Preamble

Quantum Analysis of Discrete Radiation Frequencies in a Resonate Cavity Xiaoyun Li , La Ba Sakya Genzon , Suoang Longzhou , Youping Dai , Yangsheng Xu School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen

Abstract

—According to Planck’s law, the energy density for photons with radiation frequency is based on the assumption that the photons’ frequency has a continuous distribution, with an infinitesimal frequency interval approaching zero. By integrating over the continuous frequencies, the energy density of photons with frequencies between can be derived. In this paper, it derives the energy density of discrete frequencies according to Planck’s law via critical derivation, when the frequency interval defined by Planck’s law is a non-zero constant. Then according to the cosmic resonate cavity model, there is a fundamental radiation fre- quency (minimal frequency interval) in cosmic space. Therefore, the equation for energy density of any single discrete frequency (any multiple of ) is derived consequently, which is verified by detailed traversal calculation results for any temperature . The derivation of equations for discrete radiation frequencies are helpful for more precisely quantum analysis of blackbody radiation in the future.

Keywords

s —resonant cavity, Planck’s law, blackbody radiation, energy density of discrete radiation frequency, infinitesimal frequency interval df.

ACKGROUND AND ELATED HE Planck’s law [1] describes the distribution of radia- tion from a blackbody. Planck’s law states that the energy of radiation is not continuously distributed, but rather consists of discrete energy states, which introducing the concept of quantized photon energy for the first time. The Planck’s law reveals the quantized nature of electromagnetic radiation.

The discreteness of radiation energy can be explained by grouping energy by frequency, which can be seen as a series of ”energy gradients”, where the height and width of each gradient depend on frequency. This spectrum is often referred to as the ”energy spectrum” or ”radiant energy density per unit frequency”. Due to the quantum nature of radiant frequency, the amount of energy contained in each frequency band is determined by the number of photons [2].

Research related to the Planck’s Law mainly includes both computational and experimental aspects. In recent years, the quantized energy distribution and spectral distribution have become key tools for studying thermodynamics and equations of state, in the research fields including quantum coherence and quantum processes [3][4][5].

The resonant cavity model of electromagnetic waves refers to a model where electromagnetic waves are confined within a closed space, forming a resonant mode. In short, it is a model where electromagnetic waves are confined between two Manuscript submitted Nov 4, 2025; reflective mirrors and form a pattern. In this model, electro- magnetic waves can propagate and reflect repeatedly within the space, forming resonant modes that result in standing wave patterns at specific frequencies. These standing wave patterns have discrete energy values and frequencies, and only electromagnetic waves (photons) with specific frequencies can remain in the cavity in a standing wave state. Therefore, only photons whose frequencies match the frequencies of the modes confined within the resonant cavity can be retained within the cavity. The main theories and research work regarding the frequency distribution and evolution of photons within the resonant cavity are as follows.

Optical cavity quantum electrodynamics is a theoretical method based on Maxwell’s optics and quantum electrody- namics, used to calculate and describe quantum optics and its nonlinear response within a closed resonant cavity. This approach provides a quantum mechanical explanation of high- order nonlinear phenomena within the resonant cavity and pre- dicts effects such as single-photon coupling, cavity quantum electrodynamics (CQED) energy levels, and photon coherence [6][7]. Photon coherent states are the optimal state as they minimize the product of photon number and variance. Photon statistical distribution is a probability distribution function that describes the distribution of photons within the resonant cavity, such as Bose-Einstein distribution [8]. These distribu- tions describe the probability distribution functions of photon distribution within the resonant cavity under different physical conditions. The study of photon statistical distributions mainly involves quantum optics and statistical physics.

Firstly, this paper uses the resonant cavity model to prove that the minimum frequency interval is a finite value, rather than approaching zero. Based on this, the energy density of any single discrete frequency is derived, and the correctness of the equations for discrete frequency energy density are ver- ified via derivation and traversal calculation for any radiation temperature , according to Planck’s Law.

