Particle in a box analysis of a youtube video.

 The Planck scales are just SI units rescaled as a coordinate rotation in unit space of the axis of measurements. Natural units are just where the equivalences of particles are harmonized in the Planck unit system. The Planck length is not a limit, it is just the unit of measure of length definition with SI unit c scaling removed and then scaled by scaled by t_P. l_P = c * t_P << all Planck units are just removing SI units of measurement scaling then rescaling each to Planck time. m_P = Hz_kg/t_P and T_P = Hz_K / t_P, that is the definition of second, meter, kg, and K unit scaling as Jacobean coordinates.

The no constants version of the physics. The constants are just doing this same natural unit scaling in the formulas. E_n = n^2 pi^2 h^2 / (8pi^2 m L^2 ) # unwrap the hbar, there is no pi in this formula E_n = n^2 Hz_kg ^2 c^4 / (8 m_n Hz_kg L_n c^2) # factor the formula for the definitions of length and mass in SI E_n = n^2 Hz_kg c^2 / (8 m_n L_n^2 ) # simplify E_n / (Hz_kg c^2) = n^2 / (8 m_n L_n^2 ) # move the m/f and (L/T)^2 unit scaling to other side to of the formula. E_n_natural = n^2 / (8 m_n L_n^2 ) with both sides having Hz units. This is the real physics that is happening independent of any unit system. You can multiply through by the non reduced Planck time scale t_Ph to go dimensionless and independent of any unit scaling but that doesn't give us anything here. If you exceed the limits of the box, it is not forbidden, the particle simply escapes the box. other energy levels are not forbidden, it just can't be contained in the box. k_B = K_Hz Hz_kg c^2 and h = Hz_kg c^2 because T ~ f ~ m ~ E are all equivalent measurements of the same underlying stuff, scaled to different units of measurement scales for each axis of measurement. One thing to keep in mind is that a single particle doesn't have temperature it just has a velocity. so h / k_B is just the units scaling of the frequency to a characteristic temp K_Hz unit conversion between equivalent units of measurement. if you pull the h/k_B out of the hawking formula it gives you a frequency. if you then scale that frequency by Hz_K you get the same measurement of that frequency in the temperature axis of measurement.