Wavefunctions for the Hydrogen Atom

Now let us apply the Schrodinger equation to the hydrogen atom. The hydrogen atom with its single electron lends itself to a closed form solution of the Schrodinger equation. The purpose of this exercise is to show how the quantum numbers that will be used to characterize the wavefunctions of more complicated atoms arise from pure mathematical considerations. 

Note that the first term is a function of I only therefore, it must equal be to some constant, say —m. We now have an ordinary differential equation (ODE) for I , i.e.. 

The m is the azimuthal quantum number which, for reasons that become clear later, is also called the magnetic or projection quantum number. The normalization coefficient is 

The left side is now a fimction of 9 only. We set it equal to a separation constant ( +1). Now the 0 term can be obtained by solving the ODE 

This is an associated Legendre equation whose solution may be written 

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The radial parts of the wavefunctions for the hydrogen atom can be constructed from the general form of the associated Laguerre polynomials, as developed in Section 5.5.3. However, in applications in physics and chemistry it is often the probability density that is more important (see Section 5.4.1). This quantity in this case represents the probability of finding the electron in the appropriate three-dimensional volume element. 

The complete wavefunction for the hydrogen atom is a combination of the spherical harmonic, and the exponential-associated... 

In this case, the integration limits are 0 to oo, rather than — to -Ho . Also, because the magnitude of any velocity vector is independent of its direction, each value in G(v) actually represents a spherical shell of possible velocity vectors, as demonstrated in Figure 19.3. There is thus a 4t7V component as part of the infinitesimal. (This is akin to the argument used to get a physically useful description for the Is wavefunction for the hydrogen atom in section 11.11.) Using the linear probability functions g, g, and g we get... 

The potential function for the hydrogen atom is dependent only on the spherical polar coordinate r. Thus, the separation of variables carried out in going from Equations 9.18 through 9.22 remains valid for this problem. That means we immediately know that the wavefunctions for the hydrogen atom consist of some type of radial fimction, R(r),... 

From the separation of variables, we now have that the wavefunctions for the hydrogen atom are... 

In general, the Slater function is not an exact solution of any Schrodinger equation (except the Is- wavefunction, which is the exact solution for the hydrogen-atom problem). Nevertheless, asymptotically, the orbital exponent C is directly related to the energy eigenvalue of that state. Actually, at large distances from the center of the atom, the potential is zero. Schrodinger s equation for the radial function R(r) is... 

Plot the radial portions of the Is, 2s, 2p, 3s, and 3p hydrogen like atomic wavefunctions for the Si atom using screening concepts for any inner electrons. 

A complete treatment of this derivation can be found in Ref. [19]. The first three terms in the kinetic energy operator indicates the presence of a 3D harmonic oscillator, and the final two terms indicate the presence of a 2D rotator (as for the hydrogen atom). A similar conclusion was made by Auberbach et al. [22] where they use a semi-classical quantisation method and the molecule is said to undergo a unimodal distortion and then the semi-classical Hamiltonian is found to be separated into two parts - a harmonic oscillator part with three vibrational coordinates and a rotational part with two rotational coordinates. However, more progress in terms of specifying the wavefunctions of the system can be made by following a different approach. 

Here is yet another bizarre result of quantum mechanics for you to ponder. The lx wavefunction for a hydrogen atom is unequal to zero at the origin. This means that there is a small, but nonzero probability that the electron is inside the proton. Calculation of this probability leads to the so-called hyperfine splitting —the magnetic dipoles on the proton and electron interact. This splitting is experimentally measurable. Transitions between the hyperfine levels in the lx state of hydrogen are induced by radiation at 1420.406 MHz. Since this frequency is determined by... 

There have been several papers published on a(co) and y(co) for the hydrogen atom[85]-[90]. Shelton[89] used an expansion in Sturmian functions to obtain y values for Kerr, ESHG, THG and DFWM at a number of frequencies. A more straightforward and simpler method is to use the SOS approach and a pseudo spectral series based on the wavefunctions formed by the linear combinations ... 

It has been shown that, under appropriate conditions, the momentum distribution uj py for an individtial electronic state is directly meastired by electron-momentum spectroscopy (ref. 29). Figure 3.8 compares the experimental momentum distribution for the hydrogen atom ground state with the function calculated by the Fourier transform of the hydrogen Is orbital. In general, the electron-momentum spectroscopy results serve to evaluate wavefunctions at various levels of theory for a variety of atomic (and molecular) systems. 

In particular, for the hydrogen atom, it was recognized that the confinements by elliptical cones and by dihedral angles were pending. Section 5.1 in the Preview of Ref. [9] formulated the problem of the hydrogen atom confined by a family of elliptical cones identified in its Eqs. (123 and 124), with the boundary condition that the wavefunctions vanish in such cones, Eqs. (125 and 126). The corresponding solution [8] is the subject of Section 3.3. 

This important relation holds for all the states for the Dirichlet problem (see e.g. [5], Sect. XIII.15). However, the analogous relation is wrong for the other types of boundaries [29]. For example, the constant function in a sphere (that is, the wavefunction r/r(r) = const) satisfies the Neumann boundary conditions and has a zero value for the kinetic energy. Hence the energy functional is the mean value for the potential, that is, it equals —3/(2R) for the hydrogen atom. When the sphere radius R goes to zero, the energy value decreases, in contrast to Equation (2.4). 

But this equation is the Schrodinger equation for the hydrogen atom ( e// = 1) and its lowest solution is the Is AO of a hydrogenic atom. The required Sra.lence orbital is the second solution of the equation. This second solution is, of course, orthogonal to the lowest solution what is, in fact, required is the lowest solution of this Schrodinger equation which is orthogonal to the wavefunctions of the simulated electrons. 

A wavefunction, ip, is a solution to the Schrodinger equation. For atoms, wavefunctions describe the energy and probabihty of location of the electrons in any region around the proton nucleus. The simplest wavefunctions are found for the hydrogen atom. Each of the solutions contains three integer terms called quantum numbers. They are n, the principal quantum number, I, the orbital angular momentum quantum number and mi, the magnetic quantum number. These simplest wavefunctions do not include the electron spin quantum number, m, which is introduced in more complete descriptions of atoms. Quantum numbers define the state of a system. More complex wavefunctions arise when many-electron atoms or molecules are considered. 

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