2. Naturaleza del ser humano
2.12 Procesos de decisión
Statistically 3 N one-dimensional oscillators are equivalent to N three-dimen-sional oscillators when all have the same frequency vo. Therefore, the energy of N three-dimensional oscillators is given by ( 4.10 ) with N replaced by 3N ,
and the entropy of a system of N three-dimensional oscillators is given by ( 4.12 ) with N replaced by 3 N , that is, by applied these results in 1907.
Example 4.1 Correspondence principle
Problem: What is the relation between the classical entropy and the quantum entropy of a system of N three-dimensional, linear, harmonic oscillators?
Solution: The energy ( 4.13 ) and the entropy ( 4.14 ) of a solid composed of quantized oscillators should reduce to their classical counterparts in the appro-priate limit. In particular, the quantum entropy ( 4.14 ) should reduce to the classical entropy of an ideal solid ( 3.56 ) with H replaced by h ,
of an ideal solid. Likewise the quantum energy ( 4.13 ) should reduce to the classical energy ( 3.54 ) of an ideal solid,
E 3NkNkTNN .
Indeed, these reductions obtain in the limit of high energy per oscillator E N relative to the unit hh in which energy is quantized, that is, for E Nνo 4hνo or, equivalently, for high thermal energy kT relative to hh , that is, for kTνo 4hνo. Such is one use of the correspondence principle according to which there is
4.3 Einstein solid 81
always a limit, the so-called semi-classical limit , in which quantum expres-sions reduce to classical ones with the arbitrary H replaced with Planck’s constant h .
4.3 Einstein solid
Albert Einstein was probably the i rst, in 1907, to deliberately apply the quan-tum conditions to a physical model – to what has become known as the Einstein solid . An Einstein solid is a crystalline array of atoms or molecules each one of which oscillates simple harmonically in three dimensions with a common frequency vo . Einstein derived the energy ( 4.13 ) and the entropy ( 4.14 ) of a system of N such three-dimensional oscillators.
Einstein was primarily concerned with the heat capacity C dE ddd T of a solid, that is, with
C nN k e
NA
ho kT
(
h o kT)
(
eho kT −)
3
2
2
h hν
h , (4.15)
where here n is the number of moles and NN is Avogadro’s number. The high A temperature, semi-classical regime, kT 4hνo, of this heat capacity recovers the law of Dulong and Petit, C 3nN kNA
[
=]
. However, as kT hhνo→ 0 the molar specii c heat capacity C n/ → 0 . In 1907 physical scientists were just becoming aware of molar specii c heat capacities that drop below their Dulong–Petit value at relatively low temperatures.Einstein compared the dependence of C / n on T described by ( 4.15 ) to data available on the specii c heat of diamond in one of only three graphical comparisons of theory and experiment he ever published. In doing so he used the quantity hhνo k , now known as the Einstein temperature , as a parameter with which to i t the data and found that hhνo k= 1325 K worked well for dia-mond. I have reproduced Einstein’s graph with his data converted to SI units in Figure 4.2 .
The Einstein solid is a rough model that in its details must be inaccu-rate. After all, because neighboring molecules exert the forces that restore a molecule to its equilibrium position, neighboring molecules do not oscillate independently as is assumed in deriving the Einstein solid. The resulting col-lective motion generates a whole spectrum of oscillations, not just a single frequency. Peter Debye (1884–1966) incorporated these collective oscilla-tions into a statistical model of solids that better i ts specii c heat data. But
it was Einstein’s 1907 paper that initiated the modern study of condensed matter physics.
4.4 Phonons
Because the energy of a simple harmonic oscillator is quantized in units of energy hvo where vo is the natural frequency of the oscillator, the energy of a system composed of such oscillators is also quantized. In particular, the nor-malized system energy in excess of its minimum, zero-temperature or ground state , value, that is,
(
E NhNhNNh ooo)
hhhνo is an integer. In the context of an Einstein solid hvo-sized bundles or quanta of energy are called phonons . Thus, energy l ows from one place to another in the solid as phonons diffuse from one place to another.This way of thinking about the energy of an Einstein solid suggests an alter-native derivation of its entropy that uses each of three quantum conditions: (1) determinate phase space cells, (2) quantized energy, and (3) the quantum indis-tinguishability of phonons. Imagine that each oscillator degree of freedom is a compartment that can hold any number of phonons. The Einstein solid consists of 3 N such compartments. Together these compartments hold phonons.
20
15
C/n 10
5
200 400
T
600 800 1000 1200
Figure 4.2 Molar specii c heat capacity C n of diamond in joule/(kelvin mole) versus thermodynamic temperature T in kelvin. Filled circles: Measurements available to Einstein in 1907. Curve: Einstein’s prediction ( 4.15 ) with
hνo k= 1325 K.
4.4 Phonons 83
The number of distinct ways of distributing n indistinguishable phonons among 3N compartments is the system multiplicity Ω . To motivate our cal-culation of the multiplicity let the symbol 卷 stand for a phonon and 弗 for a divider between compartments. One distribution of seven phonons into i ve compartments is shown in Figure 4.3 . The order of compartments from left to right correlates with the order of oscillator sites and degrees of freedom. Here the i rst compartment contains two phonons, the second one phonon, the third four, and the fourth and i fth no phonons.
Thus, the number of distinct ways n indistinguishable phonons may be put into 3 N compartments is n
(
3NNN−1)
! ⎡⎣⎡⎡⎡⎡⎡nn!(
3NN−1)
! – the binomial coefi -⎤⎦⎤⎤cients again. This counting procedure naturally incorporates the conservation of certain quantities: the system energy E ⎡⎣⎡⎡=hhνo
( (
n+3NN 2)
⎤⎦⎤⎤ and the number of oscillators N . Therefore, the multiplicity of a macrostate consisting of n phonons and N three-dimensional oscillators is Figure 4.3 An arrangement of seven phonons in i ve ordered compartments. The symbol 卷 stands for a phonon and 弗 for a divider between the compartments that hold the phonons. From left to right these compartments hold 2, 1, 4, 0 and 0 phonons.
The method of phonons determines the entropy of a system directly in terms of the thermodynamic variables – here energy E and oscillator number N – without maximizing the constrained entropy as a function of occupation num-ber macrostate. When such direct methods work, they often work well.