The Bohr Atom and the Correspondence Principle

     Confronted with overwhelming evidence that the amounts of internal energy that could be stored in an atomwere not arbitrary, butwere, instead, quantized, physicists attempted to explain the origin of these quantum levels. The experiments performed in Great Britain by Lord Rutherford clearly established that the atom consisted of a tiny massive positively charged nucleus surrounded by very light negatively charged electrons that orbited this nucleus. A major problem was that, according to classical electromagnetic theory, accelerating charges emit electromagnetic energy (light). Thus, an orbiting electron should lose energy as it revolves about the nucleus, thus spiraling into the nucleus. If that spiraling process were to take a very long time, say 1050 years, then there would be no problem because that is longer than the age of the universe. On the other hand, if the “lifetime” of these atoms is short, then the planetary model of the atom had to be reconciled with classical electromagnetic theory. Because it is important to understand the problem that presented itself to these pioneers of quantum physics it is worthwhile to do a simple calculation to estimate τ , the classical lifetime for a hydrogen atom, that is, the decay time due to radiation of an electron in orbit around a proton. From electromagnetic theory the famous Larmor formula gives the instantaneous power P radiated by an electron undergoing acceleration a. In SI units, which we will use throughout this book unless otherwise stated,
where the minus sign indicates that power is being radiated away. If we assume that each successive loop of the spiral toward the nucleus is a circle of radius r, then we may compute the acceleration a using Coulomb’s law:

The total mechanical energy (TME) E of the electron in the orbit is the sum of kinetic energy and the Coulomb potential energy:
The motion is assumed to be circular so we can eliminate the velocity by equating the centripetal force to the Coulomb force between the electron and proton. This results in
Now, P is the rate of loss of energy dE/dt so we may differentiate Equation 1.20 with respect to time and equate it to Equation 1.17.We obtain
 
 which, when integrated from the initial radius R to the nucleus, yields τ :
 
     From Rutherford’s experiments it was known that R ∼ 0.1nm. The other parameters in this equation for τ were reasonably well known.When inserted in Equation 1.22 the result is τ ∼ 10−11s, hardly comparable with the age of the universe. There was clearly a problem. Niels Bohr attempted to explain the quantized levels using a combination of classical ideas, quantal hypotheses, and postulates of his own [1]. This pioneering work was published in 1913 and Bohr was awarded the Nobel Prize in Physics in 1922 “for his services in the investigation of the structure of atoms and of the radiation emanating from them.”
     To deal with the problem of radiation by an accelerating charge Bohr simply avoided it by postulating his way out of it. Paraphrasing the first of his postulates:

I. An atom exists in a series of energy states such that the accelerating electron does not
radiate energy when in these states. These states are designated as stationary states.

     The designation as “stationary states” has survived time and is used today.Why the accelerating electron ignored the classical laws of electromagnetic theory by not radiating was simply finessed, that is, ignored. Bohr’s second postulate accounted for the emitted and absorbed radiation in terms of the stationary states.

II. Radiation is absorbed or emitted during a transition between two stationary states. The frequency of the absorbed or emitted radiation is given by Planck’s theory.

     Bohr’s reference here to “Planck’s theory” is the relationship between the energy and the frequency, Equation 1.1, that was used by Planck to explain blackbody radiation. The energy was taken to be the difference in the energies of the two states involved in the transition. Thus, the frequency, ν, of this radiation is given by
 
where h is Planck’s constant and E' and E" are the energies of the two states involved in the emission or absorption.
     Bohr had a third postulate, although he did not state it as such. It is the famous and ingenious correspondence principle. Loosely stated, the correspondence principle states that when quantum systems become large they behave in a manner that is consistent with classical physics. Bohr essentially used this as his third postulate, although many derivations of the consequences of the Bohr model of the atom often ignore the correspondence principle. Instead, these treatments postulate that the angular momentum must be quantized in units of (h bar). Bohr made no such postulate, although it does lead to the correct answers without appealing to the correspondence principle. These derivations usually then present the correspondence principle as a consequence of this erroneous postulate.
     It is a simple matter to obtain the relationship between the TME of the electron E and the circular orbital radius r using elementary classical mechanics and electromagnetic theory. Equating the centripetal force to the Coulomb force we have
 
  where v is the speed of the electron in the orbit of radius r . For simplicity and convenience we are assuming that the reduced mass of the electron–proton system is the same as me. From Equation 1.24 we can solve for the kinetic energy of the electron so the TME is
 
