Semiclassical Theory of Atoms

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In considering the emission spectra we assume that the excited states of the atoms are approximately in thermal equilibrium, so the number of states in any given atom is proportional to the Boltzmann factor [10]. In order to check this property a spectrum of transitions, which satisfy the relation. The problem of the proportionality coefficient has been eliminated by considering a pattern of the ratios of intensities and that of probabilities corresponding to different pairs of the electron transitions.

In general the pattern of semiclassical intensities is found to fit rather well to that obtained for the quantum-mechanical probabilities calculated for the same levels [8]. Certainly the selection rules for electron transitions, especially those dictated by the quanta of angular momentum belonging to and n, could not be included by the semiclassical theory. This raised the problem how the angular momentum parameters, for example different than those applied in [8] , can influence the relation between the semiclassical intensities and quantum-mechanical probability results.

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To answer this point the present paper examines the ratios obtained for a spectrum of transitions together with and. In any of these cases the relation of 1 is preserved. This means that we referred the above transitions to those where the state of a higher energy is associated systematically with the state of a higher angular momentum; evidently the states of the lower energy refer to those of the lower angular momentum. But in spectroscopy also an opposite emission than from to in which.

It is demonstrated that the accuracy of agreement obtained between the results of the semiclassical and quantum-mechanical formalisms depends on the assumption whether the case of 2 or 3 is taken into account. An outline of the formulae which are of use in a semiclassical theory of the energy emission in the atom has been done before [8]. Since the time quanta can be next represented by the energy quanta see [6] - [8] , the emission intensity is in fact expressed solely with the aid of the electron transition energies in the atom. The intensity ratios of different spectral lines obtained in this way can be compared with the quantum-mechanical ratios of transition probabilities characteristic for these lines.

The number represents the ground atomic state. Let us note that 4 label the energy distances between the neighbouring states. But beyond of 4 the energy quanta between more distant states than neighbouring ones can also enter the calculations. In a particular case of the present paper the energy quanta which are of use become special cases of the formula.

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A fundamental relation does exist between and where. The relation is given by the quantum aspect of the Joule-Lenz law [6] - [8] the essence of which is expressed by the formula. Because of 13 the emission intensity for transitions between the neighbouring quantum levels is. For the case of more distant quantum levels than and n, for example and n where , we have. Particular ratios of the emission intensity belonging to different pairs of the electron transitions in the hydrogen atom are represented in Table 1 and Table 3 ; see also [8].

In Table 1 are given the ratios. A characteristic point in 20 is that the angular momentum of the beginning state is larger than the angular momentum of the end state. In Table 3 are represented the intensity ratios. This is a case representing the angular momentum behaviour opposite to that given in A quantum-mechanical counterpart of the semiclassical ratios of intensity considered in Sec.

These ratios are presented in Table 2 and Table 4. In Table 2 the intensity ratios of Table 1 are compared with the quantum-mechanical ratios of the transition probabilities which are. On the other hand the semiclassical intensity ratios of Table 3 are compared with the quantum-mechanical ratios of transition probabilities [5].

In general the semiclassical intensity ratios presented in Table 1 differ solely by few percent from the quantum-mechanical ratios calculated in Table 2. The largest discrepancy between the semiclassical ratios of Table 1 and quantum-mechanical ratios of Table 2 seems to be in the case 61 where see Table 2 :. Table 1. Intensity ratios of electron transitions between the d and p states, f and d states, and g and f states of the hydrogen atom calculated in a semiclassical way.

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The applied intervals of energy are presented in 4 - 6 and 9 - The results are compared with the ratios of quantum-mechanical transition probabilities in Table 2. Table 2. Quantum-mechanical ratios of the transition probabilities between the pairs of quantum states examined in Table 1 see [5] compared with the intensity ratios calculated in Table 1. Table 3. Intensity ratios of the electron transitions between the s and p states of the atomic hydrogen calculated in a semiclassical way; states s have here higher energy than states p.

A comparison of the results of the present Table with the ratios of quantum-mechanical transition probabilities is done in Table 4. Table 4. Quantum-mechanical ratios of the transition probabilities for the pairs of states examined in Table 3 compared with the intensity ratios calculated in Table 3.

