Explain (in detail, with equations if possible) why classical and quantum theori

Question

Explain (in detail, with equations if possible) why classical and quantum theories predict similar values for thermal conductance but very different ones for the specific heat of metals.

Explanation / Answer

Thermal conductance is dominated by lattice phonons. Phonons are bosons, with the property that any number of bosons can occupy the same quantum state at the same time. At reasonable temperatures, the granularity predicted by quantum theory is lost in measurements since the resultant amplitude of a very large number of particles will be measured. Significant deviations from classical theory therefore require very low temperatures to freeze out phonon modes. Specific heat of metals is dominated by the electronic component, because of the high free-electron concentration in metals (> 10^20 per cubic cm). Electrons are fermions and obey Pauli exclusion principle. No more than 2 electrons can occupy the same state. Since lower energy levels are jam-packed, there is no room for low-energy electrons to acquire thermal energy and transition to a slightly higher energy level. Thus only electrons in the top few eigen states (those in a narrow band of energies around the Fermi level) can acquire thermal energy, and thus even at room temperature only a small fraction of electrons contribute to the specific heat. This causes a large deviation from the classical picture, which predicts that each and every free electron in the metal acquires about kT worth of energy. Thus quantum theory successfully explained why the experimental heat capacity of metals was about 2 orders of magnitude less than the classical prediction.

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