  If we now apply Postulate II assuming a transition from state n to state m, wenote that the only variable in the expression for the energy, Equation 1.25, is the orbital radius r .We must therefore attach a subscript to r to designate to which state it belongs For definiteness we assume that n > m and, applying Postulate II, we
write
 where νnm is the frequency of the photon emitted in the transition from the higher state n to the lower state m.
     At this point there were two ingenious steps taken by Bohr. The first was to note the similarity between Equation 1.26 and the generalized Balmer formula, Equation 1.9 (recall the reciprocal relationship between ν and λ). The orbital radius is the only variable in Equation 1.26 so it is clear that it is rn that is quantized. That is, each of the stationary states must have a unique orbital radius. Moreover, to be consistent with Equation 1.9 these orbital radii must be such that
where a0 has units of length. It is called the Bohr radius. To find it Bohr imposed the correspondence principle.
     We had noted that accelerating charges radiate electromagnetic energy.But that is not the whole story. If these accelerating charges are being accelerated periodically, for example, a harmonically oscillating charge or a circularly moving charge, then the frequency of the emitted radiation is the same as the frequency of the motion. Bohr therefore stated that as n and m become very large, the frequency νnm in Equation 1.26 must approach the frequency νorbit of the circular motion of the electron at the nth Bohr radius. The orbital frequency is
 
 where vn is the orbital speed in the nth Bohr orbit, Equation 1.24.Working with the square of νorbit for convenience and using Equation 1.28 we have
We can calculate ν2 (n+1)n for high values of n from Equation 1.26 by substituting Equation 1.27 for the orbital radii.We obtain
 
To apply the correspondence principle we equate ν2 orbit and ν2 (n+1)n for high n and obtain
 
Examination of Equation 1.25 shows that, because the orbital radii are quantized in accord with Equation 1.27, the total internal energy of the atom must also be quantized. We may thus write Equation 1.25, replacing r with n2a0 and E with En to indicate the nth energy level.We obtain
 
where n is called the principal quantum number. Substituting for a0 we have
Note that the minus sign is required since the electron is bound to the proton. The TME must therefore be negative. E = 0 corresponds to infinitely separated proton and electron each having zero kinetic energy.
     Equation 1.33 is called the Bohr energy. Although the Bohr model is not entirely correct, the Bohr energy is correct. It applies to any quantum level of the hydrogen atom as designated by the quantum number n. In fact, it applies to any one-electron atom, for example heliumwith one electron removed,when the number of protons in the nucleus is included.When compared with Equation 1.9, the Bohr energy yields the value of the Rydberg constant, which is found to be
 
where the numerical value given is the accepted value today. The agreement between this theoretically obtained value and that empirically determined using atomic spectroscopy was astonishing. While we know today that some of the concepts of the Bohr model are incorrect, it remains a paradigm of clear and creative thinking. Bohr’s use of known empirical facts together with his statement of the correspondence principle led to a breakthrough in physics that gave birth to quantum physics as we know it today. Although physicists know that the wave nature of matter, as exemplified by, for example, the Davisson–Germer experiment,makes precise location of particles problematic,most nevertheless envision a Bohr-like atom when thinking about atoms (even if they don’t admit it in public). Besides permitting visualization, the Bohr model also gives the correct order of magnitude and scaling with principal quantum number of parameters, such as orbital distances and electronic velocities. Most importantly, it also gives the correct quantized energies.
     It also follows from the above analysis that the electronic angular momentum must be quantized in units of (h bar), the postulate that is incorrectly attributed to Bohr. Indeed, this postulate follows as a consequence of the his two stated postulates and the correspondence principle. Note, however, that (h bar) has units of angular momentum
(as does h).
     Before leaving the Bohr energy it is useful to cast this important quantity in terms of other, more revealing, parameters. One of the most convenient ways of writing it is in terms of the fine structure constant, which is a combination of fundamental constants that results in a pure number that is very nearly 1/137. This number is of
fundamental importance in quantum physics. It is given by
The Greek letter α is universally used for the fine structure constant. Regrettably, it is also universally used for a number of other important quantities. In terms of the fine structure constant the Bohr energy is
The reason Equation 1.36 is convenient is that most physics students know that the rest mass of the electron is 0.51MeV (1MeV = 106eV). A simple calculation shows that the lowest energy state of the Bohr atom, and, consequently, hydrogen, is −13.6 eV. This energy is also called the ionization potential since it is the minimum
energy required to liberate the electron from the hydrogen atom, leaving behind a “hydrogen ion.” A hydrogen ion is simply a proton, but the term “ionization potential” is applied to all atoms and molecules. It is also convenient to remember the Bohr energy in electron-volts. For this purpose we may rewrite Equation 1.33 as
from which it is clear that the ionization potential of hydrogen is 13.6 eV.
     From the Bohr energy as given in Equation 1.37 it is a simple matter to calculate energy differences between any pair of levels. We have
 
