Bohr’s equation

Transitions between two quantised energy levels, say from E0 to , correspond to the absorption of radiant energy, and the amount of energy absorbed (AE) is determined by Bohr s equation... 

The constants in Bohr s equation and Balmer s equation are related through E = hv. 

Bohr s Equation If we consider two quantized energy levels e.g., higher as E2 and lower as Ex, the radiation given out during the transition from E2 to E, may be expressed by the following equation ... 

The expression (c) is the Bohr s equation which enables us to calculate ... 

Comparison of de Broglie s equation (see Eq. 4.20) with Bohr s equation (see Eq. 4.22) shows that the wavelength of the standing wave is related to the linear momentum, f, of the electron by the following simple formula ... 

Applying Bohr s Equation for the Energy Levels of an Atom We can use the... 

Einstein s equation for the photoelectric effect) (Bohr s equation for the hydrogen atom)... 

When Max Planck wrote his remarkable paper of 1901, and introduced what Stehle (1994) calls his time bomb of an equation, e = / v , it took a number of years before anyone seriously paid attention to the revolutionary concept of the quantisation of energy the response was as sluggish as that, a few years later, whieh greeted X-ray diffraction from crystals. It was not until Einstein, in 1905, used Planck s concepts to interpret the photoelectric effect (the work for which Einstein was actually awarded his Nobel Prize) that physicists began to sit up and take notice. Niels Bohr s thesis of 1911 which introduced the concept of the quantisation of electronic energy levels in the free atom, though in a purely empirical manner, did not consider the behaviour of atoms assembled in solids. 

The origin of electronic configuration Is frequently and inaccurately attributed to Niels Bohr, who introduced quantum theory to tire study of the atom. But Bohr essentially tidied up Thomson s pre-quantum configurations and took advantage of a more accurate knowledge erf the number of electrons each of the elements actually possessed. Furtlrer developments in quantum theory, including Pauli s occlusion principle and Schrodjtiger s equation. 

It is now shown how the abrupt changes in the eigenvalue distribution around the central critical point relate to changes in the classical mechanics, bearing in mind that the analog of quantization in classical mechanics is a transformation of the Hamiltonian from a representation in the variables pR, p, R, 0) to one in angle-action variables (/, /e, Qr, 0) such that the transformed Hamiltonian depends only on the actions 1r, /e) [37]. Hamilton s equations diR/dt = (0///00 j), etc.) then show that the actions are constants of the motion, which are related to the quantum numbers by the Bohr correspondence principle [23]. In the present case,... 

Bohr s idea of restricted energy levels was revolutionary, because scientists at that time thought that the electron in a hydrogen atom could have any energy, not just the ones described by Equation. hi contrast, Bohr interpreted the hydrogen emission spectrum to mean that electrons bound to atoms can have only certain specific energy values. 

Using Bohr s model, one could calculate the energy difference between orbits of an electron in a hydrogen atom with Planck s equation. In the example of a system with only two possible orbits, the equation of the emitted radiation as the electron went from a higher energy state 2 to a lower one j would be - E = hf, where h is Planck s constant and/is the frequency of the emitted radiation. 

The Bohr theory also permits a calculation of the total energy, E, of the atom from Hamilton s equation ... 

There are some scientists and philosophers who still claim that a model by definition "furnishes a concrete image" and "does not constitute a theory." 10 But if the model is the mathematical description, then the question of whether the model is the theory appears to become moot, since most people accept the view that rigorous mathematical deduction constitutes theory. For others, like Hesse and Kuhn, even if the model is a concrete image leading to the mathematical description, it still has explanatory or theoretical meaning, for, as Kuhn put it, "it is to Bohr s model, not to nature, that the various terms of the Schrodinger equation refer." 11 Indeed, as is especially clear from a consideration of mathematical models in social science, where social forces are modeled by functional relations or sets of mathematical entities, the mathematical model turns out to be so much simpler than the original that one immediately sees the gap between a "best theory" and the "real world." 12... 

Figure 1 shows a comparison between experimental stopping data and equations (7) and (11). It is seen that over a broad range of beam energies, Bohr s classical formula is superior to Bethe s when the same /-value is inserted. 

In equation 1.116, Uq is Bohr s radius for the hydrogen ion in the ground state... 

Hinshelwood, 1951), it is evident that / cannot be greater than — 1. It is also clear that ( ) does not depend on the total value of n but on the separate values of n and /. The correspondence between n and the principal quantum number is thus obvious the energy levels follow the prediction of Bohr s model, assuming distinct values according to quantum numbers n, /, and nti, which lead to configurations s, p, and d. Equation 1.122 is then satisfied by a product of separate functions, one dependent on / and a function of angular coordinates 0 and and the other dependent on n, /, and a function of radial distance r ... 

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. 

The angular momentum for the electron in its orbit is mrv. Bohr s postulate for quantizing the circular orbits is represented by the equation... 

Millikan s experiment did not prove, of course, that (he charge on the cathode ray. beta ray, photoelectric, or Zeeman particle was e. But if we call all such particles electrons, and assume that they have e/m = 1.76 x Hi" coulombs/kg. and e = 1.60 x 10" coulomb (and hence m =9.1 x 10 " kg), we find that they fit very well into Bohr s theory of the hydrogen atom and successive, more comprehensive atomic theories, into Richardson s equations for thermionic emission, into Fermi s theory of beta decay, and so on. In other words, a whole web of modem theory and experiment defines the electron. The best current value of e = (1.60206 0.00003) x 10 g coulomb. 

By 1903. llie wave theory of light based oil Maxwell s equations was well established, but certain phenomena would not fit in. It seemed that emission and absorption of hght occur discontinuously. This led Einstein to (lie view that the energy is concentrated in discrete particles. It was a revolutionary idea, very hard to understand, as the successes of the wave theory were undeniable. It seemed that light had to be understood sometimes as waves, sometimes as particles, and physicists had to get used to it, The idea was incorporated into Bohr s theory of the hydrogen atom and forms an essential part of it. 

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