## The Massive Gap Between Classical and Quantum Physics

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.