Angular momentum and magnetic moment of a one-electron atom

Angular momentum and magnetic moment of a one-electron atom 

8) is modified by taking into account that - (/is the kinetic energy K because of the fact that in this particular case the potential energy of free rotation is zero (K = E), and m = is the reduced mass of the rigid rotator. Since the angular part of the wavefunction is the product of two functions Y(0, (p) = O((p)0(0), we can obtain 

Note that in the last equation each term depends either on cp or on 0. They can be grouped in the following manner  

A function 0(0) is on the left-hand side and a function 0((p) is on the right-hand side. Therefore, this equality can be satisfied only if both sides are equal to a constant value. Since the function 0((p) (7.5.18) is already known, we can find that constant. In fact, this function is equal to  

This illustrates the previous statement that the wavefunction of the hydrogen atom can be presented as a product (7.5.6). 

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