phy-4600/solutions/chap7/prob8

The transformation from polar coordinates to cartesian is the set of equations

⎧    x = r sinθ cosϕ
⎪
⎨    y = r sinθ sinϕ
⎪
⎩    z = r cosθ

The transformation from cartesian to polar coordinates is the set

⎧   r = √(x²+y²+z²)
⎪
⎪   cos(θ) = z͟ =     _͟z͟
⎨            r   √(x²+y²+z²)
⎪
⎪   tan(ϕ) = y͟
⎩            x

Some differential forms may come in handy.

⎧   dcos(θ) = -sin(θ) dθ
⎨
⎪   dtan(ϕ) =  _͟1͟   dϕ
⎩            cos²(ϕ)

∂/∂θ:

⎧   ∂͟x = r cosϕ cosθ 
⎪   ∂θ
⎪
⎨   ∂͟y = r sinϕ cosθ 
⎪   ∂θ
⎪
⎪   ∂͟z = -r sinθ 
⎩   ∂θ

∂/∂ϕ:

⎧   ∂͟x = -r sinθ sinϕ 
⎪   ∂ϕ
⎪
⎨   ∂͟y = r sinθ cosϕ 
⎪   ∂ϕ    
⎪    
⎪   ∂͟z = 0 
⎩   ∂ϕ

∂/∂x: 

⎧   ∂͟r = √(x²+y²+z²) = x͟
⎪   ∂x                 r
⎪
⎪   ∂͟cos(θ) = -z͟x͟ 
⎨   ∂x         r³ 
⎪
⎪   ∂͟tan(ϕ) = -y͟
⎩   ∂x         x²

∂/∂y: 

⎧   ∂͟r = √(x²+y²+z²) = y͟
⎪   ∂y                 r
⎪
⎪   ∂͟cos(θ) = -z͟y͟ 
⎨   ∂y         r³ 
⎪
⎪   ∂͟tan(ϕ) = 1͟
⎩   ∂y        x

∂/∂z: 

⎧   ∂͟r = √(x²+y²+z²) = z͟
⎪   ∂z                 r
⎪
⎪   ∂͟cos(θ) = z͟ = 1͟  ⎛r ∂͟z͟ - z ∂͟r͟⎞ = 1͟ - z͟²
⎨   ∂z        r   r² ⎝  ∂z     ∂z⎠   r   r³
⎪
⎪   ∂͟tan(ϕ) = 0
⎩   ∂z          


These differential forms can be used to transform each cartesian differentiation operator. 

    ∂͟ = ∂͟r͟ ∂͟ + ∂͟c͟o͟s͟θ͟ ∂͟     +  ∂͟t͟a͟n͟ϕ͟ ∂͟    ;
    ∂x  ∂x ∂r  ∂x    ∂cosθ    ∂x    ∂tanϕ

    ∂͟ = x͟ ∂͟ -  z͟x͟  _͟1͟   ∂͟  - y͟ cos²(ϕ) ∂͟ ;  
    ∂x  r ∂r   r³ -sinθ ∂θ   x²        ∂ϕ

    ∂͟ = sin(θ) cos(ϕ) ∂͟ 
    ∂x                ∂r
                + 1͟ sin(θ) cos(ϕ) cos(θ) _͟1͟    ∂͟  
                  r                     sin(θ) ∂θ     
                        - r͟  s͟i͟n͟(ϕ͟)  s͟i͟n͟(θ͟)  cos²(ϕ͟) ∂͟  ;  
                          r² cos²(ϕ) sin²(θ)         ∂ϕ 

    ∂͟ = sin(θ) cos(ϕ) ∂͟  + 1͟ cos(ϕ) cos(θ) ∂͟  - 1͟ s͟i͟n͟(ϕ͟) ∂͟  .
    ∂x                ∂r   r               ∂θ   r sin(θ) ∂ϕ

    ───────────────────────────────────────────────

    ∂͟ = ∂͟r͟ ∂͟ + ∂͟c͟o͟s͟θ͟ ∂͟     +  ∂͟t͟a͟n͟ϕ͟ ∂͟    ;
    ∂y  ∂y ∂r  ∂y    ∂cosθ    ∂y    ∂tanϕ

    ∂͟ = y͟ ∂͟  - z͟y͟ ∂͟     + 1͟ ∂͟    ;
    ∂y  r ∂r   r³ ∂cosθ   x ∂tanϕ

    ∂͟ = sin(θ) sin(ϕ) ∂͟  - 1͟ sin(θ) cos(θ) s͟i͟n͟(ϕ͟) ∂͟  +  c͟o͟s͟(ϕ͟)  ∂͟  ;
    ∂y                ∂r   r              -sin(θ) ∂θ   r sin(θ) ∂ϕ

    ∂͟ = sin(θ) sin(ϕ) ∂͟  + 1͟ cos(θ) s͟i͟n͟(ϕ͟) ∂͟  +  c͟o͟s͟(ϕ͟)  ∂͟  .
    ∂y                ∂r   r               ∂θ   r sin(θ) ∂ϕ

