Angular momentum wave-mechanical components

What is the significance of these operators The electronic state is given by the wave function ifj. In order to decide the question whether definite values of the components of angular momentum round the three co-ordinate axes belong to this state, we have to apply the above operators, according to the rules of wave mechanics, to the wave function ijj, i.e. we perform the differentiations involved in the operators. There are now two possibilities either this operation reproduces the wave function except for a constant factor, or it does not. In the first case, the wave function is also at the same time a proper function of the equation of angular momentum... 

Two wave-functions with the same energy are obtained, contradicting the assumption of a non-degenerate state. Thus, the wave-function can be taken real (cp2 = 0). The orbital angular momentum is also a real quantity. On the other hand, the corresponding quantum mechanical operator of e.g. the x-component. Lx = iHy i - z- ) is imaginary. Thus, the expectation value is zero, Lx l ) = 0. 

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