This article concerns the rotation operator, as it appears in quantum mechanics.
 With every physical rotation 
  
    
      
        R
      
    
    {\displaystyle R}
  
, we postulate a quantum mechanical rotation operator 
  
    
      
        D
        (
        R
        )
      
    
    {\displaystyle D(R)}
  
 which rotates quantum mechanical states.
  In terms of the generators of rotation,
  The rotation operator 
  
    
      
        R
        ⁡
        (
        z
        ,
        θ
        )
      
    
    {\displaystyle \operatorname {R} (z,\theta )}
  
, with the first argument 
  
    
      
        z
      
    
    {\displaystyle z}
  
 indicating the rotation axis and the second 
  
    
      
        θ
      
    
    {\displaystyle \theta }
  
 the rotation angle, can operate through the translation operator 
  
    
      
        T
        ⁡
        (
        a
        )
      
    
    {\displaystyle \operatorname {T} (a)}
  
 for infinitesimal rotations as explained below. This is why, it is first shown how the translation operator is acting on a particle at position x (the particle is then in the state 
  
    
      
        
          |
        
        x
        ⟩
      
    
    {\displaystyle |x\rangle }
  
 according to Quantum Mechanics).
 Translation of the particle at position 
  
    
      
        x
      
    
    {\displaystyle x}
  
 to position 
  
    
      
        x
        +
        a
      
    
    {\displaystyle x+a}
  
: 
  
    
      
        T
        ⁡
        (
        a
        )
        
          |
        
        x
        ⟩
        =
        
          |
        
        x
        +
        a
        ⟩
      
    
    {\displaystyle \operatorname {T} (a)|x\rangle =|x+a\rangle }
  

 Because a translation of 0 does not change the position of the particle, we have (with 1 meaning the identity operator, which does nothing):
  Taylor development gives:
  From that follows:
  This is a differential equation with the solution
   Additionally, suppose a Hamiltonian 
  
    
      
        H
      
    
    {\displaystyle H}
  
 is independent of the 
  
    
      
        x
      
    
    {\displaystyle x}
  
 position. Because the translation operator can be written in terms of 
  
    
      
        
          p
          
            x
          
        
      
    
    {\displaystyle p_{x}}
  
, and 
  
    
      
        [
        
          p
          
            x
          
        
        ,
        H
        ]
        =
        0
      
    
    {\displaystyle [p_{x},H]=0}
  
, we know that 
  
    
      
        [
        H
        ,
        T
        ⁡
        (
        a
        )
        ]
        =
        0.
      
    
    {\displaystyle [H,\operatorname {T} (a)]=0.}
  
 This result means that linear momentum for the system is conserved.
  Classically we have for the angular momentum 
  
    
      
        
          L
        
        =
        
          r
        
        ×
        
          p
        
        .
      
    
    {\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} .}
  
 This is the same in quantum mechanics considering 
  
    
      
        
          r
        
      
    
    {\displaystyle \mathbf {r} }
  
 and 
  
    
      
        
          p
        
      
    
    {\displaystyle \mathbf {p} }
  
 as operators. Classically, an infinitesimal rotation 
  
    
      
        d
        t
      
    
    {\displaystyle dt}
  
 of the vector 
  
    
      
        
          r
        
        =
        (
        x
        ,
        y
        ,
        z
        )
      
    
    {\displaystyle \mathbf {r} =(x,y,z)}
  
 about the 
  
    
      
        z
      
    
    {\displaystyle z}
  
-axis to 
  
    
      
        
          
            r
          
          ′
        
        =
        (
        
          x
          ′
        
        ,
        
          y
          ′
        
        ,
        z
        )
      
    
    {\displaystyle \mathbf {r} '=(x',y',z)}
  
 leaving 
  
    
      
        z
      
    
    {\displaystyle z}
  
 unchanged can be expressed by the following infinitesimal translations (using Taylor approximation):
   From that follows for states:
  And consequently:
  Using 
  To get a rotation for the angle 
  
