An strange equation

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SE equation in (1d) [tex]a\frac{d^{2}\phi}{dx^{2}}+V(x)\phi=E_{n}\phi [/tex] (1)

WKB approach [tex]\phi=exp(iW/\hbar)[/tex] (2)

[tex]W=[2m(E_{n}-V)]^{1/2}[/tex]

then from (2) we can obtain the potential and introduce it in (1) to get a differential NOn-linear equation...

[tex]a\frac{d^{2}\phi}{dx^{2}}+(b/phi)^{2}(\frac{d\phi}{dx})^{2}+c\phi=0 [/tex] we could use this to prove that always the eigenfucntion of the Hamiltonian [tex]\phi[/tex] will exist...
 
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493
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[tex]a\frac{d^{2}\phi}{dx^2}+V(x)\phi-E_{n}\phi=0 [/tex]

[tex]\phi=exp(iW/\hbar)[/tex]

combinign both and substituting the potential of V as a function of [tex]\phi[/tex] we could form the equation:

[tex]a\frac{d^{2}\phi}{dx}+(b/\phi)^{2}(\frac{d\phi}{dx})^{2}+c\phi=0[/tex] or if we call [tex]a=-\hbar^{2}/2m [/tex]

[tex]a(\frac{d^{2}\phi}{dx}+(1\phi)(\frac{d\phi}{dx})^{2})=0[/tex]

EDIT:sorry it would be...[tex]a(\frac{d^{2}\phi}{dx}+(1/\phi)(\frac{d\phi}{dx})^{2})=0[/tex]

from this equation and applying existence theorem we would get that the eigenfunctions [tex]\phi[/tex] exist so the potential V(x) will always exist for the WKB case....
 
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