    ───────────────────────────────────────────────

    ∂͟ = ∂͟r͟ ∂͟ + ∂͟c͟o͟s͟θ͟ ∂͟     +  ∂͟t͟a͟n͟ϕ͟ ∂͟    ;
    ∂z  ∂z ∂r  ∂z    ∂cosθ    ∂z    ∂tanϕ

    ∂͟ = z͟ ∂͟ + ⎛1͟ - z͟²⎞∂͟     +  0 ;
    ∂z  r ∂r  ⎝r   r³⎠∂cosθ    

    ∂͟ = cos(θ) ∂͟ + 1͟⎛1 - cos²(θ)⎞   _͟1͟    ∂͟  ;
    ∂z         ∂r  r⎝           ⎠ -sin(θ) ∂θ    

    ∂͟ = cos(θ) ∂͟ -  _͟1͟_   ⎛1 - cos²(θ)⎞ ∂͟  ;
    ∂z         ∂r  rsin(θ)⎝           ⎠ ∂θ    

    ∂͟ = cos(θ) ∂͟ -  _͟1͟_    sin²(θ) ∂͟  ;
    ∂z         ∂r  rsin(θ)         ∂θ    

    ∂͟ = cos(θ) ∂͟ - s͟i͟n͟(θ͟) ∂͟  .
    ∂z         ∂r    r    ∂θ    


The set of differential operator transformations is

⎧   ∂͟ = sin(θ) cos(ϕ) ∂͟  + 1͟ cos(ϕ) cos(θ) ∂͟  - 1͟ s͟i͟n͟(ϕ͟) ∂͟ 
⎪   ∂x                ∂r   r               ∂θ   r sin(θ) ∂ϕ
⎪
⎨   ∂͟ = sin(θ) sin(ϕ) ∂͟  + 1͟ cos(θ) s͟i͟n͟(ϕ͟) ∂͟  +  c͟o͟s͟(ϕ͟)  ∂͟ 
⎪   ∂y                ∂r   r               ∂θ   r sin(θ) ∂ϕ
⎪
⎪   ∂͟ = cos(θ) ∂͟ - s͟i͟n͟(θ͟) ∂͟ 
⎩   ∂z         ∂r    r    ∂θ


7.47 is the set of algebraic conditions expressed by the vector definition 𝐋 = 𝐫 × 𝐩.

⎧    L̂𝓍 = yp𝓏 - zp𝓎 = -ιħ (y ∂͟_  - z ∂͟_ )
⎪                           ∂z      ∂y
⎪
⎨    L̂𝓎 = zp𝓍 - xp𝓏 = -ιħ (z ∂͟_  - x ∂͟_ )
⎪                           ∂x      ∂z
⎪
⎪    L̂𝓏 = xp𝓎 - yp𝓍 = -ιħ (x ∂͟_  - y ∂͟_ )
⎩                           ∂y      ∂x

Substituting 7.35 into 7.47,

⎧    L̂𝓍 = -ιħ (r sinθ sinϕ ∂͟_  - r cosθ ∂͟_ )
⎪                          ∂z           ∂y
⎪
⎨    L̂𝓎 = -ιħ (r cosθ ∂͟_  - r sinθ cosϕ ∂͟_ )
⎪                     ∂x                ∂z
⎪
⎪    L̂𝓏 = -ιħ (r sinθ cosϕ ∂͟_ - r sinθ sinϕ ∂͟_ )
⎩                          ∂y               ∂x

Substituting the transformed differentiation operators,

⎧   L̂𝓍 = -ιħ⎛r sinθ sinϕ ⎛cos(θ) ∂͟ - s͟i͟n͟(θ͟) ∂͟ ⎞ 
⎪           ⎝            ⎝       ∂r    r    ∂θ⎠ 
⎪               
⎪         - r cosθ ⎛sin(θ) sin(ϕ) ∂͟  + 1͟ cos(θ) s͟i͟n͟(ϕ͟) ∂͟  +  c͟o͟s͟(ϕ͟)  ∂͟ ⎞⎞
⎪                  ⎝              ∂r   r               ∂θ   r sin(θ) ∂ϕ⎠⎠
⎪
⎪
⎪   L̂𝓎 = -ιħ⎛r cosθ⎛sin(θ) cos(ϕ) ∂͟  + 1͟ cos(ϕ) cos(θ) ∂͟  - 1͟ s͟i͟n͟(ϕ͟) ∂͟ ⎞
⎪          ⎝      ⎝              ∂r   r               ∂θ   r sin(θ) ∂ϕ⎠
⎨           
⎪               - r sinθ cosϕ ⎛cos(θ) ∂͟ - s͟i͟n͟(θ͟) ∂͟ ⎞⎞           
⎪                             ⎝       ∂r    r    ∂θ⎠⎠
⎪
⎪
⎪   L̂𝓏 = -ιħ⎛r sinθ cosϕ⎛sin(θ) sin(ϕ) ∂͟  + 1͟ cos(θ) s͟i͟n͟(ϕ͟) ∂͟  +  c͟o͟s͟(ϕ͟)  ∂͟ ⎞ 
⎪          ⎝           ⎝              ∂r   r               ∂θ   r sin(θ) ∂ϕ⎠ 
⎪
⎪               - r sinθ sinϕ ⎛sin(θ) cos(ϕ) ∂͟  + 1͟ cos(ϕ) cos(θ) ∂͟  - 1͟ s͟i͟n͟(ϕ͟) ∂͟ ⎞⎞
⎩                             ⎝              ∂r   r               ∂θ   r sin(θ) ∂ϕ⎠⎠

Simplifying...

