Bohrs Classical Theory

2 Theories prior to Bohr, such as that of J. J. Thompson, are only of historical interest. 

5 e and m refer respectively to the magnitude of the charge and the mass of the electron. 

Theory of Stopping Power of Fast Charged Particles 

TABLE 2.2 Approximate LETs of Various Qualities of Radiation in Water 

The differential cross-section of this process for the range of impact parameters between b and b - db is given geometrically from Eq. (2.1) as follows  

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According to the correspondence principle as stated by N. Bohr (1928), the average behavior of a well-defined wave packet should agree with the classical-mechanical laws of motion for the particle that it represents. Thus, the expectation values of dynamical variables such as position, velocity, momentum, kinetic energy, potential energy, and force as calculated in quantum mechanics should obey the same relationships that the dynamical variables obey in classical theory. This feature of wave mechanics is illustrated by the derivation of two relationships known as Ehrenfest s theorems. 

In Bohr s theory, only estimates of maximum and minimum impact parameters are necessary. Better computations are required for determining the transverse distribution of lost energy or the effect of secondary electrons. The minimum impact parameter according to classical mechanics is ze2/mv2 from angular momentum consideration in quantum mechanics, it is h /mv. In practice, the larger of these two is taken. Also, the impulse approximation used by Bohr for the maximum impact parameter is not an absolute rule energy transfer beyond bmax falls off exponentially (Orear et al., 1956 Mozumder, 1974). 

Conceptionally the situation is much clearer in classical theory because the cross-over toward negative stopping numbers of the Bohr logarithm L ohi — In(Cmu /ZiUof) can easily be avoided [11]. Orbital motion can be incorporated into the initial conditions [17], although the actual evaluation in Ref. [17] was carried through only to the leading term in u . ... 

While more than a handful theoretical schemes are available to nonpermrba-tively evaluate the Barkas-Andersen correction quantum mechanically, binary stopping theory developed recently [32] fulfills the task on the basis of the Bohr stopping model the only quantum feature added is the inverse-Bloch correction (18) which does not differentiate between particle and antiparticle. Figure 4 demonstrates that with regard to comparison with experimental antiproton stopping data, classical theory is fully competitive with various quantum theories. 

The basic stopping power formula of Bethe has a structure similar to that of Bohr s classical theory [cf. Eq. (2)]. The kinematic factor remains the same while the stopping number is given hy B = Zln(2mv /7) for incident heavy, nonrelativistic particles. The Bethe... 

Despite the apparent similarity of the Bohr and the Bethe stopping power formulae, the conditions of their validity are rather complimentary than the same. Bloch [23] pointed out that Born approximation requires the incident particle velocity v ze jh, the speed of a Is electron around the incident electron while the requirement of Bohr s classical theory is exactly the opposite. For heavy, slow particles, for example, fission fragments penetrating light media, Bohr s formula has an inherent advantage, although the typical transition energy has to be taken as an adjustable parameter. 

The next paper was by Dirac on the Theory of the Positron. In the following discussion Niels Bohr made a long intervention on the correspondence principle in connection with the relation between the classical theory of the electron and the new theory of Dirac. 

In the classical theory of electrodynamics, electromagnetic radiation is emitted when an electron moves in its orbit but, ac cording to the Bohr theory of the atom,... 

The nuclear theory of atomic structure, put forward by Rutherford, regarded the electrons as moving in orbits round the nucleus. The dynamical theory of this system was developed by Bohr, who found it necessary to supplement classical mechanics by the quantum mechanics of Planck. According to classical theory, a system consisting of an electron moving in a circular orbit round a nucleus, to which it is attracted according to Coulomb s law, would lose energy, with the result that the electron would approach and finally collide with the nucleus. Thus on the basis of classical theory, the Rutherford atom would only be stable for about io seconds, after which time the electron would have fallen into the nucleus. 

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 ... 

Although there was early work on energy deposition using classical theory and free particle targets by Darwin [8] and Thomson [9] in 1912, the first formulation of the energy deposition problem based on the realization that the binding of electrons in a target atom is important was due to Niels Bohr in 1913 [10] and 1915 [11], when he realized the importance of... 

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. 

Thus, in Heisenberg s view, Bohr s theory fails because the fundamental ideas on which it is based (the orbit picture, the validity of the classical laws of motion, and so on) can never be put to the test. We move, therefore, in a region beyond experience, and ought not to be surprised if the theory, constructed as it is on a foundation of hypotheses which cannot be proved experimentally, partially fails in those deductions from it which can be subjected to the test of experiment. 

We have deduced this formula in accordanco with the classical vector model representation. In quantum mechanics this representation is certainly still permissible, but with this difference, that the square of the magnitude of an angular momentum, with the quantum number I, is not equal to P as in Bohr s theory, Init is given by l(l + 1). This is proved in Appendix XIX (p. 302) for the orbital angular momentum ... 

Briefly describe Bohr s theory of the hydrogen atom and how it explains the appearance of an emission spectrum. How does Bohr s theory differ from concepts of classical physics ... 

We give in conclusion a brief formulation of the ideas which have led to Bohr s atomic theory. There are two observations which are fundamental firstly the stability of atoms, secondly the validity of the classical mechanics and electrodynamics for macroscopic processes. The application of the classical theory to atomic processes... 

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