Hydrogen ground-state energy, Schrodinger

Show that the Is orbital givien in eqn 4.22 is a solution of Schrodinger s equation for the hydrogen atom with the correct ground-state energy, either by substitution into the radial equation (eqn 4.19), or, if you are feeling brave, by substitution into the full equation (eqn 4.17). You will find the latter method distinctly harder, and will need to use the result, applicable to any function /which depends on r only,... 

The conclusion of these comments is that the trial functions and the results reported in [3] do not provide a reliable description of the two-dimensional hydrogen atom ground-state energy for confinements by an angle and by a hyperbola. The solutions of the Schrodinger equation for the hydrogen atom confined by a hyperbola can also be constructed transparently and accurately using standard methods. 

Figure 6.6 Comparison of ground-state energies E[glZ scaled by I7 obtained tor hydrogen-iike atoms from Schrodinger quantum mechanics (horizontal line on top at -0.5 hartree), from Dirac theory with a Couiomb potential from a point-like nucleus (dashed line) and from Dirac theory with a finite nuclear charge distribution of Gaussian form (thin black line). The highest energy of the positronic continuum states, -2meC, appears as a thick black line, which is bent because of the l/Z scaling.

We consider the problem of s-state energy shift according to the perturbation theory. Such analysis was performed for the pionic hydrogen in Ref. (Lyubovitskji and Rusetsky, 2000). Let Ho + Hc be the unperturbed Hamiltonian, whereas V is considered as a perturbation. The ground-state solution of the unperturbed Schrodinger equation in the center of mass (CM) system frame (E — Ho — Hc) To(0)) = 0, with E = M + m + E, is given by... 

Unfortunately, the Schrodinger equation can be solved exactly only for one-electron systems such as the hydrogen atom. If it could be solved exactly for molecules containing two or more electrons,3 we would have a precise picture of the shape of the orbitals available to each electron (especially for the important ground state) and the energy for each orbital. Since exact solutions are not available, drastic approximations must be made. There are two chief general methods of approximation the molecular-orbital method and the valence-bond method. 

Table 4 Convergence of the symmetrized Rayleigh-Schrodinger perturbation expansion for the interaction of a ground-state helium atom with a hydrogen molecule at R = 6.5 bohr. The B83 basis set was used. (n) and 6(n) are defined as in Table 1. Energies are in /ihartree.

We consider a hydrogen-like atom, with nuclear charge Z, enclosed in a spherical well, of radius R, with an impenetrable wall. The nucleus is assumed fixed at the centre of the well and we note that, for finite R, it is not therefore possible to separate out the translational motion of the centre of mass of the system. Pupyshev [18] proved that, for the ground state, the energy is a minimum when the nucleus is at r = 0. In a non-relativistic approximation the Schrodinger equation for the electronic motion is1... 

For systems of two or more electrons, we are not able to obtain analytical solutions in closed form, as in the case of hydrogen. Fortunately, a variational principle exists for the Schrodinger equation. According to this principle, any trial for the ground state always yields an upper bound to the exact energy E ... 

The first ionisation limit of a many-electron atom corresponds to the ground state of the corresponding or parent ion. Higher thresholds correspond to excited states of the parent ion. Apart from the special case of He, which has a hydrogenic parent ion, they are not simply related to fundamental constants. The many-electron Schrodinger equation must also be solved for the parent ion in order to determine the energies of the thresholds. 

Like the Schrodinger equation of the H atom, the Schrddinger equation of the hydrogen molecular ion has infinitely many solutions for each value of R. We shall, however, only discuss the ground state and the first excited state. The electronic energies of these states as functions of R are shown in Fig. 7.3. 

In [194] the authors presented the comparison of three variational approaches for solving the radial Schrodinger equation. In the comparison special attention was spent to accurate computation of the vibrational energy levels lying close to the dissociation limit in bound electronic states of diatomic molecules. The above mentioned methodologies are studied on the ground state of the hydrogen molecule. 

The orbitals j/ and ij/2 can be found by solving two onq-electron Schrodinger equations. Since each of these involves only the three coordinates of one electron, they can be solved. The solutions are a set of orbitals similar in shape to those of hydrogen we can designate them by the same kind of symbol. The ground state of helium is naturally one in which both electrons occupy the orbital of lowest energy, i.e., the Is orbital we accordingly depict this state as (Is). ... 

---