The Massive Gap Between Classical and Quantum Physics

To illustrate how complete a break there is between quantum physics and classical mechanics exists, or how dangerous it is to mix them, suppose we try to calculate the temperature inside the nucleus.
Any particle inside the nucleus must have a wavelength of about the same order as the size of the nucleus  
\[{} \simeq 10^{-15}\]
m. Suppose then that Consider a proton in the nucleus has this wavelength. Its momentum is then  
\[p= \frac{h}{\lambda}=\frac{6.626 \times 10^{-34}}{10^{-15}}=6.626 \times 10^{-19} \]
kg m/s.
The kinetic energy  
\[\frac{1}{2}mv^2 = \frac{p^2}{2m}\]
  can also be considered to be  
\[\frac{3}{2}kT\]
  (according to the kinetic theory of gases), so  
\[T=\frac{p^2}{2mk}= \frac{h^2}{2mk \lambda^2}\]
.
Inside the nucleus the temperature is then  
\[T=\frac{(6.626 \times 10^{-34})^2}{2 \times 1.66 \times 10^{-27} \times 1.38 \times 10^{-23} \times (10^{-15})^2}=9.58 \times 10^{12}\]
K.
This is far hotter than the temperature at the centre of the Sun, which is only  
\[1.5 \times 10^7\]
K.

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