Hydrogen-atom wave functions radial factors

The Radial Factor in the Hydrogen Atom Wave Function. The Energy Levels of the Hydrogen Atom... 

Apply the particle-in-a-box basis functions to the radial equation for the hydrogen atom for the / = 0 states. Recall that in Section 6.9, we expressed the radial factor in the H-atom wave function as R r) = r F r), where F(r) = 0 at r = 0. The variation function in this problem will have the form = r F r)Yf 6, ) take the dimensionless function F,(r,) to be a linear combination of 28 pib basis functions, where the box goes from r, = 0 to 27, where r, is defined in Section 6.9. Work out the proper forms for the integrals //y and 5. Find the estimates for the lowest three / = 0 energies. For the ground-state variation function, how many pib basis functions appear with a coefficient greater than 0.1 ... 

The radial factors of the hydrogen-like atom total wave functions ip r, 0, tp) are related to the functions Sni(p) by equation (6.23). Thus, we have... 

The wave function F for a single electron, in a hydrogen atom for example, may be written as a product of four factors. These are the radial function R(r), which is dependent only on the radial distance rfrom the nucleus two angular functions 0(0) and ( ), which depend only on the angles 0 and (j>... 

Figure 1.3 The normalized [see Chapter 3] radial wave function, Ri(r), for the Is atomic orbital in the hydrogen atom [Z = 1 ] presented as an EXCEL graph, constructed as described in the text. Note, that since Yqo = 1 for s-orbitals, the only difference between this function and the total Is atomic orbital is the factor (l/4jrf- due to the normalization constant over the angular coordinates.

FIGURE 6.8 Graphs of the radial factor in the hydrogen-atom (Z = 1) wave functions.The same scale is used in all graphs. (In some texts, these functions are not properly drawn to scale.)... 

The radial factor R(r) is also called the radial wave function, and the angular factor Y (9, cf) is also called the angular wave function. Each orbital, f, has three quantum numbers to define it since the hydrogen atom is a three-dimensional system. The particular set of quantum numbers confers particular functional forms to R(r) and Y(0,4>), which are most conveniently represented in graphical form. In Section 8-8, we will use various graphical representations of orbitals to deepen our understanding of the description of electrons in atoms. 

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