Relative Schrodinger equation spherical polar coordinates

Here, r denotes the scalar distance of the electron measured from the centre of mass (assumed to be at rest) of the complete system in the infinite proton mass approximation this corresponds to the electron-proton distance. The resulting reduced Schrodinger equation for the relative electronic motion is found to be soluble in several different orthogonal coordinate systems. In particular, in spherical polar coordinates (r, 0, ) referred to the centre of mass, the natural unconfined ranges of these variables are... 

The kinetic energy operator,however,is almost separable in spherical polar coordinates, and the actual method of solving the differential equation can be found in a number of textbooks. The bound solutions (negative total energy) are called orbitals and can be classified in terms of three quantum numbers, n, I and m, corresponding to the three spatial variables r, d and q>. The quantum numbers arise from the boundary conditions on the wave function, i.e. it must be periodic in the 0 and q> variables, and must decay to zero as r oo. Since the Schrodinger equation is not completely separable in spherical polar coordinates, there exist the restrictions n > /> m. The n quantum number describes the size of the orbital, the / quantum number describes the shape of the orbital, while the m quantum number describes the orientation of the orbital relative to a fixed coordinate system. The / quantum number translates into names for the orbitals ... 

The relative Schrodinger equation can he solved in spherical polar coordinates hy separation of variables, assuming that... 

The relative Schrodinger equation cannot be solved in Cartesian coordinates. We transform to spherical polar coordinates in order to have an expression for the potential energy that contains only one coordinate. Spherical polar coordinates are depicted in Figure 17.3. The expression for the Laplacian operator in spherical polar coordinates is found in Eq. (B-47) of Appendix B. The relative Schrodinger equation is now... 

The time-independent Schrodinger equation for a hydrogen atom was separated into a one-particle Schrodinger equation for the motion of the center of mass of the two particles and a one-particle Schrodinger equation for the motion of the electron relative to the nucleus. The motion of the center of mass is the same as that of a free particle. The Schrodinger equation for the relative motion was solved by separation of variables in spherical polar coordinates, assuming the trial function... 

We shall be concerned mainly with the wavefunctions of electrons in atoms and in this case the predominant contribution to the potential comes from the Coulomb attraction of the nucleus. This potential is. spherically symmetric and therefore V(r) is a function of tlie radial coordinate r alone, This enables the Schrodinger equation to be separated into three differential equations which involve r, 0, and (j> separately. If we consider the motion of a single electron of mass m about a nucleus of mass M we can separate off the centre-of-mass motion and consider only the relative motion of the electron. In spherical polar coordinates equation C3.8) becomes (Problem 3.3)... 

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