Bohr postulates

The first application of quantum theory to a problem in chemistry was to account for the emission spectrum of hydrogen and at the same time explain the stability of the nuclear atom, which seemed to require accelerated electrons in orbital motion. This planetary model is rendered unstable by continuous radiation of energy. The Bohr postulate that electronic angular momentum should be quantized in order to stabilize unique orbits solved both problems in principle. The Bohr condition requires that... 

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. 

Equation 4.7 is the Bohr postulate, that electrons can defy Maxwell s laws provided they occupy an orbit whose angular momentum (corresponding to an orbit of appropriate radius) satisfies Eq. 4.7. The Bohr postulate is not based on a whim, as most textbooks imply, but rather follows from (1) the Plank equation Eq. 4.3, AE = hv and (2) starting with an orbit of large radius such that the motion is essentially linear and classical physics applies, as no acceleration is involved, then extrapolating to small-radius orbits. The fading of quantum-mechanical equations into their classical analogues as macroscopic conditions are approached is called the correspondence principle [11]. 

Bohr postulated circular orbits for the electrons in an atom and developed a mathematical model to represent the energies of the orbits, as well as then-distances from the atom s nucleus. His model worked very well for the hydrogen atom. It could be used to calculate the energy of the emitted and absorbed light, as well as the radius of the atom. However, the intensity of the various wavelengths of fight involved was not explained well. Moreover, no other atom was explained well at all. Bohr s theory has since been replaced by a quantum mechanical model, but it was a milestone because Bohr was the first to postulate energy levels in atoms. 

Bohr postulated that the electrons in an atom revolved about the nucleus in circular orbits and absorbed or emitted light only when they changed from one orbit to another. 

The experimental foundation of the quantum theory of atomic structure as put forward by Bohr, lies in the stability of the atom and in the existence of discrete energy levels and the ability of the atom to absorb and emit energy only in quanta, as demonstrated by the discontinuous nature of atomic spectra and by the critical potential measurements of Franck and Hertz. Bohr postulated that the atom could only exist in a limited number of orbits or stationary states, which were defined by the quantum condition that the angular momentum can assume only certain limited values which are given by the expressioiT ... 

The next step in the development of the Bohr model was his assertion that the angular momentum of the electron is quantized. This was an ad hoc assumption designed to produce stable orbits for the electron it had no basis in either classical theory or experimental evidence. The linear momentum of an electron is the product of its mass and its velocity, mgV. The angular momentum, L, is a different quantity that describes rotational motion about an axis. An introduction to angular momentum is provided in Appendix B. For the circular paths of the Bohr model, the angular momentum of the electron is the product of its mass, its velocity, and the radius of the orbit (L = meVr). Bohr postulated that the angular momentum is quantized in integral multiples of / /2tt, where h is Planck s constant ... 

The Bohr model of one-electron atoms Bohr postulated quantization of the angular momentum, L = m vr = nh/lir, substituted the result in the classical equations of motion, and correctly accounted for the spectrum of all one-electron atoms. E = —Z jrd (rydbergs). The model could not, however, account for the spectra of many-electron atoms. 

In addition to the fact that the electrons fly on circular paths, Bohr also postulated that only certain orbits are permitted. Transitions between these orbits occurred immediately or spontaneously without the possibility to take any intermediate positions. It is impossible to understand these postulates with the laws of classical physics. Only quantum mechanics could explain the second Bohr postulation. 

It is possible to find in the history of science many vivid examples illustrating the relativity of the concept fundamental . For instance, the Planck postulate of energy quantization and the Bohr postulate on the quantization on angular momentum made a revolution in physics and were actually axioms at that time. At present from the formal viewpoint, they are only ordinary consequences of Schroedinger s equation [4], Another vivid example is provided by the four famous Maxwell electrodynamic equations which, as was found later, can be derived from Coulomb s law and Einstein s relativity principle [5]. 

