Solution of Schrodingers Equation for the Kepler Problem

Solution of Schrodinger s Equation for the Kepler Problem 

Changing to three-dimensional polar co-ordinates r, 0, cf), we obtain the equation 

The second equation is the equation defining the spherical harmonic Pf GOs0), when A has the value 1) and m Z for other values the equation has no finite one-valued solution. We shall prove this generally by combining the 6 and -relationships in the general surface harmonic Yi 9, cf)). If for brevity we introduce the notation 

A general solution of this equation is obtained in the following way. We consider a homogeneous polynomial Ui of the Z-th degree in x, y, z, which satisfies Laplace s equation 

It is possible to prov( . tliat no other vaf iu s of Agiv( - finite and continuous one-valued solutions of the diifer( ntial ( quation. The ])r op(U values of the equation AY.+AF,= 0 

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