Wave equation and angular momentum

nie and Wn represent the electron mass and the mass of the nucleus, respectively V is the potential energy corresponding to the electron-nucleus attraction  

By making use of the relations between cartesian coordinates and spherical polar coordinates (Fig. 3.1), 

Problem 3.1 Consider a free particle (F = 0) moving in a plane xy. 

We can then explore the possibility of expressing the wavefunction by a product of two functions, one only dependent on r and the other a function 

Since the left side of this equation is a function only of r (i.e. it does not depend on the values given to 6 and 4 ) and the right side is a function only of 6 and 0 (i.e. it does not depend on the value of r), this equality requires that both members of the equation are equal to a constant, /x. Rearranging leads to  

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