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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 the principle quantum number 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
Energy Levels Probability Density respectively; the 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".

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