Thermodynamics of the harmonic oscillator: derivation of the Planck blackbody spectrum from pure thermodynamics

Abstract

In 1893, Wien applied the first two laws of thermodynamics to blackbody radiation and derived his displacement theorem. Believing that the information from thermodynamics had been exhausted, Planck turned to statistical ideas in 1900 in order to provide a physical understanding for his experimental-data-based interpolation giving the Planck spectrum without zero-point radiation. Here we point out that the third law of thermodynamics (which was developed in the early years of the 20th century) introduces additional thermodynamic information regarding thermal radiation. The Planck spectrum for thermal radiation can be derived from purely thermodynamic ideas applied to the classical simple harmonic oscillator, since every radiation mode takes a simple oscillator form. Thermodynamics alone implies the Planck spectrum including zero-point energy without any need for quantum theory or statistical ideas. It is pointed out that the inclusion of zero-point energy involves more natural thermodynamics than its exclusion.

Quantum harmonic oscillators and thermalization

“…For example, δΩ I = ω I δS I in ( 19) is determined by Eq. (13). As an example, when Ω I is just a function of ω only, the nonadiabaticity is automatically determined as…”

“…Here, as a probe to study thermalization, we consider a quantum oscillator undergoing thermalization. A quantum harmonic oscillator is a useful model in a variety of fields in physics due to its simplicity and broad applicability [10][11][12][13]. An example of the above interaction Hamiltonian for the combined system is Ĥcoupling = mx k xk , where a set of an infinite number of coupled quantum oscillators approximates a heat bath [10].…”

Thermal Radiation Equilibrium: (Nonrelativistic) Classical Mechanics versus (Relativistic) Classical Electrodynamics

“…If we consider only the first two laws of thermodynamics, then the function f (ω 0 /T ) giving the transition between these extremes is unknown. Only if we add further criteria such as "smoothness" [5] or inclusion of the third law, [6] do we obtain the Planck spectrum with zero-point energy given in Eq. (1).…”

Dirac's Classical-Quantum Analogy for the Harmonic Oscillator: Classical Aspects in Thermal Radiation Including Zero-Point Radiation

“…where τ = (2e 2 )/(3mc 3 ) is associated with radiation damping, and eE T x (x, t) is the force on the oscillator due to the random classical thermal radiation. The equation of motion (24) has been solved (for the steady-state solution) many times [20] going back to Planck's work. [15] The result for the equilibrium phase space distribution P T (x, p x ) for the oscillator in random classical radiation (corresponding to the Planck spectrum at temperature T and including zero-point radiation) is independent of the (small) charge e and takes the form…”

“…This viewpoint that the thermal spectrum vanishes at zero temperature is quite different from the classical view which includes classical zero-point radiation as an integral part of the spectrum of classical thermal radiation. [24] In classical theory, the random classical zero-point radiation leads to the zero-point fluctuations of a classical mechanical oscillator which must match the average radiation energy at the natural frequency of the oscillator in order to be in equilibrium with the random classical radiation. On the other hand, quantum theory has an entirely different basis for thermal equilibrium between radiation and matter.…”