1)⊗…⊗𝑆(𝑁)
formed the individual single particle spaces.

41. So for a single spinless particle
𝐽 = ℓ
Generator of Rotations = Total Angular Momentum
Note that this allows us to write the rotation operators for a spinless particle in the explicit form
So what happens with a collection of spinless particles?
Recall: the state of a collection of 𝑁 such particles is an element of the direct product space
𝑆=𝑆(1)⊗…⊗𝑆(𝑁)
formed from the individual single particle spaces.

42. Since operators from different spaces commute with each other, the unitary operator 𝑈 𝐑 (𝛽) that “rotates” particle 𝛽 will commute with those that rotate all the others.
Under these circumstances the operator 𝑈 𝐑 that rotates the entire state vector |𝜓〉 is simply the product of the rotation operators for each particle.
Suppose, e.g., that |𝜓〉 is a direct product state, i.e.,
|𝜓〉=|𝜓₁,𝜓₂,⋯, 𝜓 𝑁 〉.
Under a rotation 𝐑, the state |𝜓〉 is taken onto the state
|𝜓′〉=|𝜓′₁,𝜓′₂,⋯, 𝜓 𝑁 ′ 〉
= 𝑈 𝐑 (1) 𝜓 1 𝑈 𝐑 𝜓 2 ⋯ 𝑈 𝐑 (𝑁) | 𝜓 𝑁 〉 = 𝑈 𝐑 (1) 𝑈 𝐑 2 ⋯ 𝑈 𝐑 (𝑁) |𝜓₁,𝜓₂,⋯, 𝜓 𝑁 〉 = 𝑈 𝐑 |𝜓〉, where …

43. Since operators from different spaces commute with each other, the unitary operator 𝑈 𝐑 (𝛽) that “rotates” particle 𝛽 will commute with those that rotate all the others.
Under these circumstances the operator 𝑈 𝐑 that rotates the entire state vector |𝜓〉 is simply the product of the rotation operators for each particle.
Suppose, e.g., that |𝜓〉 is a direct product state, i.e.,
|𝜓〉=|𝜓₁,𝜓₂,⋯, 𝜓 𝑁 〉.
Under a rotation 𝐑, the state |𝜓〉 is taken onto the state
|𝜓′〉=|𝜓′₁,𝜓′₂,⋯, 𝜓 𝑁 ′ 〉
= 𝑈 𝐑 (1) 𝜓 1 𝑈 𝐑 𝜓 2 ⋯ 𝑈 𝐑 (𝑁) | 𝜓 𝑁 〉 = 𝑈 𝐑 (1) 𝑈 𝐑 2 ⋯ 𝑈 𝐑 (𝑁) |𝜓₁,𝜓₂,⋯, 𝜓 𝑁 〉 = 𝑈 𝐑 |𝜓〉, where …

44. Since operators from different spaces commute with each other, the unitary operator 𝑈 𝐑 (𝛽) that “rotates” particle 𝛽 will commute with those that rotate all the others.
Under these circumstances the operator 𝑈 𝐑 that rotates the entire state vector |𝜓〉 is simply the product of the rotation operators for each particle.
Suppose, e.g., that |𝜓〉 is a direct product state, i.e.,
|𝜓〉=|𝜓₁,𝜓₂,⋯, 𝜓 𝑁 〉.
Under a rotation 𝐑, the state |𝜓〉 is taken onto the state
|𝜓′〉=|𝜓′₁,𝜓′₂,⋯, 𝜓 𝑁 ′ 〉
= 𝑈 𝐑 (1) 𝜓 1 𝑈 𝐑 𝜓 2 ⋯ 𝑈 𝐑 (𝑁) | 𝜓 𝑁 〉 = 𝑈 𝐑 (1) 𝑈 𝐑 2 ⋯ 𝑈 𝐑 (𝑁) |𝜓₁,𝜓₂,⋯, 𝜓 𝑁 〉 = 𝑈 𝐑 |𝜓〉, where …

