\( \nabla ^{ 2 }\psi +\left( \cfrac { n }{ x } \right) ^{ 2 }\psi =0\)
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. We have,
\(\cfrac{1}{\psi}\cfrac{\partial^2\psi}{\partial\,x^2}=- \left( \cfrac { n }{ x } \right) ^{ 2 } \)
\(\int { \cfrac { 1 }{ \psi } \cfrac { \partial }{ \partial \, x } \left\{ \cfrac { \partial \, \psi }{ \partial \, x } \right\} } \, d\, x=-\int { \left( \cfrac { n }{ x } \right) ^{ 2 } } \, d\, x\)
Integration by parts,
\(\cfrac { 1 }{ \psi } \cfrac { \partial \, \psi }{ \partial \, x } -\int { \cfrac { \partial \, \psi }{ \partial \, x } \cfrac { \partial }{ \partial \, x } \left\{ \cfrac { 1 }{ \psi } \right\} } d\, x=\cfrac { n^{ 2 } }{ x } +A\)
Integrate again along \(x\),
\( \int { \cfrac { 1 }{ \psi } \cfrac { \partial \, \psi }{ \partial \, x } d\, x+\int { \int { \cfrac { 1 }{ \psi ^{ 2 } } \cfrac { \partial \, \psi }{ \partial \, x } } } d\, \psi } \, d\,x=\int { \cfrac { n^{ 2 } }{ x } +A } \,d\, x\)
\( \int { \cfrac { 1 }{ \psi } } d\, \psi +\int { \int { \cfrac { 1 }{ \psi ^{ 2 } } } } \left( d\, \psi \right) ^{ 2 }=n^{ 2 }ln(x)+Ax+C\)
Let \( B={ e }^{ \cfrac { C }{ n^{ 2 } } }\),
\( ln(\psi )-\int { \cfrac { 1 }{ \psi } } d\, \psi =n^{ 2 }ln(Bx)+ln(e^{ { Ax } })\)
\(n^{ 2 }ln(Bx)+ln(e^{ { Ax } })=0\)
\(Bx=e^{ -\cfrac { Ax }{ n^{ 2 } } }\)
\(x=e^{ -\cfrac { Ax+C }{ n^{ 2 } } }\)
A solution plot of the above \(y=x\) and \(y=e^{ -\cfrac { Ax+C }{ n^{ 2 } } }\) is given below,
A zoomed version of these discrete solutions to \(x\) is given below,
Discrete values of \(x\) bunches up in a characteristic way for high values of \(n\), but the dependence of \(x\) on \(n\) is exponential of the reciprocal squared; not \(\cfrac{1}{n}\).
\(x\) is not unbounded, for \(\psi\) to be positive,
\(x\le x_a\),
and
\(x\le2x_z\)
since, \(x_a=2x_z\).
\(x\) is not unbounded, for \(\psi\) to be positive,
\(x\le x_a\),
and
\(x\le2x_z\)
since, \(x_a=2x_z\).
