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The ground state wavefunction for the free hydrogen atom in spherical coordinates is given as

where r' is the electron-nucleus distance and A is a normalization constant.

Without loss of generality, we can assume that the nucleus is located on the z axis. The trial wavefunction for the ground state can be written as

where r0 is the radius of the spherical confining domain, A is a normalization constant, and

Here, 6 is the usual polar angle of the spherical coordinates.

The results of the variational calculations of the ground state energy with this trial wavefunction are displayed in Table 1. A comparison with the results obtained in [85, 86] is also shown. Figure 21 shows the ground state energy for different sizes of the confining sphere as a function of (a/r0).

In the case of cylindrical coordinates, it is obvious that the Schrodinger equation for the Coulomb potential is nonsep-arable; however, the application of the variational method is still possible in the same context as could be done for spherical coordinates, as we shall see.

If the nucleus of the atom is located at the x-axis, at a distance b (as depicted in Fig. 22), the electron-nucleus distance r' and the position of the electron relative to the origin on the axis of the cylinder r are related by r' = \r'\ = \r — b\ (140)

That is, r' = [(x — b)2 + y2 + z2 ]1/2 = [p2 + b2 — 2pb cos 0 + z2 ]1/2

The trial wavefunction can be written as

where p0 is the radius of the cylinder and \r — b\ is given by Eq. (141). The ground state energies resulting from the

Table 1. Ground state energy for the off-center hydrogen atom within an impenetrable spherical box as a function of the position of the nucleus (relative to the center of the confining sphere) for various radii of the box.

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