Wavefunctions spherical wave boundary condition

Equ. (7.20) describes for an out -state the asymptotic behaviour of the stationary wavefunction. As discussed above, the characteristic property of this state is that the incoming spherical waves e Kr/r have the scattering amplitude /(-)( ). It is this minus sign in the exponential term of the incoming spherical waves which is kept as a superscript to characterize the out -state, and the relation described by equ. (7.20) is frequently called the incoming spherical waves boundary condition. Hence, one should not mix up the state with the waves. 

There are now two independent angular coordinates, 6 and cp (besides the fixed distance R to the centre of the sphere) and wavefunctions that can be expressed as products of 0(0) and (p). The wave equation can be split into two, one for each variable 6 and cp. Each one of the functions 0(0) and (0) is subject to boundary conditions, in a similar way to the motion on a circular ring. Accordingly, two quantum numbers arise. The complete solutions Q 6) (p) are known as spherical harmonic functions and the allowed energies are given by an expression that resembles Eq. (2.72) ... 

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