Schrodinger equation hydrogen-like atom

Stark effect of a hydrogen-like atom, using the Schrodinger wave mechanics. Their equation, obtained independently and by different methods, is... 

In the asymptotic region, an electron approximately experiences a Z /f potential, where Z is the charge of the molecule-minus-one-electron ( Z = 1 in the case of a neutral molecule) and r the distance between the electron and the center of the charge repartition of the molecule-minus -one-electron. Thus the ip orbital describing the state of that electron must be close to the asymptotic form of the irregular solution of the Schrodinger equation for the hydrogen-like atom with atomic number Z. ... 

Other atoms with only a single electron (He+, Li2+, etc.) are known as hydrogen-like atoms. The Schrodinger equation for such a system is the same as that for the hydrogen atom... 

The requirement that the basis functions should describe as closely as possible the correct distribution of electrons in the vicinity of nuclei is easily satisfied by choosing hydrogen-like atom wave functions, t], the solutions to the Schrodinger equation for one-electron atoms for which exact solutions are available ... 

The simplest analytical radial orbitals may be found by solving the radial Schrodinger equation for a one-electron hydrogen-like atom with arbitrary Z. They are usually called Coulomb functions and are expressed... 

The hydrogen atom, consisting of a proton and only one electron, occupies a very important position in the development of quantum mechanics because the Schrodinger equation may be solved exactly for this system. This is true also for the hydrogen-like atomic ions He+, Li2+, Be3+, etc., and simple one-electron molecular ions such as Hj. 

The non-relativistic Schrodinger equation for a confined hydrogen-like atom 205... 

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... 

This is the Schrodinger equation for a hydrogen-like atom. The solution for E < 0, i.e. for the bound states, has the form... 

The standard approach to solving the Schrodinger equation for hydrogen-like atoms involves transforming it from Cartesian (x, y, z) to polar coordinates (r, 6, (p), since these accord more naturally with the spherical symmetry of the system. This makes it possible to separate the equation into three simpler equations, /(r) = 0, f(9) = 0, and /(0) = 0. Solution of the f[r) equation gives rise to the n quantum number, solution of the f(9) equation to the / quantum number, and solution of the f(quantum number. For each specific n = n, l = V and... 

These individual functions R, , and give rise to the three orbital quantum numbers n, i, and m. We have seen that solutions of the Schrodinger equation are possible only for certain values of the total energy E. For hydrogen-like atoms, the permitted total energy values are given by the equation... 

---