Bohr’s frequency condition

Transition probabilities. The interaction of quantum systems with light may be studied with the help of Schrodinger s time-dependent perturbation theory. A molecular complex may be in an initial state i), an eigenstate of the unperturbed Hamiltonian, Jfo I ) = E 10- If the system is irradiated by electromagnetic radiation of frequency v = co/2nc, transitions to other quantum states /) of the complex occur if the frequency is sufficiently close to Bohr s frequency condition,... 

The probability is a function of the incident energy per unit time, per unit area, I (co) Aco of the incident radiation in the frequency interval between co and co + Acu. We will also refer to /(co) as the spectral intensity of the incident radiation. The matrix element represents the expectation value of the dipole moment operator between initial and final state, hcofi = Ef—Ei is Bohr s frequency condition it is related to the energies of the initial and final states, i), /), and n designates the refractive index. 

The integration over / replaces the <5 function of frequency by h and fixes Ef at the value given by Bohr s frequency condition, Eq. 2.80. The integration over , must be done numerically. For each fixed value /,-, the summation over //is over only two terms (selection rules),... 

In this equation, the energies , and / of the initial and final states, i) and I/), and the dipole moment all refer to a pair of diatomic molecules hcvij = Ef — Ei is Bohr s frequency condition. With isotropic interaction, rotation and translation may be assumed to be independent so that the rotational and translational wavefuntions, population factors, etc., factorize. Furthermore, we express the position coordinates of the pair in terms of center-of-mass and relative coordinates as this was done in Chapter 5. 

If a molecule interacts with an electromagnetic field, a transfer of energy from the field to the molecule can occur only when Bohr s frequency condition is satisfied. Namely,... 

Here we conclude our account of Bohr s theory. Although it has led to an enormous advance in our knowledge of the atom, and in particular of the laws of line spectra, it involves many difficulties of principle. At the very outset, the fundamental assumption of the validity of Bohr s frequency condition amounts to a. direct and unexplained contradiction of the laws of the classical theory. Again, the purely formal quantisation rule, which stands at the head of the theory, is a foreign element which in the first instance is absolutely unintelligible from the physical point of view. We shall see later how both of these difficulties are removed in a perfectly natural way in wave mechanics. 

Another application of this idea occurs in a new derivation of Planck s radiation formula this is duo to Einstein, and has given effective support to the ideas of the quantum theory and in particular to Bohr s frequency condition. 

As mentioned in the introduction, the interaction of the atomic systems with the radiation is governed by a further independent quantum principle, Bohr s frequency condition,... 

A much more rigorous test of the quantum rules is made possible by applying Bohr s frequency condition to the frequencies of spectral lines. 

To this is added, as the second quantum principle, Bohr s frequency condition hv=WW—W<2>. 

At time t = 0, we have p = Xa in accordance with the boundary conditions (Figure 7.1). For t = tt/2Hi2, we have p = Xb and for t = fttt/Hi2, p = Xa again. Apparently, the electronic density oscillates between the two protons with the frequency H,2/ tt = (E+ - E )/h. This frequency is the same as the frequency for the electromagnetic (EM) radiation that can excite the system from E+ to E (Bohr s frequency condition). When the protons are close to each other and IH12I is large, the... 

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