45. Since operators from different spaces commute with each other, the unitary operator 𝑈 𝐑 (𝛽) that “rotates” particle 𝛽 will commute with those that rotate all the others.
Under these circumstances the operator 𝑈 𝐑 that rotates the entire state vector |𝜓〉 is simply the product of the rotation operators for each particle.
Suppose, e.g., that |𝜓〉 is a direct product state, i.e.,
|𝜓〉=|𝜓₁,𝜓₂,⋯, 𝜓 𝑁 〉.
Under a rotation 𝐑, the state |𝜓〉 is taken onto the state
|𝜓′〉=|𝜓′₁,𝜓′₂,⋯, 𝜓 𝑁 ′ 〉
= 𝑈 𝐑 (1) 𝜓 1 𝑈 𝐑 𝜓 2 ⋯ 𝑈 𝐑 (𝑁) | 𝜓 𝑁 〉 = 𝑈 𝐑 (1) 𝑈 𝐑 2 ⋯ 𝑈 𝐑 (𝑁) |𝜓₁,𝜓₂,⋯, 𝜓 𝑁 〉 = 𝑈 𝐑 |𝜓〉, where …

46. Since operators from different spaces commute with each other, the unitary operator 𝑈 𝐑 (𝛽) that “rotates” particle 𝛽 will commute with those that rotate all the others.
Under these circumstances the operator 𝑈 𝐑 that rotates the entire state vector |𝜓〉 is simply the product of the rotation operators for each particle.
Suppose, e.g., that |𝜓〉 is a direct product state, i.e.,
|𝜓〉=|𝜓₁,𝜓₂,⋯, 𝜓 𝑁 〉.
Under a rotation 𝐑, the state |𝜓〉 is taken onto the state
|𝜓′〉=|𝜓′₁,𝜓′₂,⋯, 𝜓 𝑁 ′ 〉
= 𝑈 𝐑 (1) 𝜓 1 𝑈 𝐑 𝜓 2 ⋯ 𝑈 𝐑 (𝑁) | 𝜓 𝑁 〉 = 𝑈 𝐑 (1) 𝑈 𝐑 2 ⋯ 𝑈 𝐑 (𝑁) |𝜓₁,𝜓₂,⋯, 𝜓 𝑁 〉 = 𝑈 𝐑 |𝜓〉, where …

47. Since operators from different spaces commute with each other, the unitary operator 𝑈 𝐑 (𝛽) that “rotates” particle 𝛽 will commute with those that rotate all the others.
Under these circumstances the operator 𝑈 𝐑 that rotates the entire state vector |𝜓〉 is simply the product of the rotation operators for each particle.
Suppose, e.g., that |𝜓〉 is a direct product state, i.e.,
|𝜓〉=|𝜓₁,𝜓₂,⋯, 𝜓 𝑁 〉.
Under a rotation 𝐑, the state |𝜓〉 is taken onto the state
|𝜓′〉=|𝜓′₁,𝜓′₂,⋯, 𝜓 𝑁 ′ 〉
= 𝑈 𝐑 (1) 𝜓 1 𝑈 𝐑 𝜓 2 ⋯ 𝑈 𝐑 (𝑁) | 𝜓 𝑁 〉 = 𝑈 𝐑 (1) 𝑈 𝐑 2 ⋯ 𝑈 𝐑 (𝑁) |𝜓₁,𝜓₂,⋯, 𝜓 𝑁 〉 = 𝑈 𝐑 |𝜓〉, where …

48. Since operators from different spaces commute with each other, the unitary operator 𝑈 𝐑 (𝛽) that “rotates” particle 𝛽 will commute with those that rotate all the others.
Under these circumstances the operator 𝑈 𝐑 that rotates the entire state vector |𝜓〉 is simply the product of the rotation operators for each particle.
Suppose, e.g., that |𝜓〉 is a direct product state, i.e.,
|𝜓〉=|𝜓₁,𝜓₂,⋯, 𝜓 𝑁 〉.
Under a rotation 𝐑, the state |𝜓〉 is taken onto the state
|𝜓′〉=|𝜓′₁,𝜓′₂,⋯, 𝜓 𝑁 ′ 〉
= 𝑈 𝐑 (1) 𝜓 1 𝑈 𝐑 𝜓 2 ⋯ 𝑈 𝐑 (𝑁) | 𝜓 𝑁 〉 = 𝑈 𝐑 (1) 𝑈 𝐑 2 ⋯ 𝑈 𝐑 (𝑁) |𝜓₁,𝜓₂,⋯, 𝜓 𝑁 〉 = 𝑈 𝐑 |𝜓〉, where …