In average this is evidently a much better agreement between the semiclassical and quantum-mechanical theory than attained for transitions considered in [8]. A different situation is represented, however, by the semiclassical data collected in Table 3 compared with the quantum-mechanical results in Table 4.

An agreement between these sets of the data is evidently poorer than attained in case of Table 1 and Table 2 of the present paper, as well as for the data collected in [8]. In 16 cases [ 2 , 4 , 5 , 7 - 10 , 12 , 13 , 23 , 25 - 27 , 33 , 38 and 39 ] the quantum-mechanical ratio divided by the semiclassical one or vice versa, so the resulted value is always obtained exceeds 2. Maximal ratios are given by cases 5 and 8 for which the results are respectively.

Because of a qualitatively different reference between the angular momenta in the beginning and end quantum states entering transitions examined respectively in Table 1 and Table 3 , it can be supposed that not only the energy levels but also the values of the angular momenta can influence the ratio of the semiclassical intensities.

At present we have no insight into the angular-momentum contribution to the semiclassical results for the intensity relations. A well-known reference to the correspodence principle concerns the energy spectrum of the Bohr hydrogen atom [11] [12]. Let us examine a difference which exists between the frequency of the electron motion about the atomic nucleus in state n represented by. The and in 26 and 27 are the time periods of the circular motion about the nucleus and. Both quantities and become coupled together by the relation. For low m and n, and given in 30 the result 36 does not hold because in this case.

The idea of the correspondence principle represented by the formula 36 was that it works only on condition. This is due to the property that a good accuracy of the formula. In fact the interval 40 is the time difference between two situations represented respectively by the neighbouring quantum values of the energy, and.

The result that.

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In practical calculations, especially for the low-energy transitions, the formula 41 can be neglected but only 38 applied. This is done in the present and the former paper [8]. Since 38 does hold solely for the neighbouring energy states any time interval. It seems useful to be noted that the orbital velocity in 34 can be obtained as a drift velocity [13]. In consequence from 44 - 46 we obtain. The time seems to be a not favourite parameter for the quantum physicists.

In fact, the quantum events occupy so short intervals of time that their accurate measurement seems to be hardly possible. A similar difficulty concerns a precise definition of the beginning or end time of the quantum process. In effect the time as a measurable observable enters quite seldom the quantum-theoretical analysis or an empirical observation. This led to situation that the patterns of the emission intensity between the states of the hydrogen atom calculated with the aid of quantum mechanics could be compared with the semiclassical patterns of intensity data obtained from the time intervals which are characteristic for the electron transitions between the energy levels in the atom.

It should be added that all transitions entering calculations have been selected according to the well- known rules of quantum mechanics concerning the electron angular momentum. It was a running joke that any theory of atomic and molecular spectra could be destroyed by throwing a book of data at it, so complex were the spectra. Figure 2. Part a shows, from left to right, a discharge tube, slit, and diffraction grating producing a line spectrum.

Part b shows the emission line spectrum for iron. The discrete lines imply quantized energy states for the atoms that produce them. The line spectrum for each element is unique, providing a powerful and much used analytical tool, and many line spectra were well known for many years before they could be explained with physics.

1. Introduction

In some cases, it had been possible to devise formulas that described the emission spectra. As you might expect, the simplest atom—hydrogen, with its single electron—has a relatively simple spectrum. The hydrogen spectrum had been observed in the infrared IR , visible, and ultraviolet UV , and several series of spectral lines had been observed. See Figure 3. These series are named after early researchers who studied them in particular depth.

The observed hydrogen-spectrum wavelengths can be calculated using the following formula:. The constant n f is a positive integer associated with a specific series. The Paschen series and all the rest are entirely IR.

There are apparently an unlimited number of series, although they lie progressively farther into the infrared and become difficult to observe as n f increases. The constant n i is a positive integer, but it must be greater than n f. Note that n i can approach infinity. While the formula in the wavelengths equation was just a recipe designed to fit data and was not based on physical principles, it did imply a deeper meaning.

Balmer first devised the formula for his series alone, and it was later found to describe all the other series by using different values of n f. Bohr was the first to comprehend the deeper meaning.