II. T HE FREQUENCY OF PHOTONS IN A RESONANT CAVITY

The radiation frequency interval is called the spectral line resolution in classical physics, which is limited by the radi- ation wavelength and the harmonics of the spectrum. From the perspective of quantum mechanics, the frequency interval of spectral lines is limited by the quantization of photon energy. For the frequency interval of spectral lines, the energy

difference between photons in adjacent spectral lines can only be an integer multiple of Planck constant multiplied by the frequency difference , that is, , where an integer [1][2].

Therefore, when the energy of the photon is small enough, it cannot be further divided into smaller energy units, and the frequency difference at this point approaches the fundamental frequency. Therefore, it can be assumed that when the resonant cavity radius is large enough, the frequency interval of spectral lines approaches the fundamental frequency light [9][10]. This can be derived by combining wave optics and quantum mechanics.

Assuming that there is a monochromatic light beam with a frequency of , a wavelength of , and a corresponding energy , when it passes through a medium, its velocity changes, resulting in a change in frequency and wavelength. According to Fermat’s principle [11], the propagation path is determined based on the principle of minimum optical path length, so the path of the incident ray and the reflected ray must be equal in the medium, that is:

2 nd = Nλ

→ λ = 2 nd N

→ f = Nc 2 nd (1)

where is the refractive index of the medium, is the thickness of the medium, and is an integer.

Quantum mechanics analysis reveals that only allowed en- ergy states in the medium can be occupied by photons. These allowed energy states correspond strictly to the wavelength and frequency of light. Therefore, when a photon passes through a medium, it can only occupy those energy states that are integer multiples of the fundamental frequency of the incident light. This phenomenon can be explained as follows. Only energy states that are integer multiples of the fundamental frequency can be occupied by photons. When photons occupy non-integer multiples of energy states, their propagation paths in the medium do not satisfy the principle of minimum optical path length, thus they will be scattered or absorbed [10][11][12].

This article proposes a resonant cavity model for the cosmic space. Based on the Big Bang model [13], the cosmic space is originated from a finite-sized closed space. Since the beginning of Big Bang, the age of the universe is a finite time, estimated to be about 13.8 billion years until now [14][15]. According to different models of cosmic expansion, the current cosmic radius is on the order of tens of billions of light-years [14][15][16], which is a finite value rather than tending to- wards . Therefore the cosmic space is a finite closed space, which conforms to the characteristics of a resonant cavity.

According to equation represents the thickness of the medium. In the cosmic resonant cavity model, the thickness of the medium can be replaced by the cosmic radius yielding:

→ f = Nc 2 nR (2)

Let integer N = 1 for fundamental frequency f 0 . The refractive index n = 1 in vacuum.

→ ∆ f = f 0 = c 2 nR = c 2 R (3)

Since the cosmic radius is a finite value. Therefore, according to equation , the minimum frequency interval is a non-zero finite value. Furthermore, based on Planck’s Law, the following sections derive the energy density equation for any single discrete frequency , where an integer. Then the correctness of the derived equations is verified through simulations for any radiation temperature III. R ADIATION NERGY ENSITY OF ISCRETE REQUENCY CCORDING TO LANCK A. The Distribution of Discrete Radiation Frequencies The following energy density equations (4) and (5) accord- ing to Planck’s law are based on Bose-Einstein statistics, in which the number of photons per unit volume is extremely large, which could be equivalent to the number of infinite in statistics.

When performing statistical analysis on a large number of photons’ frequency distribution, the minimum frequency interval is assumed to be infinitesimally small, results in the assumption that frequencies being a continuous distribution.

This simplifies the mathematical expressions and facilitates obtaining analytical solutions.

However, the assumption of continuous frequency distri- bution is merely an idealized approximation and does not represent the real-world scenario where photon frequencies are not entirely continuous. In the original Bose-Einstein statistical analysis, each photon has a discrete frequency, implying the presence of discrete intervals between photon frequencies. The minimum interval between frequencies a non-zero constant. If more accurate computational results are required, the discreteness of photon frequencies still needs to be considered.

