Oscillator mean energy

For radiation in equilibrium with electromagnetic oscillators consisting of the charges on material ions we can combine the formula for the mean energy of a resonator (Planck, Wcirme-strahluny, p. 124) ... 

For a harmonic oscillator, the mean energy is given in terms of the frequency, v, by... 

It may be recognized that E ui,T) is, formally, the expression for the mean energy of an oscillator of natural frequency lo, at temperature T. 

To determine the optimal value of quantum correction y, several criteria have been proposed, all of which are based on the idea that an appropriate classical theory should correctly reproduce long-time hmits of the electronic populations. (Since the populations are proportional to the mean energy of the corresponding electronic oscillator, this condition also conserves the ZPE of this oscillator.) Employing phase-space theory, it has been shown that this requirement leads to the condition that the state-specihc level densities... 

Planck showed that the mean energy of a great number of oscillators, each with a characteristic angular frequency w, = 2ttv,, is given by... 

The most general vibrational motion of our solid is one in which each overtone vibrates simultaneously, with an arbitrary amplitude and phase. But in thermal equilibrium at temperature T, the various vibrations will be excited to quite definite extents. It proves to be mathematically the case that each of the overtones behaves just like an independent oscillator, whose frequency is the acoustical frequency of the overtone. Thus we can make immediate connections with the theory of the specific heats of oscillators, as we have done in Chap. XIII, Sec. 4. If the atoms vibrated according to the classical theory, then we should have equipartition, and at temperature T each oscillation would have the mean energy kT. This means that each of the N overtones would have equal... 

The most obvious difficulty is that the numbers of oscillators in the classical RRK theory required to fit both the low pressure limit and the fall-off are smaller than the actual number of oscillators in the molecule, typically by a factor of about 2, although there can be wide variations [14]. Thus 14 oscillators have been used in the calculation presented in Fig. 5, but cyclobutane actually contains 30 oscillators. The problem is that the use of classical statistical mechanics overestimates the populations of excited states relative to the ground state. For example, in classical theory the mean energy of an oscillator is kT,... 

Here X is the wave length emitted by the oscillator and c is the velocity of light. The mean energy of the oscillator is therefore. proportional to the intensity of that part of the complete radiation which has the wave length emitted by the oscillator itself. 

In light of these claims, it is useful to commence our study of the thermodynamic properties of the harmonic oscillator from the statistical mechanical perspective. In particular, our task is to consider a single harmonic oscillator in presumed contact with a heat reservoir and to seek the various properties of this system, such as its mean energy and specific heat. As we found in the section on quantum mechanics, such an oscillator is characterized by a series of equally spaced energy levels of the form E = (n + )hu>. From the point of view of the previous section on the formalism tied to the canonical distribution, we see that the consideration of this problem amounts to deriving the partition function. In this case it is given by... 

This result should be contrasted with expectations founded on the classical equipartition theorem which asserts that an oscillator should have a mean energy of kT. Here we note two ideas. First, in the low-temperature limit, the mean energy approaches a constant value that is just the zero point energy observed earlier in our quantum analysis of the same problem. Secondly, it is seen that in the high-temperature limit, the mean energy approaches the classical equipartition result E) = kT. This can be seen by expanding the exponential in the denominator of the second term to leading order in the inverse temperature (i.e. — 1 ... 

If therefore we know the mean energy of an oscillator, we know also the spectral intensity distribution of the cavity radiation. 

A year or two later (1907) Einstein showed that Planck s formula for the mean energy of an oscillator,... 

Einstein s explanation of these deviations is that it is not permissible in this case to use the classical expression for the mean energy of the oscillators, but that we must apply the expression obtained by Planck for the mean energy of a quantised oscillator. In that case the mean energy of the oscillators per mole is... 

It is now only consistent to assume tkat every proper vibration behaves like a Planck oscillator witk tke mean energy... 

Although this metkod of deduction is formally extremely simple, it contains all tke same a serious difficulty of principle. Tke formula used for tke mean energy of an oscillator is bound up witk tke idea... 

In addition, the atoms within the coniines of the molecule will oscillate to and from one another at temperatures sufficiently high to exclude the quantum theory, each atom has an energy of vibration 3RT (law of Kopp-Neumann) at lower temperatures the mean energy of vibration per... 

The behavior of the individual terms in the momentum balance (right) is similar to that in the power balance. Now the normalized momentum loss in elastic collisions r (z)/I oo) oscillates around the oscillating momentum gain /(z)// (oo), and the somewhat lesser deviations between these quantities are compensated for to a large extent by the normalized source term d/dz) [(2/3m )u z)]/P oQ) of the momentum balance (dotted-dashed curve) containing the spatial derivative of the mean energy density , (z). 

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