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Again, we see the interplay between experiment and theory in physics. Experimentally, the spectra were well established, an equation was found to fit the experimental data, but the theoretical foundation was missing. Figure 3. A schematic of the hydrogen spectrum shows several series named for those who contributed most to their determination. Part of the Balmer series is in the visible spectrum, while the Lyman series is entirely in the UV, and the Paschen series and others are in the IR. Values of n f and n i are shown for some of the lines.

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For an Integrated Concept problem, we must first identify the physical principles involved. In this example, we need to know two things:. Hydrogen spectrum wavelength. The calculation is a straightforward application of the wavelength equation.

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Entering the determined values for n f and n i yields. This is indeed the experimentally observed wavelength, corresponding to the second blue-green line in the Balmer series. More impressive is the fact that the same simple recipe predicts all of the hydrogen spectrum lines, including new ones observed in subsequent experiments. What is nature telling us? Double-slit interference Wave Optics. To obtain constructive interference for a double slit, the path length difference from two slits must be an integral multiple of the wavelength.

Solving for d and entering known values yields.

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This number is similar to those used in the interference examples of Introduction to Quantum Physics and is close to the spacing between slits in commonly used diffraction glasses. Bohr was able to derive the formula for the hydrogen spectrum using basic physics, the planetary model of the atom, and some very important new proposals. His first proposal is that only certain orbits are allowed: we say that the orbits of electrons in atoms are quantized. Each orbit has a different energy, and electrons can move to a higher orbit by absorbing energy and drop to a lower orbit by emitting energy.

If the orbits are quantized, the amount of energy absorbed or emitted is also quantized, producing discrete spectra. Photon absorption and emission are among the primary methods of transferring energy into and out of atoms. The energies of the photons are quantized, and their energy is explained as being equal to the change in energy of the electron when it moves from one orbit to another. Figure 4. The planetary model of the atom, as modified by Bohr, has the orbits of the electrons quantized. Only certain orbits are allowed, explaining why atomic spectra are discrete quantized.

The energy carried away from an atom by a photon comes from the electron dropping from one allowed orbit to another and is thus quantized. This is likewise true for atomic absorption of photons. It is quite logical that is, expected from our everyday experience that energy is involved in changing orbits. A blast of energy is required for the space shuttle, for example, to climb to a higher orbit.

What is not expected is that atomic orbits should be quantized. This is not observed for satellites or planets, which can have any orbit given the proper energy. See Figure 4. In the present discussion, we take these to be the allowed energy levels of the electron. Energy is plotted vertically with the lowest or ground state at the bottom and with excited states above. Given the energies of the lines in an atomic spectrum, it is possible although sometimes very difficult to determine the energy levels of an atom.

Energy-level diagrams are used for many systems, including molecules and nuclei. A theory of the atom or any other system must predict its energies based on the physics of the system. Figure 5. An energy-level diagram plots energy vertically and is useful in visualizing the energy states of a system and the transitions between them. This diagram is for the hydrogen-atom electrons, showing a transition between two orbits having energies E 4 and E 2.

Bohr was clever enough to find a way to calculate the electron orbital energies in hydrogen. This was an important first step that has been improved upon, but it is well worth repeating here, because it does correctly describe many characteristics of hydrogen. Assuming circular orbits, Bohr proposed that the angular momentum L of an electron in its orbit is quantized , that is, it has only specific, discrete values. At the time, Bohr himself did not know why angular momentum should be quantized, but using this assumption he was able to calculate the energies in the hydrogen spectrum, something no one else had done at the time.

We start by noting the centripetal force causing the electron to follow a circular path is supplied by the Coulomb force. To be more general, we note that this analysis is valid for any single-electron atom. The spectra of hydrogen-like ions are similar to hydrogen, but shifted to higher energy by the greater attractive force between the electron and nucleus.

The tacit assumption here is that the nucleus is more massive than the stationary electron, and the electron orbits about it. This is consistent with the planetary model of the atom. Equating these,. Angular momentum quantization is stated in an earlier equation. We solve that equation for v , substitute it into the above, and rearrange the expression to obtain the radius of the orbit.