B. Energy density of discrete frequencies According to Planck’s law, the equation for energy density at frequency is as follows [1]:

U ( f, T ) V = 8 πhf 3

The energy density is the energy of photons within frequencies between per unit volume , where represents an infinitesimal change in frequency according to Plank’s law [1].

Therefore, the total energy density of all photons within frequency where is given as follows.

U ( f, T ) V = � ∞

U ( f, T ) V df = 8 π 5 ( kT ) 4

U ( T ) V =

Since is constant for , therefore where are consecutive integers 0,1,2,3... , as shown in figure Since each frequency equals the discrete frequency hence it is the differential distribution of discrete frequencies, being the differential naturalization factor. Let the naturalization factor coefficient for . Then, equation can be rewritten as follows.

U ( f, T ) V = U ( Ndf, T ) V = C df 8 πh ( Ndf ) 3

U ( Ndf, T ) V = C df 8 πh ( df ) 3

→ U ( T ) V =

Let x = hNdf

, then,

U ( T ) V = C df 8 πh ( df ) 3

e x − 1 = π 4

Therefore,

U ( Ndf, T ) V = C df 8 πh ( df ) 3

U ( T ) V =

→ U ( T ) V = C df 8 π 5 ( kT ) 4

Since U ( T ) V = 8 π 5 ( kT ) 4

according to Planck’s law, hence,

→ C df = df for any df > 0 (8)

Equation (8) proves that, the naturalization factor coeffi- cient equals the naturalization factor Hence, for any naturalization factor (frequency interval) then,

U ( f, T ) V = U ( Ndf, T ) V = df 8 πh ( Ndf ) 3

→ U ( f, T ) V = U ( Ndf, T ) V = 8 πh ( df ) 4

→ U ( T ) V = df 8 π 5 ( kT ) 4

15( hc ) 3 df = 8 π 5 ( kT ) 4

For instance, let df = 1 Hz , then,

U ( Ndf, T ) V = U ( N, T ) V = 8 πh c 3 · N 3

The traversal calculation for any radiation temperature also verifies that , which is consistent with equation (11), as described in the following simulation results in Appendix.

Energy density Vs frequency for T = 7 · 10 − 11 K, f 0 = df

Energy density Vs frequency for T = 7 · 10 − 11 K, f 0 = 1 M , M = 4

IV. T HE ENERGY DENSITY OF A SINGLE DISCRETE FREQUENCY According to equation , the minimum frequency interval in the cosmic resonate cavity is a non-zero finite value, where is the fundamental frequency in a resonate cav- ity. Therefore all radiation frequencies in the cosmic resonate cavity are discrete frequencies , where are consecutive integers 0,1,2,3... . Hence the minimum frequency interval between these discrete frequencies is , then the energy density of any single discrete frequency is as follows, according to equation (6), where Ndf, T

→ U ( Nf 0 , T ) V = 8 πhf 4 0 c 3 · N 3

according to equation (11), where , and . It shows that the energy density distribution of frequency spectrum is discrete, rather than a continuously smooth curve. , where is a real number where Figure shows an example for energy densities of every discrete frequencies according to equation (12), where , and . Therefore, there are 4 discrete frequencies per . Note that is an example in figure

2. The exact value of

is still unknown, which need to be measured in future experiments.

The following traversal calculation for any temperature and any in the Appendix, where is a real number , also verifies that which is consistent with equations (10) and (12).

ONCLUSION This paper derives the energy density of discrete frequencies according to Planck’s law via critical derivation, when the frequency interval defined by Planck’s law is a non-zero constant. Then according to the cosmic resonate cavity model, there is a fundamental radiation frequency (minimal frequency interval) in cosmic space. Therefore, the equation for energy density of any single discrete frequency (any multiple of is derived consequently, which is verified by detailed traversal calculation results for any temperature . The derivation of equations for discrete radiation frequencies are helpful for more precisely quantum analysis of blackbody radiation in the future.

EFERENCES Planck, M. Ueber das Gesetz der Energieverteilung im Normalspectrum.

