Spherical Harmonics Applied To Electric Potential

To review, the term spherical harmonics  derives from the simplest type of geometry (the sphere) which can be applied to boundary value problems of potential theory.  Hence, applied to everything from soap bubbles, to balloons and  even the Sun.  The important and distinguishing aspect is that the relevant functions are encountered when using systems of orthogonal curvilinear coordinates.  The relevant spherical equation will then have a form similar to:

å ^¥ m= _ℓ     (C _m   r ^m   + D _m r  ^{- (m+1)}   )  P _m  cos q )

Where by orthogonality:

ò ¹  _-1    P _n P _m    d m   =   {0,    n  ≠  m

                                             { z/ z _n + 1,  n =  m

Where P _n   and P _m   are solutions to Legendre’s equation.  As noted by N.N. Lebedev:   “by  spherical harmonics we mean solutions of the linear differential equation”[1]:

[(1 – z ² ) u’ – 2 zu’ + [n  (n + 1) -   m ² / 1 – z ² ] u  =  0

Where z is a complex variable, i.e. z = x +iy, and   m  ,  n are parameters (not constants) which can take real or complex values.

As in the case of solar oscillation modes for a spherical configuration, see e.g.

Solar Oscillations Revisited

and wave vibrations for an atomic system, like the 1-electron atom,  we can use spherical harmonics for electric potential situations.

Consider  a spherical volume for which we wish to find the potential  V(r,  q  ) if r = 1 and if the upper half of the sphere is charged to potential V _o   and the lower half is at potential 0.

We write for the potential, i.e. in terms  of a series of orthogonal functions:

V(r,  q )    =      

å ^¥ n=0    (A _n   r _n    +  B _n /  r ⁿ⁺¹   ) (C _n P _n  cos q )  +

 D _n  Q _n   (cos q)

And:   D _n   =  0,  so that:

V(r,  q )    =      

å ^¥ n=0    (A _n   r _n    +  B _n /  r ⁿ⁺¹   ) (C _n P _n  cos q ) 

Then inside the sphere (r < a):

   B _n   =  0

And:

V(r,  q )   =  f(q) = å ^¥ n=0    A _n   r _n    P _n   (  cos q )    

To determine  A _n  :

 A _n  =   (2n+1) / 2  ò ^2p _o    f(q) P _n   ( cos q )  sin q dq  

A _n  =    ò ¹  _-1    f(cos ^-1   m) P _n  (m) dm

Outside the sphere (r >  a):  A _n  =     0

V(r,  q )    =  å ^¥ n=0    B _n /  r ⁿ⁺¹    P _n   ( cos q )      

If r = 1:

V(1,  q )    =  å ^¥ n=0    A _n P _n   ( cos q )      

Then: A _n  =    

(2n+1) / 2  ò ^p _o    cos q sin q dq        =

(2n+1) / 2 ò ¹  _-1   m P _n  (m) dm

è

å^¥ n=0   [ (2n+1) / 2 ò¹  _-1   m P _n  (m) dm] r ⁿ P _n   (cos q )      

 The preceding is based on the use of Laplace’s equation:

Ñ ²   V  =  0

Which in spherical coordinates becomes:

1/ r ²  ¶ / ¶ r (r ²   ¶ V/¶ r  )  

+ 1/( r ² sin q)  ¶ / ¶ q (sin q  ¶ V/¶ q)  + 

1/( r ² sin q)   ¶  ² V/¶ j²     =  0

Which  equation may be separated by letting:

  V  = R  (r) Q (q)F( j)  

Analogous to the variables separation for the hydrogen atom wave function, e.g.

  y  = R  (r) Q (q)F( j)

(see e.g. Sec. 4 in the post:  Introduction to Quantum Mechanical Operators (Pt. ...   )

Two of the ordinary differential equations which result are:

1) d ² F / d f ²    =   - m²  F    

And:

2)d² Q / d ² q  + cot q (d Q / d  q )

+ (a²  -  m²/ sin² q ) Q = 0  

For which the constants  a²  and - m²   may be real or complex.  For (1) when m is an integer  F  is periodic with a period  2 p  as we saw with the analogous  equation for  the hydrogen atom.  We call (2) the associated Legendre equation.

 Suggested Problems:

1) A Spherical electromagnetic wave front exhibits a mode featuring an associated Legendre polynomial that can be described:

P m  _ℓ   (cos q ) =  - 3/ 2  sin q (5 cos ²  q  - 1 )

If m =   1  and  ℓ = 3, show that the function is accurate.

2)Write the full Laplace equation in spherical coordinates for a homogeneous medium with magnetic permeability m  ,  permittivity e, _, conductivity, s    and frequency w.

3) For the case of an  electromagnetic  system for which the associated Legendre equation applies, rewrite it if the constant  a²  is  real  and also has the form:    a²    =  n(n + 1)

  In your equation let x = cos q,  sin² q = 1 – x ²   and    d/ dq  = - sin  q d/ dx .

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[1] Lebedev, N.N.:1972, Special Functions and Their Applications, Dover Publications, p. 161.