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Actually I was learning about Wien's displacement law. It states that $$\lambda T=2.898×10^{-3} mK$$

This is actually a part of Planck's law where the Planck's constant originated.

Now Planck's temperature is given as $$T_{p}=\sqrt{\frac{hc^5}{2\pi G k^2_b}}=1.416×10^{32} K$$

Now Planck's length is $1.616×10^{-35} m$

Now since the smallest possible wavelength is Planck's length, we can say wavelength of the electromagnetic radiation is Planck's length (Assume the energy doesn't create a black hole).

Now according to Wien's displacement law,

$$l_p T=2.898×10^{-3} mK$$

Now solving this we get $1.79×10^{32} K$, which is higher than the actual Planck's temperature.

Since this displacement law is completely derived from Planck's law, it bit frustrated me. I'm a bit confused. Is it the limit of the displacement law or my flaw?

Please rectify this.

(Sorry if I made any mistake. I'm new to this one. Please explain my mistake. I'm glad to hear that.)

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    $\begingroup$ Two things - (1) Wein's displacement law is an approximation of sorts so you need to be careful there. (2) The Planck length/energy/time/temperature are not rigourous cut-offs. They are more like ball-park values. Factors of $2$, $\pi$, etc. are not carefully kept track of when evaluating their values. Additionally, they are not even real cutoffs. The real interpretation is that above the Planck temperature there must be new physics (and by new, I mean drastically new. It's not enough to simply add new particles). What that physics is, we simply do not know (though we have some guesses). $\endgroup$
    – Prahar
    Commented Feb 9, 2022 at 12:58
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    $\begingroup$ I guess, there exist a temperature on which the temperature velocity of molecules become larger than speed of light. So, it will be a temperature limit $\endgroup$
    – Robotex
    Commented Feb 9, 2022 at 13:23
  • $\begingroup$ @Robotex maybe... but if particles start to get hyperenergetic, their relativistic mass increases. I'd have to do some careful reading/research to figure out whether the total kinetic energy is ever limited. $\endgroup$
    – Carl Witthoft
    Commented Feb 9, 2022 at 15:21
  • $\begingroup$ @CarlWitthoft In that case the increasing of temperature will increase the mass of matter. I'm interesting, is it possible to detect this mass changing in lab? $\endgroup$
    – Robotex
    Commented Feb 10, 2022 at 14:30
  • $\begingroup$ @CarlWitthoft In some temperature the molecules and atoms will be completely destroyed and only quarks will left $\endgroup$
    – Robotex
    Commented Feb 10, 2022 at 14:32

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When you derive a formula for some quantity, the result you get is often a product of powers of the parameters of the problem and fundamental constants of the theory and some real number that tends to be close to $1$. For example, the Newtonian escape velocity is $\sqrt{2Gm/r}$; the real factor there is $\sqrt2\approx 1$.

There's a general expectation that quantities of interest in quantum gravity are likely to be products of powers of the fundamental constants $\hbar, c, G$ and some real factor close to $1$. The actual factor, and the actual meaning of the quantities, depends on the theory.

So maybe in the correct theory of quantum gravity, lengths/temperatures in the vicinity of the Planck length/temperature have some significance, but it probably won't be precisely the Planck length/temperature, and the significance probably won't be that it is the minimum length / maximum temperature.

You got a value close to the Planck temperature from the Planck length and Wien's constant because they're all equal to products of fundamental constants times a unitless factor close to $1$, but the value you got is a bit different because the unitless factors aren't quite the same. The Planck length isn't the smallest possible wavelength (probably).

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