📂Quantum Mechanics

Electrons Cannot Be Constituents of the Nucleus

Theorem

An electron cannot be a component of a nucleus.

Explanation

$10^{-14}\mathrm{m}$ A nucleus with a scale of $1\mathrm{MeV} \sim10\mathrm{MeV}$ emits an electron with energy in the range of $1\mathrm{MeV} \sim10\mathrm{MeV}$. In the early days of nuclear physics, it was believed that electrons existed within the nucleus. By using the uncertainty principle, it can be shown that an electron with such energy cannot be confined within the nucleus.

Proof

By assuming the electron is inside the nucleus, and using the uncertainty principle,

$$p2r \simeq \hbar$$

Here, $2r$ is the diameter of the nucleus.

$$K_{E}=\frac{p^2}{2m}=\frac{1}{2m}\frac{\hbar^2}{4r^2}=\frac{\hbar^2}{8mr^2}$$

If we actually calculate the values,

$$\frac{\hbar^2}{8mr^2}\simeq 100\mathrm{MeV}$$

Experimental results indicate that $1\mathrm{MeV}$, so we see that $r$ must be much larger. Thus, we conclude that electrons must be far away from the nucleus.

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