Schrodinger energy hydrogen-like atom

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... 

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... 

The direct variational method has been used to solve Schrodinger equation (4.2) with respect to Eqs. (4.3), (4.4), and (4.5). Hydrogen-like atom wave functions 2p and 3p have been taken as trial functions for ground and exited states respectively. Two-fermion wave functions were written in conventional form as a product of coordinate part and symmetric or asymmetric spin part for triplet (spin S = 1) or singlet (spin E = 0) state. The energy level positions 23 (S = 0), 33 (S = 0),... 

From this equation we understand that the nonielativistic Schrodinger energy eigenvalue for hydrogen-like atoms with point-like Coulomb nucleus, Eq. (6.19), is obtained only after subtraction of the rest energy,... 

To translate the physics of atoms to that of quantum dots, it is necessary to modify the mass of the electron using an effective electron mass. Calculation of the quantized energy values of a quantum dot can be accomplished numerically by solving the Schrodinger equation of the hydrogen-like atom with an effective electron mass. ... 

Erwin Schrodinger developed an equation to describe the electron in the hydrogen atom as having both wavelike and particle-like behaviour. Solution of the Schrodinger wave equation by application of the so-called quantum mechanics or wave mechanics shows that electronic energy levels within atoms are quantised that is, only certain specific electronic energy levels are allowed. 

Under the first assumption, each electron moves as an independent particle and is described by a one-electron orbital similar to those of the hydrogen atom. The wave function for the atom then becomes a product of these one-electron orbitals, which we denote P (r,). For example, the wave function for lithium (Li) has the form i/ atom = Pa ri) Pp r2) Py r3). This product form is called the orbital approximation for atoms. The second and third assumptions in effect convert the exact Schrodinger equation for the atom into a set of simultaneous equations for the unknown effective field and the unknown one-electron orbitals. These equations must be solved by iteration until a self-consistent solution is obtained. (In spirit, this approach is identical to the solution of complicated algebraic equations by the method of iteration described in Appendix C.) Like any other method for solving the Schrodinger equation, Hartree s method produces two principal results energy levels and orbitals. 

The preceding sections have been devoted to hydrogen-like species containing one electron, the energy of which depends only on n (equation 1.16) the atomic spectra of such species contain only a few lines associated with changes in the value of n (Figure 1.3). It is only for such species that the Schrodinger equation has been solved exactly. 

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.

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