Consider the total energy \(\psi\),
\(\psi =\psi _{ 1 }+i\psi _{ 2 }\)
\( |\psi |^{ 2 }=\psi\psi^*=\left( \psi _{ 1 }+i\psi _{ 2 } \right) \left( \psi _{ 1 }-i\psi _{ 2 } \right) \)
\( |\psi |^{ 2 }=\psi _{ 1 }^{ 2 }+\psi _{ 2 }^{ 2 }\) --- (1)
If \(\psi_1\) is sinusoidal,
\( \psi _{ 1 }=|\psi |cos(\omega t)\)
then to satisfy (1)
\( \psi _{ 2 }=|\psi |sin(\omega t)\)
Therefore,
\( \psi =|\psi |\left\{ cos(\omega t)+isin(\omega t) \right\}\)
\( \psi =|\psi |e^{ iwt }\) --- (*)
From the post "Standing Waves, Particles, Time Invariant Fields", \(\psi\) is a circular standing wave about a center,
\(\cfrac { \partial ^{ 2 }\, \psi }{ \partial \, t^{ 2 }_{ c } } =(ic)^{ 2 }\cfrac { \partial ^{ 2 }\, \psi }{ \partial \, x^{ 2 } } =-c^{ 2 }\cfrac { \partial ^{ 2 }\, \psi }{ \partial \, x^{ 2 } }\)
or
\(\ddot{\psi}=(ic)^2\nabla^2\psi\)
And
\(2\pi x=i.n\lambda\) --- (2)
where \(n=1,2,3,4...\) and when \(x=0\), \(n=0\)
Differentiating (*) wrt \(t\),
\( \dot { \psi } =i\omega |\psi |e^{ iwt }=i\omega\psi\)
Differentiating wrt \(t\) again,
\( \ddot { \psi } =-\omega ^{ 2 }|\psi |e^{ iwt }=-\omega ^{ 2 }\psi\)
And the wave equation becomes,
\( -\omega ^{ 2 }\psi =(ic)^{ 2 }\nabla ^{ 2 }\psi \)
\( \nabla ^{ 2 }\psi+\cfrac { \omega ^{ 2 } }{ (ic)^{ 2 } } \psi =0\)
As,
\(\lambda=\cfrac{c}{f}\)
We have Helmholtz equation of a complex wave \(\psi\),
\( \nabla ^{ 2 }\psi +\left( \cfrac { 2\pi }{ i\lambda } \right) ^{ 2 }\psi =0\)
Substitute (2) into the above,
\( \nabla ^{ 2 }\psi +\left( \cfrac { n }{ x } \right) ^{ 2 }\psi =0\)
where \(n=1,2,3,4...\) and when \(x=0\), \(n=0\) such that
\(\lim\limits_{x\rightarrow0}{\left( \cfrac { n }{ x } \right)}=1\)
\(\psi\) is the total energy of the system, quantized.
And God said: "I'll just decide. No Dice."