⎧   L̂𝓍 = -ιħ⎛r sinθ sinϕ cosθ ∂͟ - sinϕ sin²θ ∂͟  
⎪          ⎝                 ∂r             ∂θ 
⎪               
⎪         -  r cosθ sinθ sinϕ ∂͟  - cos²θ sinϕ ∂͟  - c͟o͟s͟θ͟ cosϕ ∂͟ ⎞
⎪                             ∂r              ∂θ   sinθ      ∂ϕ⎠
⎪
⎪
⎪   L̂𝓎 = -ιħ⎛r cosθ sinθ cosϕ ∂͟  + cosθ cosϕ cosθ ∂͟  - c͟o͟s͟θ͟ sinϕ ∂͟ 
⎪          ⎝                 ∂r                  ∂θ   sinθ      ∂ϕ
⎨           
⎪               - r sinθ cosϕ cosθ ∂͟ + sinθ cosϕ sinθ ∂͟ ⎞           
⎪                                  ∂r                 ∂θ⎠
⎪
⎪
⎪    L̂𝓏 = -ιħ⎛r sin²θ cosϕ sinϕ ∂͟  + sinθ cosθ sinϕ cosϕ ∂͟  + cosϕ c͟o͟s͟ϕ͟  ∂͟  
⎪           ⎝                  ∂r                       ∂θ              ∂ϕ 
⎪
⎪               - r sin²θ sinϕ cosϕ ∂͟  - sinθ cosθ sinϕ cosϕ  ∂͟  + sinθ sinϕ s͟i͟n͟ϕ͟ ∂͟ ⎞
⎩                                   ∂r                        ∂θ             sinθ ∂ϕ⎠


⎧   L̂𝓍 = -ιħ⎛- sinϕ sin²θ ∂͟  - cos²θ sinϕ ∂͟  - c͟o͟s͟θ͟ cosϕ ∂͟ ⎞
⎪          ⎝             ∂θ              ∂θ   sinθ      ∂ϕ⎠
⎪
⎨   L̂𝓎 = -ιħ⎛cosθ cosϕ cosθ ∂͟  - c͟o͟s͟θ͟ sinϕ ∂͟  + sinθ cosϕ sinθ ∂͟ ⎞
⎪          ⎝               ∂θ   sinθ      ∂ϕ                  ∂θ⎠
⎪           
⎪    L̂𝓏 = -ιħ⎛cosϕ c͟o͟s͟ϕ͟  ∂͟  + sinθ sinϕ s͟i͟n͟ϕ͟ ∂͟ ⎞
⎩           ⎝           ∂ϕ             sinθ ∂ϕ⎠


⎧   L̂𝓍 = ιħ⎛ sinϕ sin²θ ∂͟  + cos²θ sinϕ ∂͟  + c͟o͟s͟θ͟ cosϕ ∂͟ ⎞
⎪         ⎝            ∂θ              ∂θ   sinθ      ∂ϕ⎠
⎪
⎨   L̂𝓎 = ιħ⎛-cos²θ cosϕ  ∂͟ - sin²θ cosϕ  ∂͟ + c͟o͟s͟θ͟ sinϕ ∂͟  ⎞
⎪         ⎝             ∂θ              ∂θ  sinθ      ∂ϕ ⎠
⎪           
⎪    L̂𝓏 = -ιħ⎛cos²ϕ  ∂͟  + sin²ϕ ∂͟ ⎞
⎩           ⎝       ∂ϕ         ∂ϕ⎠


⎧   L̂𝓍 = ιħ⎛ sinϕ  ∂͟ + cotθ cosϕ ∂͟ ⎞
⎪         ⎝       ∂θ            ∂ϕ⎠
⎪
⎨   L̂𝓎 = ιħ⎛-cosϕ ∂͟ + cotθ sinϕ ∂͟  ⎞
⎪         ⎝      ∂θ            ∂ϕ ⎠
⎪           
⎪   L̂𝓏 = -ιħ∂͟ 
⎩          ∂ϕ 

The spherical representation is the following set, which perfectly matches the obtained result.

⎧    L̂𝓍 = ιħ (sinϕ ∂͟_  + cosϕ cotθ ∂͟_ )
⎪                 ∂θ              ∂ϕ
⎪
⎨    L̂𝓎 = ιħ (-cosϕ ∂͟_ + sinϕ cotθ ∂͟_ )
⎪                  ∂θ             ∂ϕ
⎪
⎪    L̂𝓏 = -ιħ ∂͟_ 
⎩            ∂ϕ