Bohr frequency

Atomic and Molecular Energy Levels. Absorption and emission of electromagnetic radiation can occur by any of several mechanisms. Those important in spectroscopy are resonant interactions in which the photon energy matches the energy difference between discrete stationary energy states (eigenstates) of an atomic or molecular system = hv. This is known as the Bohr frequency condition. Transitions between... 

This relation is called the Bohr frequency condition. If the energies on the right of this expression are each proportional to h ln2, then we have accounted for Rydberg s formula. We still have to explain why the energies have this form, but we have made progress. 

Bohr frequency condition The relation between the change in energy of an atom or molecule and the frequency of radiation emitted or absorbed ... 

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In order to understand these observations it is necessary to resort to quantum mechanics, based on Planck s postulate that energy is quantized in units of E = hv and the Bohr frequency condition that requires an exact match between level spacings and the frequency of emitted radiation, hv = Eupper — Ei0wer. The mathematical models are comparatively simple and in all cases appropriate energy levels can be obtained from one-dimensional wave equations. 

According to the Bohr frequency condition the emitted radiation should have the exact frequency to excite a second Fe atom from its ground state, in the reverse process... 

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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,... 

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... 

This relation is called the Bohr frequency condition. Each spectral line arises from a specific transition (Fig. 1.18). 

Here is an empirical constant now known as the Rydberg constant its value is 3.29 X 101J Hz. This empirical formula for the lines, together with the Bohr frequency condition, strongly suggests that the energy levels themselves are proportional to rg[n1. 

The observation of discrete spectral lines suggests that an electron in an atom can have only certain energies. Transitions between these energy levels generate or absorb photons in accord with the Bohr frequency condition. 

Bohr frequency condition The relation between the change in energy of an atom or molecule and the frequency of radiation emitted or absorbed AE = hv. Bohr radius a0 In an early model of the hydrogen atom, the radius of the lowest energy orbit now a specific combination of fundamental constants (aG =... 

The Bohr frequency condition relates the characteristic frequencies of an atom to a set of characteristic energies... 

Fig. 5.4.1b). If the nucleus is supplied with sufficient energy to fulfill the Bohr frequency condition or resonance condition, AE = hv0, a transition becomes possible. This condition can be achieved by applying an alternating radio frequency field (rf), B, perpendicular to B0 and rotating in the xy plane with a frequency, v, equal to v0. In this situation, v, = v = ( >/2re)B0 [Eq. (3)] defines the resonance condition. As the nucleus absorbs energy, its magnetic moment fi rotates away from B0 toward the xy plane (Fig. 5.4.1a). 

Since Ms = 1/2 only two energy states are available, which are degenerate in the absence of a magnetic field, but as B increases this degeneracy is lifted linearly as illustrated in Figure 1.1. The separation of the two levels can be matched to a quantum of radiahon through the Bohr frequency condihon ... 

If the perturbation frequency co is approximately in resonance with co o, thus satisfying the Bohr frequency condition hco = E — Eq, then Eq (4.88) reduces to... 

This is called the Bohr frequency condition. We now understand that the atotnic transition energy AE is equal to the energy of a photon, as proposed earlier by Planck and Einstein. 

The electronic transitions in atoms and molecules (and in solids) are associated with two electronic states of the system by the Bohr frequency condition... 

Processes that are resonant at zero held (i.e., with a atomic Bohr frequency that is an integer multiple of the laser frequency) can be investigated through an effective Hamiltonian of the model constructed from a multilevel atom driven by a quasi-resonant pulsed and chirped radiation held (referred to as a pump held). If one considers an w-photon process between the considered atomic states 1) and 2) (of respective energy E and Ef), one can construct an effective Hamiltonian with the two dressed states 11 0) (dressed with 0 photon) and 2 —n) (dressed with n photons) coupled by the w-photon Rabi frequency (2(f) (of order n with respect to the held amplitude and that we assume real and positive) and a dynamical Stark shift of the energies. It reads in the two-photon RWA [see Section III.E and the Hamiltonian (190)], where we assume 12 real and positive for simplicity,... 

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