Nonlinear Schrödinger equation and classical-field description of thermal radiation

Abstract

It is shown that the thermal radiation can be described without quantization of energy in the framework of classical field theory using the nonlinear Schrödinger equation which is considered as a classical field equation. Planck’s law for the spectral energy density of thermal radiation and the Einstein A-coefficient for spontaneous emission are derived without using the concept of the energy quanta. It is shown that the spectral energy density of thermal radiation is apparently not a universal function of frequency, as follows from the Planck’s law, but depends weakly on the nature of atoms, while Planck’s law is valid only as an approximation in the limit of weak excitation of atoms. Spin and relativistic effects are not considered in this paper.

Appendix

Using the spectral energy density of radiation (90), the Stefan–Boltzmann constant is defined by the expression

$$\sigma = \frac{{k^{4} }}{{\hbar^{3} \pi^{2} c^{3} }}\mathop \int \limits_{0}^{\infty } \frac{{x^{3} }}{{\left[ {\exp \left( x \right) - 1} \right]\left[ {1 + \exp \left( { - x} \right)} \right]}}dx$$

(99)

Let us calculate the integral in the expression (99). Obviously,

$$\mathop \int \limits_{0}^{\infty } \frac{{x^{3} }}{{\left[ {\exp \left( x \right) - 1} \right]\left[ {1 + \exp \left( { - x} \right)} \right]}}dx = \mathop \int \limits_{0}^{\infty } \frac{{x^{3} \exp \left( { - x} \right)}}{{\left[ {1 - \exp \left( { - 2x} \right)} \right]}}dx = \mathop \int \limits_{0}^{\infty } x^{3} \mathop \sum \limits_{n = 0}^{\infty } e^{{ - \left( {2n + 1} \right)x}} dx = \mathop \sum \limits_{n = 0}^{\infty } \frac{1}{{\left( {2n + 1} \right)^{4} }}\mathop \int \limits_{0}^{\infty } y^{3} e^{ - y} dy = \frac{{\pi^{4} }}{16}$$

(100)

because the last integral in expression (100) is equal to 6, and

$$\mathop \sum \limits_{n = 0}^{\infty } \frac{1}{{\left( {2n + 1} \right)^{2p} }} = \frac{{2^{2p} - 1}}{{2 \cdot \left( {2p} \right)!}}B_{p}$$

where $p = 1,2,3, \ldots$; $B_{p}$ are the Bernoulli numbers; in particular $B_{2} = 1/30$.