Bohr frequency rule

Two postulates that are fundamental to the interpretation of spectra are the existence of stationary states and the Bohr frequency rule. They were enunciated by Bohr in 1913 in the famous paper2 that led in a few years to the complete elucidation of spectral phenomena. Planck8 had previously announced (in 1900) that the amount of energy dW in unit volume (1 cm8) and contained between the frequencies v and v + dv in empty space in equilibrium with matter at temperature Ty as measured experimentally, could be represented by the equation... 

II. The Bohr frequency rule. The frequency of the radiation absorbed by a system and associaled with the transition from an initial stale with energy W to a final stale with energy W% is... 

The frequencies of the spectral lines emitted by a hydrogen atom when it undergoes transition from one stationary state to a lower stationary state can be calculated by the Bohr frequency rule, with use of this expression for the energy values of the stationary state. It is seen, for example, that the frequencies for the lines corresponding to the transitions indicated by arrows in Figure 2-2, corresponding to transitions from states with n 3, 4, 5, to the state with n = 2,... 

II. The Bohr Frequency Rule. The frequency of the radiation emitted by a system on transition from an initial state of energy Wt to a final state of lower energy Wi (or absorbed on transition from the state of energy W to that of energy Wi) is given by the equation1... 

The modern student, to whom the Bohr frequency rule has become commonplace, might consider that this rule is clearly evident in the work of Planck and Einstein. This is not so, however the confusing identity of the mechanical frequencies of the harmonic oscillator (the only system discussed) and the frequency of the radiation absorbed and emitted by this quantized system delayed recognition of the fact that a fundamental violation of electromagnetic theory was imperative. 

Let us consider two non-degenerate stationary states m and n of a system, with energy values Wm and Wn such that Wm is greater than Wn. According to the Bohr frequency rule, transition from one state to another will be accompanied by the emission or absorption of radiation of frequency... 

By the Bohr frequency rule [Eq. (1), Sec. 1-5], the frequency of the light absorbed or emitted by a molecule is given by... 

Making use of the Bohr frequency rule, we obtain from this expression the following equation for the wavelength of the light emitted or absorbed on transition between the n stationary state and the n stationary state ... 

Emission and absorption of photons are governed by the Bohr frequency rule ... 

Time-dependent perturbation theory provides an approximate formula that gives the coefficients as functions of time. If the radiation is polarized with its electric field in the z direction, a (t )p is proportional to the intensity of the radiation of the wavelength that satisfies the Bohr frequency rule and is also proportional to the square of the following integral ... 

In this regard, we should notice that the time evolution of a quantum system is ruled by two different types of eigenvalues corresponding to the wave function and the statistical descriptions. On the one hand, we have the eigenenergies of the Hamiltonian within the wave function description. On the other hand, we have the eigenvalues of the Landau-von Neumann superoperator in the Liouville formulation of quantum mechanics. These quantum Liouvillian eigenvalues j are related to the Bohr frequencies according to... 

The selection rules are derived through time-dependent perturbation theory [1, 2]. Two points will be made in the following material. First, the Bohr frequency condition states that the photon energy of absorption or emission is equal... 

Bohr frequency condition. The second important feature is that iVi 2 rnust be nonzero for an allowed transition, which is how the selection rules are determined. 

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),... 

Electron Motion Around the Nucleus. The first approach to a treatment of these problems was made by Niels Bohr in 1913 when he formulated and applied rules for quantization of electron motion around the nucleus. Bohr postulated states of motion of the electron, satisfying these quantum rules, as peculiarly stable. In fact, one of them would be really permanently stable and would represent the ground state of the atom, The others would be only approximately stable. Occasionally an atom would leave one such state for another and, in the process, would radiate light of a frequency proportional to the difference in energy between the two states. By this means, Bohr was able to account for the spectrum of atomic hydrogen in a spectacular way. Bohr s paper in 1913 may well be said to have set the course of atomic physics on its latest path. 

The authors do not mention the values of the nuclear g-factors, but we may take them to be gF = +2.628 87 and gN = +0.403 76 nuclear Bohr magnetons. Consequently it is now a simple matter to calculate the energies of the 30 levels for a range of magnetic fields between 9400 and 10 600 G the magnetic resonance transitions are those which obey the selection rules AM/ = 1, AMn = AMp = 0 and their frequencies may also be calculated. 

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. 

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

Therefore, the rule is that as much the quantum levels are higher as the quantum and classical frequencies approaches each other, establishing the so called Bohr correspondence principle between the quantum and classical worlds . 

When IR radiation containing a broad range of frequencies passes through a sample, which can be represented as a system of oscillators with resonance frequencies vq then according to the Bohr rule. 

---