49. 𝑈 𝐑 = 𝑈 𝐑 (1) 𝑈 𝐑 2 ⋯ 𝑈 𝐑 (𝑁)
is the product of rotation operators for each single particle state space, all corresponding to the same rotation R.
These individual operators can all be written in the same form, i.e.,
Where is the orbital angular momentum for particle 𝛽.
It follows that the total rotation operator for the space takes the form
where is the total angular momentum of the system of 𝑁 particles

50. 𝑈 𝐑 = 𝑈 𝐑 (1) 𝑈 𝐑 2 ⋯ 𝑈 𝐑 (𝑁)
is the product of rotation operators for each single particle state space, all corresponding to the same rotation R.
These individual operators can all be written in the same form, i.e.,
Where is the orbital angular momentum for particle 𝛽.
It follows that the total rotation operator for the space takes the form
where is the total angular momentum of the system of 𝑁 particles

51. 𝑈 𝐑 = 𝑈 𝐑 (1) 𝑈 𝐑 2 ⋯ 𝑈 𝐑 (𝑁)
is the product of rotation operators for each single particle state space, all corresponding to the same rotation R.
These individual operators can all be written in the same form, i.e.,
where is the orbital angular momentum for particle 𝛽.
It follows that the total rotation operator for the space takes the form
where is the total angular momentum of the system of 𝑁 particles

52. 𝑈 𝐑 = 𝑈 𝐑 (1) 𝑈 𝐑 2 ⋯ 𝑈 𝐑 (𝑁)
is the product of rotation operators for each single particle state space, all corresponding to the same rotation R.
These individual operators can all be written in the same form, i.e.,
where is the orbital angular momentum for particle 𝛽.
It follows that the total rotation operator for the space takes the form
where is the total angular momentum of the system of 𝑁 particles

53. 𝑈 𝐑 = 𝑈 𝐑 (1) 𝑈 𝐑 2 ⋯ 𝑈 𝐑 (𝑁)
is the product of rotation operators for each single particle state space, all corresponding to the same rotation R.
These individual operators can all be written in the same form, i.e.,
where is the orbital angular momentum for particle 𝛽.
It follows that the total rotation operator for the space takes the form
where is the total angular momentum of the system of 𝑁 particles

54. 𝑈 𝐑 = 𝑈 𝐑 (1) 𝑈 𝐑 2 ⋯ 𝑈 𝐑 (𝑁)
is the product of rotation operators for each single particle state space, all corresponding to the same rotation R.
These individual operators can all be written in the same form, i.e.,
where is the orbital angular momentum for particle 𝛽.
It follows that the total rotation operator for the space takes the form
where is the total angular momentum of the system of 𝑁 particles

55. 𝑈 𝐑 = 𝑈 𝐑 (1) 𝑈 𝐑 2 ⋯ 𝑈 𝐑 (𝑁)
is the product of rotation operators for each single particle state space, all corresponding to the same rotation R.
These individual operators can all be written in the same form, i.e.,
where is the orbital angular momentum for particle 𝛽.
It follows that the total rotation operator for the space takes the form
where is the total angular momentum of the system of 𝑁 particles

56. Thus, we are led naturally to the point of view that the generator of rotations for the whole system is the sum of the generators for each part thereof, hence for a collection of spinless particles
= 𝐿
Generator of Rotations = Total Angular Momentum
Clearly, due to the exponential form of the rotation operator, the generators for any composite system formed from the direct product of other subsystems is always the sum
of the generators for each subsystem being combined.

57. Thus, we are led naturally to the point of view that the generator of rotations for the whole system is the sum of the generators for each part thereof, hence for a collection of spinless particles
= 𝐿
Generator of Rotations = Total Angular Momentum
Clearly, due to the exponential form of the rotation operator, the generators for any composite system formed from the direct product of other subsystems is always the sum
of the generators for each subsystem being combined.

58. Thus, we are led naturally to the point of view that the generator of rotations for the whole system is the sum of the generators for each part thereof, hence for a collection of spinless particles
= 𝐿
Generator of Rotations = Total Angular Momentum
Clearly, due to the exponential form of the rotation operator, the generators for any composite system formed from the direct product of other subsystems is always the sum
of the generators for each subsystem being combined.

59. Thus, we are led naturally to the point of view that the generator of rotations for the whole system is the sum of the generators for each part thereof, hence for a collection of spinless particles
= 𝐿
Generator of Rotations = Total Angular Momentum
Clearly, due to the exponential form of the rotation operator, the generators for any composite system formed from the direct product of other subsystems is always the sum
of the generators for each subsystem being combined.

