Hydrogen atom wave equation

The most obvious defect of the Thomas-Fermi model is the neglect of interaction between electrons, but even in the most advanced modern methods this interaction still presents the most difficult problem. The most useful practical procedure to calculate the electronic structure of complex atoms is by means of the Hartree-Fock procedure, which is not by solution of the atomic wave equation, but by iterative numerical procedures, based on the hydrogen model. In this method the exact Hamiltonian is replaced by... 

It is interesting to notice that when ) = 4, the number of linearly independent hyperspherical harmonics belonging to a given value of A is (A -t- 1), i.e., 1,4,9,16,.. and so on - exactly the same as the degeneracy of the solutions to the Schrodinger equation for a hydrogen atom. V. Fock was, in fact, able to show that the Fourier transforms of the hydrogen atom wave functions can be written in the form [27] ... 

The miderstanding of the quantum mechanics of atoms was pioneered by Bohr, in his theory of the hydrogen atom. This combined the classical ideas on planetary motion—applicable to the atom because of the fomial similarity of tlie gravitational potential to tlie Coulomb potential between an electron and nucleus—with the quantum ideas that had recently been introduced by Planck and Einstein. This led eventually to the fomial theory of quaiitum mechanics, first discovered by Heisenberg, and most conveniently expressed by Schrodinger in the wave equation that bears his name. 

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. 

By this time, we have introduced so many approximations and restrictions on our wave function and energy spectrum that is no longer quite legitimate to call it a Schroedinger equation (Schroedinger s initial paper treated the hydrogen atom only.) We now write... 

Now that the wave and particle pictures were reconciled it became clear why the electron in the hydrogen atom may be only in particular orbits with angular momentum given by Equation (1.8). In the wave picture the circumference 2nr of an orbit of radius r must contain an integral number of wavelengths... 

In 1926 Erwin Schrodinger (1887-1961), an Austrian physicist, made a major contribution to quantum mechanics. He wrote down a rather complex differential equation to express the wave properties of an electron in an atom. This equation can be solved, at least in principle, to find the amplitude (height) of the electron wave at various points in space. The quantity ip (psi) is known as the wave function. Although we will not use the Schrodinger wave equation in any calculations, you should realize that much of our discussion of electronic structure is based on solutions to that equation for the electron in the hydrogen atom. 

The quantum number ms was introduced to make theory consistent with experiment. In that sense, it differs from the first three quantum numbers, which came from the solution to the Schrodinger wave equation for the hydrogen atom. This quantum number is not related to n, , or mi. It can have either of two possible values ... 

Actually Schrddinger s original paper on quantum mechanics already contained a relativistic wave equation, which, however, gave the wrong answer for the spectrum of the hydrogen atom. Due to this fact, and because of problems connected with the physical interpretation of this equation, which is of second order in the spaoe and time variables, it was temporarily discarded. Dirac took seriously the notion of first... 

The Interaction of Simple Atoms.—The discussion of the wave equation for the hydrogen molecule by Heitler and London,2 Sugiura,3 and Wang4 showed that two normal hydrogen atoms can interact in either of two ways, one of which gives rise to repulsion with no molecule formation, the other... 

The parameter Z is an effective atomic number whose value is determined by the minimization of in equation (9.2). Since the hydrogen-like wave functions 01 and 02 are normalized, we have... 

In many problems for which no direct solution can be obtained, there is a wave equation which differs but slightly from one that can be solved analytically. As an example, consider die hydrogen atom, a problem that was resolved in Section 6.6. Suppose now that an electric field is applied to the atom. The energy levels of the atom are affected by the field, an example of the Stark effect. If die field (due to the potential difference between two electrodes, for example) is gradually reduced, the system approaches that of the unperturbed hydrogen atom. 

The interaction between an electron and a nucleus in a hydrogen atom gives rise to a potential energy that can be described by the relationship -e2/r. Therefore, using the Hamiltonian operator and postulate IV, the wave equation can be written as... 

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. 

In summary, the wave mechanical treatment of the hydrogen atom assumes that the electronic motion is described by tp, obtained as solutions of the equation (in atomic units)... 

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