Problem — Hydrogen levels as horizontal lines

Problem

On a diagram with energy on the vertical axis you see horizontal lines at $-13{,}6$, $-3{,}40$, $-1{,}51$, $-0{,}85$ eV. Explain what they represent, why they crowd together as they approach zero, and what happens above $E = 0$.

Solution

What they represent. Each horizontal line is an allowed energy level of the electron in the hydrogen atom, labelled by the principal quantum number $n$. These are the only energy values the bound electron can take: intermediate energies are forbidden (quantised energy). The lowest level, $n=1$ at $-13{,}6$ eV, is the ground state.

Why they are negative. The energy is negative because the electron is bound: energy must be supplied to remove it. The reference $E=0$ corresponds to the electron just freed, at rest, infinitely far from the nucleus.

Why they crowd together. The energy follows

$E_n = -\frac{13{,}6}{n^2}\;\text{eV}$

Since $1/n^2$ decreases ever more slowly, the gaps between consecutive levels shrink as $n$ grows ($10{,}2$ eV between $n=1$ and $n=2$, only $0{,}66$ eV between $n=3$ and $n=4$). Infinitely many levels accumulate just below zero.

What happens above $E=0$. At $E=0$ the electron reaches ionisation: it detaches from the nucleus. For $E > 0$ the electron is free and its energy is no longer quantised: it can take any value. This region is the continuum. The transition from bound (discrete lines) to free (continuum) occurs precisely at $E=0$.

Links

Topics: Fisica quantistica Concepts: Modello di Bohr Skills: Lettura dei grafici