60. As we have seen, for particles with spin, the individual single particle spaces can themselves be considered direct products of a spatial part and a spin part.
For a single particle with spin 𝑠, the vector operator 𝑆 with components 𝑆 x , 𝑆 y and 𝑆 z forms the generator of rotations for the spin degrees of freedom, the orbital angular momentum ℓ forms the generator of rotations for the spatial part of the state, and the vector operator
= ℓ + 𝑆
forms the generator of rotations for the total single particle state space.
Thus, as we build complex systems out of subsytems, if the generator of rotations for each subsystems corresponds its total angular momentum, that will also be true for the combined systems as well. Thus, we identify the total angular momentum of any quantum system with the generator of rotations for that system.

61. As we have seen, for particles with spin, the individual single particle spaces can themselves be considered direct products of a spatial part and a spin part.
For a single particle with spin 𝑠, the vector operator 𝑆 with components 𝑆 x , 𝑆 y and 𝑆 z forms the generator of rotations for the spin degrees of freedom, the orbital angular momentum ℓ forms the generator of rotations for the spatial part of the state, and the vector operator
= ℓ + 𝑆
forms the generator of rotations for the total single particle state space.
Thus, as we build complex systems out of subsytems, if the generator of rotations for each subsystems corresponds its total angular momentum, that will also be true for the combined systems as well. Thus, we identify the total angular momentum of any quantum system with the generator of rotations for that system.

62. As we have seen, for particles with spin, the individual single particle spaces can themselves be considered direct products of a spatial part and a spin part.
For a single particle with spin 𝑠, the vector operator 𝑆 with components 𝑆 x , 𝑆 y and 𝑆 z forms the generator of rotations for the spin degrees of freedom, the orbital angular momentum ℓ forms the generator of rotations for the spatial part of the state, and the vector operator
= ℓ + 𝑆
forms the generator of rotations for the total single particle state space.
Thus, as we build complex systems out of subsytems, if the generator of rotations for each subsystems corresponds its total angular momentum, that will also be true for the combined systems as well. Thus, we identify the total angular momentum of any quantum system with the generator of rotations for that system.

63. As we have seen, for particles with spin, the individual single particle spaces can themselves be considered direct products of a spatial part and a spin part.
For a single particle with spin 𝑠, the vector operator 𝑆 with components 𝑆 x , 𝑆 y and 𝑆 z forms the generator of rotations for the spin degrees of freedom, the orbital angular momentum ℓ forms the generator of rotations for the spatial part of the state, and the total orbital and spin angular momentum
= ℓ + 𝑆
forms the generator of rotations for the total single particle state space.
Thus, as we build complex systems out of subsytems, if the generator of rotations for each subsystems corresponds its total angular momentum, that will also be true for the combined systems as well. Thus, we identify the total angular momentum of any quantum system with the generator of rotations for that system.

64. As we have seen, for particles with spin, the individual single particle spaces can themselves be considered direct products of a spatial part and a spin part.
For a single particle with spin 𝑠, the vector operator 𝑆 with components 𝑆 x , 𝑆 y and 𝑆 z forms the generator of rotations for the spin degrees of freedom, the orbital angular momentum ℓ forms the generator of rotations for the spatial part of the state, and the total orbital and spin angular momentum
= ℓ + 𝑆
forms the generator of rotations for the total single particle state space.
Thus, as we build complex systems out of subsystems, if the generator of rotations for each subsystem corresponds to its total angular momentum, that will also be true for the combined systems as well. Thus, we identify the total angular momentum of any quantum system with the generator of rotations for that system.

65. As we have seen, for particles with spin, the individual single particle spaces can themselves be considered direct products of a spatial part and a spin part.
For a single particle with spin 𝑠, the vector operator 𝑆 with components 𝑆 x , 𝑆 y and 𝑆 z forms the generator of rotations for the spin degrees of freedom, the orbital angular momentum ℓ forms the generator of rotations for the spatial part of the state, and the total orbital and spin angular momentum
= ℓ + 𝑆
forms the generator of rotations for the total single particle state space.
Thus, as we build complex systems out of subsystems, if the generator of rotations for each subsystem corresponds to its total angular momentum, that will also be true for the combined systems as well. Thus, we are justified in asserting that the total angular momentum of any quantum system is the generator of rotations for that system.