The Normal Hydrogen Atom

18 A more detailed account of the Bohr theory of the hydrogen atom is given in Chap. 2 and Apps. II and III. 

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The eigenfunction 100, the electron density p = s10o, and the electron distribution function D = 4 x rJ p of the normal hydrogen atom as functions of the distance r from the nucleus. 

Fig. 1-2.—The wave function u, its square, and the radial probability distribution function 47rrVi fo the normal hydrogen atom.

Fig. 1-3.—A drawing illustrating the decrease in Electron density with increasing distance from the nucleus in the normal hydrogen atom.

The electron distribution function as given by quantum mechanics for the normal hydrogen atom has been discussed briefly in Chapter 1. The corresponding electron distribution functions for other orbitals will be discussed in the following chapter. 

These two equations are easily solved. It is found that the radius of the circular Bohr orbit for quantum number n is equal to W/4xaZnt. This can be written as n ao/Zy in which a0 has the value 0.530 A. The speed of the electron in its orbit is found to be v = 2irZe2/nh. For the normal hydrogen atom, with Z = 1 and n = 1, this speed is 2.18 X 108 cm/sec, about 0.7 percent that of the speed of light. 

The dimensions of the polarizability a are those of volume. The polarizability of a metallic Bphere is equal to the volume of the sphere, and we may anticipate that the polarizabilities of atoms and ions will be roughly equal to the atomic or molecular volumes. The polarizability of the normal hydrogen atom is found by an accurate quantum-mechanical calculation to be 4.5 ao that is, very nearly the volume of a sphere with radius equal to the Bohr-orbit radius a0 (4.19 a ). 

The index number refers to the principal quantum number and corresponds to the K shell designation often used for the electron of the normal hydrogen atom. The principal quantum number 2 corresponds to the L shell, 3 to the M shell, and so on. The notation s (also p, cl, f to come later) has been carried over from the early days of atomic spectroscopy and was derived from descriptions of spectroscopic lines as sharp, principal, diffuse, and fundamental, which once were used to identify transitions from particular atomic states. 

The total energy required to remove the electron from the normal hydrogen atom to infinity is hence... 

Flo. 21-1.—The functions p, V, and 4irr2i/ V for the normal hydrogen atom. The dashed curve represents the probability distribution function for a Bohr orbit. 

The correct value of a for the normal hydrogen atom, given by the second-order perturbation theory (footnote at end of preceding chapter) is... 

Let us now consider a simple example,1 the second-order Stark effect of the normal hydrogen atom, using essentially the method of Epstein (mentioned above). This will also enable us to introduce and discuss a useful set of orthogonal functions. 

It is interesting to note that if, in discussing the normal state of a system, we take as the zero of energy the first unperturbed excited level, then the sum is necessarily positive and the approximate treatment gives a lower limit to WIn the problem of the normal hydrogen atom this leads to... 

Problem 64-2. Evaluate the momentum wave function for the normal hydrogen atom,... 

Because of the nature of the equations of quantum mechanics that describe the electron in the normal hydrogen atom, it has been decided that it is not right to say that the electron moves about the nucleus in an orbit. Instead, the electron is said to occupy an orbital. The orbital that is occupied by the electron in the normal state (most stable state) of the hydrogen atom is called the s orbital. The number 1 is the value of the principal quantum number n. 

The Bohr orbit for the normal hydrogen atom has radius 53.0 pm. What is the radius for the first excited orbit, with n = 2 For the orbit with n = 3 ... 

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