Home Page Overview Site Map Index Appendix Illustration About Contact Update FAQ


Molecules


Hydrogen Atom

Beside some idealized cases, analytical solution of the Schrodinger equation can be obtained only for hydrogen atom which has one electron with mass m, and charge e moving about an atomic nucleus. The energy levels are given by:

En = - me4 / (3222n2) = - 13.6 / n2 ev

where n is any integer from 1 to . Some of the energy levels are shown in Figure 12-07a while the probability density corresponding to different quantum states are shown in Figure 12-07b where n, l, and m are the total, orbital, and magnetic quantum number respectively; the
Energy Levels Probability Density probability densities have either spherical symmetry or rotational symmetry about the z axis. When the electron jumps from a higher energy level En+1 to a lower one En, a photon with frequency = (En+1 - En)/h is released (see the atomic transition series in Figure 12-07a). Excitation is the reversed process when the electron in energy level En absorbs a photon with frequency . Incidentally, the semi-classical Bohr model of H atom can also account for such H atom structure as well. The theory portrays the electron moving around the atomic nucleus similar to the Sun-Earth system but imposed a condition that the electron wave closed on itself to become standing wave.

Figure 12-07a Energy Levels
[view large image]

Figure 12-07b Probability Density [view large image]



The state of the electron in an atom is specified by four quantum numbers. The principal quantum number n determines the energy level; its value runs from 1, 2, 3, ... For each n, the orbital quantum number l = 0, 1, 2, ... (n-1); it is related to the magnitude of angular momentum. Then for each l, the magnetic quantum number m can be -l, -l+1, ...l-1, l; it is related to the z component of the angular momentum (see Figure 12-07a). The spin quantum number s is either +1/2 or -1/2. The energy level En is said to be degenerate since it corresponds to two or more different measurable states of a quantum system, in other word, En is the same for the corresponding quantum numbers l and m.

For n = 1, l = 0, m = 0, there is only 2 possible quantum states for the electron, with s = +1/2 and -1/2 respectively.
For n = 2, l = 0, m = 0 and l =1, m = -1, 0, +1; there is a total of 2 + 6 = 8 possible quantum states. Therefore, it requires 2 electrons to complete the shell for n = 1, and 8 electrons to complete the shell for n = 2, ...and so on (see Table 13-01 in Topic Atom). The orbital quantum number l is often designated by a letter, s for l = 0, p for l = 1, d for l = 2, and f for l = 3 ...

The quantum number l is non-additive, (e.g., two of the quantum numbers l1, l2 are added as vectors, they can take on the values of l1+l2, l1+l2-1, ..., |l1-l2| ) while m is additive (e.g., m' = m1 + m2 only) and relates to an Abelian group (e.g., the two dimensional rotation about the z-axis). States having the same non-additive quantum numbers but differing from each other by their additive quantum numbers are said to belong to the same multiplet. The number of members of a multiplet is called its multiplicity. For a given multiplet l the multiplictiy is equal to 2l+1.

See some "Hydrogen Atom Images".

Go to Next Section
 or to Top of Page to Select
 or to Main Menu

.