where it is assumed that n < n'. If we let n = 2 and use the relation E = hc/λ we immediately recover the Balmer formula, Equation 1.9, the formula that predicts the wavelengths of emitted radiation for which the lower state is n = 2, the Balmer series. There are, however, other series that are observed. For example, if we let n = 1 we obtain a formula that predicts the wavelengths of the Lyman series. Because the ground state lies much lower than n = 2 these energy differences are considerably greater than those of the Balmer series. Consequently, transitions in the Lyman series yield radiation in the ultraviolet region of the spectrum.
     Let us now investigate the relationship between the quantum number n and the
angular momentum. From Equations 1.24, 1.31, and 1.27 the electronic velocity in
the nth orbit is
 so the angular momentum of the electron in the nth orbit is
 We have therefore resurrected the “postulate” that the orbital angular momentum is quantized in units of (h bar). Interestingly, this result is incorrect because, as we will learn later, the states of hydrogen can have any integer multiple of (h bar) or zero as long as it is less than the principal quantum number n. This means that the electronic angular momentum in the ground state is zero, not unity as predicted by Equation 1.40. Nonetheless, the Bohr model of the atom provides us with quantities that give the correct order of magnitude of actual atomic parameters. For this reason it is extremely useful. For example, a bit of algebra permits us to write vn in the form
(see Problem 5)
 Notice that Equation 1.41 tell us that the highest orbital velocity for an electron occurs in the ground state, but, even then, this velocity is more than two orders of magnitude smaller than the speed of light, thus justifying the nonrelativistic treatment. It is often convenient to express Bohr parameters in terms of the fine structure constant, so we present in Table 1.1 a partial listing.
     Before leaving the subject of the Bohr atom we discuss another of the conveniences afforded by it. Since the model is that of an electron circling a proton, the electric current that is the result of the electronic motion is the source of a magnetic field. Therefore, a Bohr atom has a magnetic dipole moment associated with the orbital motion of the electron about the proton and the atom generates a magnetic field identical to that of a bar magnet. The magnitude of this magnetic moment for the ground state of the Bohr atom is referred to as the Bohr magneton, and is designated by the symbol μB. Magnetic moments are often measured in terms of the Bohr magneton so we calculate its value. Figure 1.5 is a schematic diagram of the Bohr atom with the magnetic field lines due to the orbital motion of the electron.
     Also indicated in this figure are the relevant parameters. The magnetic moment of a current-carrying loop is given by the product of the area of the loop and the current. The current is the electronic charge divided by the
period of the motion, T = 2πa0/v. Therefore, the magnetic moment is
 
 
which may be written in terms of the orbital angular momentum L = meva0 as
 where the vector nature of the angular momentum has been taken into account. Because the electronic charge is negative, the angular momentum and the magnetic moment are in opposite directions. From Equation 1.43 it is clear that there is a direct relationship between the magnetic moment and the angular momentum. Because the angular momentum is quantized in units of (h bar) (see Equation 1.40), the magnitude of the magnetic moment in the first Bohr orbit, the Bohr magneton, is
 
reference: 
C.E. Burkhardt, J.J. Leventhal, Foundations of Quantum Physics, 1
DOI: 10.1007/978-0-387-77652-1 1, C Springer Science+Business Media, LLC 2008

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