Question

Planck's quantum theory.explain?

Answer 1

Basically, Planck's quantum theory was able to properly describe blackbody radiation in the early 1900s that classical theory could not explain in the late 1800s.

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Rayleigh and Jeans proposed a classically-derived spectral radiance function that varies with wavelength and temperature:

`rho(lambda,T) = (2ck_BT)/(lambda^4)`,

where `c` is the speed of light in `"m/s"`, `k_B` the Boltzmann constant in `"J/K"`, `T` temperature in `"K"`, and `lambda` the wavelength in `"m"`. So, `B` is in `"J/m"^3cdot"s"`.

The problem with that is that at the limit of low wavelength (i.e. high-energy light), the function goes to infinity, what classical physicists call the "ultraviolet catastrophe".

To address that, in 1900, Max Planck did an experiment with oscillating electrons in an oven, that radiate energy. As it turns out, the energy radiated could be described as chunks of light, known as photons, corresponding to quanta of energy `hnu`.

In doing so, he derived an empirical expression that is well-behaved at low wavelength (high frequency):

`barul|stackrel(" ")(" "rho(lambda,T) = (2hc^2)/(lambda^5) 1/(e^(hc//lambdak_BT) - 1)" ")|`

where `h` is Planck's constant, and all other variables are as seen above.

At low wavelength, `e^(hc//lambdak_BT) -> oo`, so `rho(lambda,T) -> 0` as it should. It is shown here:

Here we can see that

• the Rayleigh-Jeans law goes to infinity near `"200 THz"` (corresponding to about `"1500 nm"`), which isn't even that low.
• Wien's law does pretty well at low wavelength (particularly after `"600 THz"`, or about `"500 nm"`), but is a little off at high wavelength.
• Both Planck and Wien's laws go to zero at low wavelength (`"1400 THz" harr ~~ "214 nm"`).

If Planck is right, his expression should reduce to the classical result (low wavelength). At low wavelength, expanding as a Taylor series to first order,

`e^(hc//lambdak_BT) ~~ 1 + (hc)/(lambdak_BT) + cancel(((hc)/(lambdak_BT))^2/(2!))^"small" + ...`

So,

`lim_(lambda->epsilon) rho(lambda,T) = (2hc^2)/(lambda^5) 1/(1 + (hc)/(lambdak_BT) - 1)`

`= (2hc^2)/(lambda^5) (lambdak_BT)/(hc) = (2ck_BT)/(lambda^4)` `color(blue)(sqrt"")`

which is indeed the expression given by Rayleigh and Jeans that classically describes the behavior of blackbody radiation (low wavelength `->` correspondence principle `->` classical limit).