    
      
        t
      
    
    {\displaystyle t}
  
, we construct the following differential equation using the condition 
  
    
      
        R
        ⁡
        (
        z
        ,
        0
        )
        =
        1
      
    
    {\displaystyle \operatorname {R} (z,0)=1}
  
:
   Similar to the translation operator, if we are given a Hamiltonian 
  
    
      
        H
      
    
    {\displaystyle H}
  
 which rotationally symmetric about the 
  
    
      
        z
      
    
    {\displaystyle z}
  
-axis, 
  
    
      
        [
        
          L
          
            z
          
        
        ,
        H
        ]
        =
        0
      
    
    {\displaystyle [L_{z},H]=0}
  
 implies 
  
    
      
        [
        R
        ⁡
        (
        z
        ,
        t
        )
        ,
        H
        ]
        =
        0
      
    
    {\displaystyle [\operatorname {R} (z,t),H]=0}
  
. This result means that angular momentum is conserved.
 For the spin angular momentum about for example the 
  
    
      
        y
      
    
    {\displaystyle y}
  
-axis we just replace 
  
    
      
        
          L
          
            z
          
        
      
    
    {\displaystyle L_{z}}
  
 with 
  
    
      
        
          S
          
            y
          
        
        =
        
          
            ℏ
            2
          
        
        
          σ
          
            y
          
        
      
    
    {\textstyle S_{y}={\frac {\hbar }{2}}\sigma _{y}}
  
 (where 
  
    
      
        
          σ
          
            y
          
        
      
    
    {\displaystyle \sigma _{y}}
  
 is the Pauli Y matrix) and we get the spin rotation operator 
  Operators can be represented by matrices. From linear algebra one knows that a certain matrix 
  
    
      
        A
      
    
    {\displaystyle A}
  
 can be represented in another basis through the transformation
  From standard quantum mechanics we have the known results 
  
    
      
        
          S
          
            b
          
        
        
          |
        
        b
        +
        ⟩
        =
        
          
            ℏ
            2
          
        
        
          |
        
        b
        +
        ⟩
      
    
    {\textstyle S_{b}|b+\rangle ={\frac {\hbar }{2}}|b+\rangle }
  
 and 
  
    
      
        
          S
          
            c
          
        
        
          |
        
        c
        +
        ⟩
        =
        
          
            ℏ
            2
          
        
        
          |
        
        c
        +
        ⟩
      
    
    {\textstyle S_{c}|c+\rangle ={\frac {\hbar }{2}}|c+\rangle }
  
 where  
  
    
      
        
          |
        
        b
        +
        ⟩
      
    
    {\displaystyle |b+\rangle }
  
 and 
  
    
      
        
          |
        
        c
        +
        ⟩
      
    
    {\displaystyle |c+\rangle }
  
 are the top spins in their corresponding bases. So we have:
  Comparison with 
  
    
      
        
          S
          
            b
          
        
        
          |
        
        b
        +
        ⟩
        =
        
          
            ℏ
            2
          
        
        
          |
        
        b
        +
        ⟩
      
    
    {\textstyle S_{b}|b+\rangle ={\frac {\hbar }{2}}|b+\rangle }
  
 yields 
  
    
      
        
          |
        
        b
        +
        ⟩
        =
        
          D
          
            −
            1
          
        
        (
        y
        ,
        t
        )
        
          |
        
        c
        +
        ⟩
      
    
    {\displaystyle |b+\rangle =D^{-1}(y,t)|c+\rangle }
  
.
 This means that if the state 
  
    
      
        
          |
        
        c
        +
        ⟩
      
    
    {\displaystyle |c+\rangle }
  
 is rotated about the 
  
    
      
        y
      
    
    {\displaystyle y}
  
-axis by an angle 
  
    
      
        t
      
    
    {\displaystyle t}
  
, it becomes the state 
  
    
      
        
          |
        
        b
        +
        ⟩
      
    
    {\displaystyle |b+\rangle }
  
, a result that can be generalized to arbitrary axes.