Hydrogen-like atom Hamiltonian operator

The //hl(1) and //hl(2) terms in the Hamiltonian operator give ground-state energy eigenvalues for a hydrogen-like atom after operating and integrating. 

A transition metal with the configuration t/ is an example of a hydrogen-like atom in that we consider the behaviour of a single (d) electron outside of any closed shells. This electron possesses kinetic energy and is attracted to the shielded nucleus. The appropriate energy operator (Hamiltonian) for this is shown in Eq. (3.4). 

If we replace the z-component of the classical angular momentum in equation (6.87) by its quantum-mechanical operator, then the Hamiltonian operator Hb for the hydrogen-like atom in a magnetic field B becomes... 

A well known operator is the Hamiltonian of an electron in centre-of-mass coordinates of a hydrogen-like atom... 

The Hamiltonian operator for a hydrogen-like atom (nuclear charge of Z) can in Cartesian coordinates and atomic units be written as eq. (1.33), with M being the nuclear and m the electron mass (m = 1 in atomic units). 

The Dirac Hamiltonian for hydrogen-like atoms, h, and the square as well as any component of the total angular momentum operator commute. 

These familiar relativistic one-electron operators, whose expectation values often served to define measures for relativistic effects, are only an approximation to the Dirac Hamiltonian of hydrogen-like atoms discussed in chapter 6. Nevertheless, they are also employed in the context of many-electron systems where the one-electron part in / of Eq. (11.1), i.e., Vnuo is approximated solely by the 2 x 2-analogous expressions for Ejoj - -... 

The reason a single equation = ( can describe all real or hypothetical mechanical systems is that the Hamiltonian operator H takes a different form for each new system. There is a limitation that accompanies the generality of the Hamiltonian and the Schroedinger equation We cannot find the exact location of any election, even in simple systems like the hydrogen atom. We must be satisfied with a probability distribution for the electron s whereabouts, governed by a function (1/ called the wave function. 

The hydrogen molecule, like the He atom, poses a real problem the Hamiltonian operator contains a term representing the repulsion between the electrons, and the presence of this term makes an exact solution of the Schrodinger equation impossible. In the case of the helium atom we turned to the hydrogen atom for guidance in the choice of approximate wavefunctions. In the case of the hydrogen molecule we turn to the hJ ion and assume that the wavefunction may be approximated by the product of two molecular orbitals... 

In order to solve the wave equation for the hydrogen atom, it is necessary to transform the Laplacian into polar coordinates. That transformation allows the distance of the electron from the nucleus to be expressed in terms of r, 9, and (p, which in turn allows the separation of variables technique to be used. Examination of Eq. (2.40) shows that the first and third terms in the Hamiltonian are exactly like the two terms in the operator for the hydrogen atom. Likewise, the second and fourth terms are also equivalent to those for a hydrogen atom. However, the last term, e2/r12, is the troublesome part of the Hamiltonian. In fact, even after polar coordinates are employed, that term prevents the separation of variables from being accomplished. Not being able to separate the variables to obtain three simpler equations prevents an exact solution of Eq. (2.40) from being carried out. 

Since Hj does not have spherical symmetry like the hydrogen atom the angular momentum operator L2 does not commute with the Hamiltonian, [L2,H] 7 0. However, Hj does have axial symmetry and therefore Lz commutes with H. The operator Lz = —ih(d/d) involves only the 0 coordinate and hence, in order to calculate the commutator, only that part of H that involves need be considered, i.e. 

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