Annalen der Physik, 4(3), 553-563 (1901). Einstein, A. The Quantum Theory of Radiation. Physikalische Zeitschrift, 18, 121-128 (1917).

Feynman, R. P., Leighton, R. B., & Sands, M. The Feynman Lectures on Physics Volume III (Ch. 37). Addison-Wesley (1965).

C. Kittel. Introduction to Solid State Physics, John Wiley & Sons, 2005.

D. R. Lide. CRC Handbook of Chemistry and Physics (CRC Press, 1996).

Haroche, S., & Raimond, J. M. Exploring the quantum: atoms, cavities, and photons (No. 26). OUP Oxford (2008).

H. Mabuchi and A. C. Doherty, ”Cavity quantum electrodynamics: coherence in context”, Science 298, 1372-1377 (2002).

G. Bj?rk and Y. Yamamoto. Photon antibunching and sub-poissonian photon statistics. In Progress in Optics, E. Wolf, ed., Vol. 33, pp. 787..840.

Elsevier, 1994 Sakurai, J. J., & Napolitano, J. (2014). Modern quantum mechanics.

Pearson. Loudon, R. (2000). The quantum theory of light. Oxford University Press Feynman, R. P. (1965). The Character of Physical Law (Chapter 6). MIT Press.

Scully, M. O., & Zubairy, M. S. (1997). Quantum Optics. Cambridge University Press.

Gamow, G. (1948) The Origin of the Elements and the Separation of Galaxies. Physical Review, 74(4), 505..506 Guth, A.H. (1981) Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems. Physical Review D, 23, 347..356 Linde, A. Inflationary Cosmology. arXiv preprint arXiv:1512.00205, Planck Collaboration et al. Planck 2015 results. XIII. Cosmological parameters. Astronomy & Astrophysics.

Peebles, P.J.E. and Ratra, B. (2003) The Cosmological Constant and Dark Energy. Reviews of Modern Physics, 75, 559..60 PPENDIX Simulation Results Traverse calculations for any temperature are performed according to equtions . The calculation results are shown in Table I and Table II.

As shown in Table I, when radiation temperature , and the minimum frequency interval , therefore every discrete frequency , where N are consecutive integers 0,1,2,3... . The calculation result shows that,

U ( N, T ) V = U ( T ) V = 8 π 5 ( kT ) 4

where is the sum energy density of each discrete frequency , N are consecutive integer from 0 to is the total energy density of all frequencies according to Planck’s law [1].

Table II also shows that, when then for temperature

U ( N, T ) V = U ( T ) V = 8 π 5 ( kT ) 4

which is consistent with the derivation of equations Therefore it verifies that for any temperature , where is the frequency interval defined by Planck’s law [1].

Table II shows that, when , and temperature , then,

U ( N, T ) V < 0 . 9999 · 8 π 5 ( kT ) 4

This is because the minimum frequency interval too sparse for such low temperature , causing the distribution of all discrete frequencies, is not equivalent to the distribution of frequencies from 0 to according to the derivation of equation . Hence, when , and , if temperature , then,

e x − 1 < π 4

U ( N, T ) V < 8 π 5 ( kT ) 4

U ( N, T ) V =

Similarly, as shown in Table II, for any , then,

U ( NL, T ) V < 0 . 9999 · 8 π 5 ( kT ) 4

However, as shown in Table II, for any L > 0 and f 0 = df = L Hz , if T ≥ L · 10 − 10 K , then,

U ( NL, T ) V = 8 π 5 ( kT ) 4

U ( Nf 0 , T ) V =

which is consistent with equation (13). Therefore the energy density discrete frequency according to equations (10 and 12) are verified.

TABLE I T = 7 · 10 − 11 K , M = 1 , f 0 = df = 1 Hz

Frequency

U ( T ) V = 8 π 5 ( kT ) 4

15( hc ) 3 1816.508 � 27 N =0 U ( N,T ) U ( T )

1816 . 236 1816 . 508 = 0 . 9999

End of Appendix TABLE II < L < 1.17E-15 1.93E-06