The peak of the radial probability distribution for the ground-state H atom appears at the same distance from the nucleus (0.529A, or 5.29x10 " m) as Bohr postulated for the closest orbit. Thus, at least for the ground state, the Schrddinger model predicts that the electron spends most of its time at the same distance that the Bohr model predicted it spent all of its time. The difference between most and all reflects the uncertainty of the electron s location in the Schrddinger model. 

Bohr postulated that the hydrogen atom consisted of a central positive nucleus round which ait electron moved in atomic orbit and that an electron may only be found in one of a limited number of those orbits. 

Danish physicist Niels Bohr postulated that the electron in hydrogen traveis around the nucleus in the manner in which a planet orbits the Sun. Hence his model was called the "planetary model."... 

Finally, an explanation was provided in 1913 by Niels Bohr (1885-1962), a Danish physicist who tackled the problem of trying to understand fundamental atomic structure. Bohr postulated that the electron in hydrogen travels around the nucleus in the manner in which a planet orbits the Sun. Hence his model was called the planetary model. The important distinction between the orbit of a planet around the Sun and the orbit of an electron around a nucleus is that the distance of a planet from the Sun is arbitrary, whereas in the Bohr model an electron cannot exist at just any distance from a nucleus. An electron can orbit the nucleus only at particular fixed, or discrete, distances from the nucleus. 

Accepting the Bohr postulate of an electronic orbit on a hydrogen atom, stabilized by a balance between mechanical and electrostatic forces. 

An orbit refers to a definite, exact circular pathway around the nucleus, in which Bohr postulated that an electron would be found. An orbital represents a region of space in which there is a high probability of finding the electron. 

In 1913, Bohr proposed a model for the hydrogen atom that appeared to explain the line spectra discussed in Section 6.2. The motion of the electron around the nucleus was considered to be similar to the motion of a planet around the sun, the gravitational attraction that keeps the planet in a circular or an elliptical orbit being replaced by the coulom-bic attraction between the electron and the positively charged nucleus. To account for the line spectra, Bohr postulated that the angular momentum of the electron was restricted to multiple values of fl. This was an arbitrary postulate at the time it was made, but it comes naturally from the quantum mechanical description of a particle moving in a circle, as we have already seen in Section 5.1.3. 

The fourth Bohr postulate requires that the angular momentum of the electron in allowed orbits be restricted to integral multiples of hjln. De Broglie, in 1925, pointed out that the fourth Bohr postulate requires that the wavelength properties of the electron be similar to that shown in Figure 2-4, where r is the radius of the electron orbit and A is an associated wavelength. 

As such Einstein accepts at once both the wave quantification as well as the Bohr postulate of transitions between stationary states while his approach will results in the Planek spectral density the Bohr postulate of quantum transitions follows as being with this occasion demonstrated. 

Bohr postulated that there can be only certain discrete orbits for the electron around a nucleus—called stationary states—and that to go from one state to another, an atom must absorb or emit a packet of just the right amount of energy—a quantum. He then proceeded to predict the position of the lines in the hydrogen spectrum based on Balmer s formula, Planck s energy packets, the mass and charge on an electron, and his quantized orbits. 

We then stndy Bohr s theory for the emission spectrum of the hydrogen atom. In particnlar, Bohr postulated that the energies of an electron in the atom are quantized and transitions from higher levels to lower ones account for the emission lines. (7.3)... 

The generalized coordinate q and its conjugate momentum p vary periodically, and the line integral is evaluated over one cycle. (A similar quantization condition on angular momentum led to the famous Bohr postulate L = nh for the hydrogen atom in the old quantum theory [16].) For vibrational motion subject to a potential U K) in a diatomic, the classical energy is E = p llp -f- U R). The integral in (4.80) then translates into... 

Bohr postulated that a photon is emitted or absorbed only when the electron makes a transition from one energy level to another. The energy of an emitted or absorbed photon is equal to the difference between two quantized energies of the atom ... 

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