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Homework Statement
hey im vaibhav,16 an 12th grade..just as pastime i tried to solve schrodinger equation in the 1D 2D 3D spaces. i got the 1D solution(not quantum oscillator), i seperated in 2D by polar coordinates but there is a problem in the radial equation
as for 3D i know that the solutions are such that i wont be doing em till im a wee bit older
but im just curious as the polar and radial eqautions are simply horrifying to contemplate solving
Homework Equations
2D RADIAL EQUATION:
[tex]\hbar \frac{d}{dr} (\frac{rdR}{dr})+\frac{2m(Er+I)R}{\hbar}=[/tex][tex]\hbar[/tex]a^{2}
where;
R"=second Derivative of R wrt r..similar for R'..R being radial wavefunction
r=radial distance from origin
I=intensity factor for a central force field ie of circular/spherical symmetry..(basically a constant explaining the field for a particular body)
The Attempt at a Solution
after solving,normalizing(0,2\pi) i got the angular part of the 2D wavefunction as
Y_{a}([tex]\theta[/tex])=[tex]\frac{exp(ia\theta)}{\sqrt{2\pi}}[/tex]
where "a" is quantized as any integer
exp(x)=e^{x}
please see the attachments for the radial,polar equations in 2D;3D
also,
it would be helpful if anyone could suggest an easier way to get to the solutions of the radial, polar wavefunctions(im not that experienced..)..and also quantizing them..
thanks,
Vaibhav
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