mecànica cuàntica

( h^{2}/(2ml) )·d_{x}[f(x)] = ( E_{n}+q·A(x) )·f(x)

f(x) = sum[ (1/k!)·( ( (2ml)/h^{2} )·x [o(x)o] ( E_{n}·x+q·int[ A(x) ] d[x] ) [o(x)o] x )^{k} ]

( h^{2}/(2m) )·d_{xx}^{2}[f(x)] = ( E_{n}+q·A(x) )·f(x)

f(x) = ...

... sum[ (1/k!)·( ( (2m)/h^{2} )^{(1/3)}·x [o(x)o] ( E_{n}+q·A(x) )^{[o(x)o](1/3)} [o(x)o] x )^{k} ]

d_{x}[ ( f(x) [o(x)o] g(x) )^{k} ] = ...

... ( f(x) [o(x)o] g(x) )^{k} [o(x)o] ( f(x) [o(x)o] g(x) )^{[o(x)o]2}

d_{x}[ ( f(x) [o(x)o] g(x) )^{k} ] = ...

... k·( f(x) [o(x)o] g(x) )^{k+(-1)}·( d_{x}[f(x)]·d_{x}[g(x)] )

demostració per isomorfisme de grups.

f( d_{x}[f(x)]·d_{x}[g(x)] ) = int[ d_{x}[f(x)]·d_{x}[g(x)] ] d[ f(x) [o(x)o] g(x) ]

f( k·( f(x) [o(x)o] g(x) )^{k+(-1)} ) = ( f(x) [o(x)o] g(x) )^{k}