Welcome to the interactive demo calculator for calculating and plotting ultra-diluted gas optical transmittance.

This calculator is designed to help you visualize and understand how changes in ultra-diluted gas cloud properties, affect its ability to transmit light. With this tool, you can adjust various parameters such as the particles total cross-section, as well as the particles' wavefunction spread (degree of coherence), and see how they affect the transmittance.

The first diagram shows a gas cloud made of a few 2D particles. Their wavefunctions have the Gaussian distribution with uniform standard deviation $$stdev$$. Each particle is offset from the $$x=0$$ axis by $$o_n$$. The light direction, position of the light detector, and visibility tunnel (volume of integration) are marked red. The detector is centered and placed parallel to the X-axis. On the right-hand side, there is a plot of gas particles' probability distributions projected onto the detector plane. In this way, the distributions dimension is reduced by 1.

The transmittance chart shows the sample dependence of transmittance on the particles' wavefunction spread (measured with standard deviation) for the measurement with a fixed-size detector. The length unit is equal to the detector radius r. The detector width is 2 ($$r=1$$). A 1D (a 2D cloud cast onto the detector plane) cloud is $$N=61$$ particles in total, and they are evenly spaced every $$2r$$. The $$G=1-TR_{cl}$$ coefficient may be adjusted using a transmittance slider. Initially, it is set to 0.9 after $$TR_{cl}=10\%$$. The particles have a 1D normal distribution where the standard deviation is denominated in detector radius units ($$r$$). The horizontal axis is logarithmic for convenience.

To use this calculator, simply use the sliders to adjust the gas cloud properties. The calculator will automatically generate the transmittance value based on your inputs. You can also see a visual representation of the transmittance value with the plot.

It is interesting to note that the transmittance increases beyond the predictions of classical laws, even though the system's mass remains conserved. It is visible in the middle part of the transmittance plot - in the "closed system range". We see that the transmittance of a smeared gas cloud may be much greater than that of a non-spread (ideal gas) cloud at the same concentration. Moreover, by adjusting the slider "Classical transmittance" to 0, it is possible to see that a completely opaque cloud (as predicted classically) may be partially transparent. This phenomenon can be attributed solely to the wavefunction spread.

The equations used in this interactive demo calculator are briefly explained below along with the bibliography. The equations take into account the particle cross-section and wavefunction spread to calculate the transmittance of the gas cloud.

By using this demo, you can gain a deeper understanding of the principles of ultra-diluted gas optical transmittance and how it is affected by changes in gas cloud properties. The bibliography provides further resources for exploring this topic in more detail.

Enjoy!

 Particles spread (stdev): Classic transmittance: Cross section (or length):

The gas cloud transmittance $$TR=I_{out}/I_{in}$$ is the probability that a photon (that the detector would have detected in the absence of a cloud) passes not-absorbed the entire $$N$$-element cloud, and is detected by the detector: $$TR = \prod _{n=1}^N \big ( 1-\,P_n \big )~,$$ where $$P_n$$ is the probability $$n$$-th particle scatters a photon (within the tunnel of visibility). This probability renders as: $$P_n = G \int _{-r}^{r} \big|\Psi_n (x)\big|^2 dx = G \int _{o_n-r}^{o_n+r} \big|\Psi (x)\big|^2 dx~,$$ where

• $$G\in(0,1)$$ is a constant coefficient, which encodes the probability of passing a photon from the source to the detector in the presence of a classical cloud, it depends on the physical properties of a cloud: density, thickness, and particles’ likelihood to absorb given wavelength (its total cross-section): $$G = 1-TR_{cl} = 1-e^{-\tau}$$ , where $$TR_{cl}$$ is the transmittance of the gas cloud as it would be a classic (non-coherent, non-spread), ideal gas cloud.
• $$\Psi_n (x)$$ is the $$n$$-th particle wavefunction; for simplicity (but without loss of generality) it is assumed all wavefunctions are identical except that they are located in different places $$\Psi_n (x) = \Psi (x-o_n)$$,
• $$r$$ is detector radius and the detector is placed centrally (at $$x=0$$),
• $$o_n = \left \langle \big|\Psi_n (x)\big|^2 \right \rangle$$ is the position of $$n$$-th particle.

Note that the above equation for $$P_n$$ probability is a subject of quantum mechanics interpretation, namely interpretation of the Born rule. In our model, we assume that this is the probability that $$\Psi_n$$ interacts somewhere in the region $$x \in (-r,r)$$ . This is an important difference from the usual understanding that $$P_n$$ is the probability that $$n$$-th particle is found in this region. See  for details.

Given that each wavefunction $$\Psi_n$$ is subject to the solution of the well-known free particle Schrödinger equation (for some free time $$t$$), its probability distribution is Gaussian. We assume, again, without loss of generality, the gas cloud is 2-dimensional (for 3D case see  ). We may cast 2D Gaussians toward the detector plane (see the top-right chart), which yields 1D normal distributions: $$\big|\Psi_n (x)\big|^2 = \mathcal{N}\big(o_n, stdev(t)\big) = \frac{1}{\sqrt{2\pi}\,stdev(t) }exp \left( \frac{x-o_n}{\sqrt{2}\,stdev(t)} \right)^2~.$$ We know the analytical solution of the definite integral: $$\int_a^b \mathcal{N}(\mu, stdev)\,dx=\frac{1}{2}\left[ erf \left( \frac{a-\mu}{\sqrt{2}\,stdev} \right) - erf \left( \frac{b-\mu}{\sqrt{2}\,stdev} \right) \right]~,$$ where $$erf$$ is the Gauss error function.

After plugging the equations, we obtain the analytical form of the transmittance formula: $$TR = \frac{I_{out}}{I_{in}}= \prod _{n=1}^N \left ( 1-\, \frac{G}{2}\left[ erf \left( \frac{o_n-r}{\sqrt{2}\,stdev} \right) - erf \left( \frac{o_n+r}{\sqrt{2}\,stdev} \right) \right] \right )~,$$ which is used to plot the chart above.

For the plot above we assume for simplicity (but without loss of generality) there are 61 particles in the cloud ($$N=61$$), they are spaced evenly ($$o_n = r( \frac{2n-N-1}{2})$$), and the detector radius is of unit length ($$r=1$$).

###### Relation to the classic transmittance:

In the range of the classical system ($$stdev \ll r$$), the classic equation of transmittance (the so-called Beer-Lambert law): $$TR_{cl}=e^{-\tau}=e^{-n_dl\sigma}~$$ is the first-order $$TR$$ equation approximation. We use the standard notation: $$\tau$$ is the classic optical depth, $$n_d$$ is the number density of the attenuating particles, $$l$$ is the path length of the beam of light through the cloud, and $$\sigma$$ is the attenuation cross section of the single particle. For deriving classical equation details, see Eqs. (19-30) in .

###### References:
1. Ratajczak, J. M., The dark form of matter, on optical transmittance of ultra diluted gas. Results Phys. 19, 103674. https://doi.org/10.1016/j.rinp.2020.103674 (2020)
2. Ratajczak, J. M., Ultra-diluted gas transmittance revisited. Sci Rep 12, 19859. https://doi.org/10.1038/s41598-022-23657-0 (2022)
3. Ratajczak, J. M., On the falsification of the pilot-wave interpretation of quantum mechanics and the meaning of the Born rule. arXiv:2008.12124. https://doi.org/10.48550/arXiv.2008.12124 (2022)