Selection rules for the hydrogen atom

Again this relation requires A J to be odd if the integral is to be invariant under reflection. Detailed calculation using the recurrence formula shows that AJ = 1 only. 

The types of transitions and the polarizations that produce them are shown in Fig. 25.13 the vertical axis represents the energy the horizontal axis is simply used to space out the values of m corresponding to a particular value of J. 

The wave functions for the hydrogen atom have the form of a product of a radial function (a function of r only) and the rigid rotor functions. The selection rules for the rotor functions must be the same as those obtained above, namely. 

Thus the only remaining question concerns the radial functions. The transition-moment integral has the form 

However, the functions in the integral depend only on r, which is unaffected by any rotation of the axes or by any reflection in a plane. We conclude that any of the symmetry operations will leave r and functions of r unchanged. 

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Verify the selection rules for the hydrogen atom as given in the last paragraph of Section 12.3.3. 

The integrals over the angular functions associated with the integrals over z are the same as those involved in determining the selection rules for the hydrogen atom, where it was shown that the integrals vanish unless... 

VIII. Time-Dependent Perturbations Radiation Theory Time-Dependent Perturbations, 107. The Wave Equation for a System of Charged Particles under the Influence of an External Electric or Magnetic Field, 108. Induced Emission and Absorption of Radiation, 110. The Einstein Transition Probabilities, 114. Selection Rules for the Hydrogen Atom, 116. Selection Rules for the Harmonic Oscillator, 117. Polarizability Rayleigh and Raman Scattering, 118. 

As a second example of the determination of selection rules from the properties of special functions, consider the hydrogen atom. At any given instant the dipole moment is ft = er, where r describes the position of the electron with respect to the proton and e is the electronic charge. The wavefiinctions for the hydrogen atom are given by... 

For the hydrogen atom, the parity is determined by /. (See Section 1.17.) The selection rule that A/= 1 is in agreement with the rule that parity changes in electric-dipole transitions. 

The selection rule for l, discussed in Section 40d, allows only transitions with Al = +1 for the hydrogen atom. The lines of the Lyman series, with lower state that with n = 1 and l — 0, are in consequence due to transitions from upper states with 1 = 1. The radial electric dipole moment integral... 

In its application to hydrogen-like spectra the most important result of the Correspondence Principle was the prediction of a selection rule for the quantum number k. When the atom... 

The selection rules for the different types of polarization are, of course, significant only when there is a unique z direction, due, for example, to the presence of a uniform magnetic field. This subject will be discussed more fully in the following chapter, where the Zeeman effect is considered. It is apparent from the derivation of the above selection rules that they are not limited to the hydrogen atom but are valid for any central field problem where the angular portion of the wave function is identical with that of the hydrogen atom. 

So it is the number of electrons and not the number of atoms which determines the selection rule. Therefore, the selection rules for hydrogen migration in thermal sigmatropic shifts can be summarized as follows as given in the table ... 

The ene reaction has proved to be particularly powerful in synthesis when carried out intramolecularly. The usual increase in rate for an intramolecular reaction allows relatively unreactive partners to combine. Thus the diene 6.13 gives largely (14 1) the cis disubstituted cyclopentane 6.15 by way of a transition structure 6.14. It is important to recognize that the selective formation of the ci j-disubstituted cyclopentane has nothing to do with the rules for pericyclic reactions. It is a consequence of the lower energy when the trimethylene chain spans the two double bonds in such a way as to leave the hydrogen atoms on the same side of the folded bicyclic structure. This... 

This is the first example we will encounter of selection rules for allowed transitions. Physically, the selection rules arise because the electric field needs a dipole moment in order to interact with the atom, and only these specific changes in the quantum numbers create a dipole moment. More generally, the selection rules for absorption in a hydrogen atom are... 

According to the selection rules, one-photon absorption occurs only if the change in angular momentum (change in L) is +1 or -1 (Al = 1, A/ = 0, 1 (0 o 0 not allowed), AL = 0, 1, AS = 0) (Al is according to the hydrogenic atom model, whereas AL is for multielectron atoms). The selection rules allow transition in one-photon absorption only to the p states from the s ground state as a result only even-to-odd parity is allowed. 

The operation of the selection rule for l for hydrogen and hydrogenlike ions can be seen by the study of the fine structure of the lines. The phenomena are complicated, however, by the influence of electron spin.1 In alkali atoms the levels with given n and varying l are widely separated, and the selection rule for l plays an important part in determining the nature of their spectra. Theoretical calculations have also been made of the intensities of lines in these spectra with the use of wave functions such as those described in Chapter IX, leading to results in approximate agreement with experiment. 

We have discussed the energy of the hydrogen atom earlier. The energy level system for the lithium atom is shown in Fig. 24.8. The lowest filled level, Is, is not shown. In the ground state, the term is 2 S. Since the selection rule requires Al = AL = +1, the only... 

Atoms and molecules can adsorb and emit radiation to change their internal energy states. The electronic transitions of the hydrogen-like atoms have already been mentioned. The quantization of the energy levels restricts the possible wavelengths of the radiation to discrete spectral lines. Only certain transitions are allowed and these are given by separate selection rules for electronic. 

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