Energy Levels — Core Principles

Core Principles

Energy levels in the Bohr model describe the discrete, quantized amounts of energy an electron can possess within an atom. Instead of orbiting arbitrarily, electrons are restricted to specific 'stationary states,' each with a unique energy value.

The lowest energy state is the ground state ($n=1$), and higher states are excited states ($n=2, 3, ...$). These energy levels are negative, indicating the electron is bound to the nucleus, with zero energy representing a free electron.

The energy of an electron in the $n$-th orbit of a hydrogen-like atom is given by $E_n = -\frac{13.6 Z^2}{n^2} \text{ eV}$. Transitions between these levels involve the absorption or emission of photons with energy precisely equal to the energy difference between the levels, explaining the characteristic line spectra of atoms.

As the principal quantum number $n$ increases, the energy levels become less negative and are spaced more closely together, eventually converging to zero at the ionization limit.

Important Differences

vs Classical Model of Atom

Aspect	This Topic	Classical Model of Atom
Electron Orbits	Electrons can orbit at any radius and possess any energy (continuous spectrum).	Electrons can only exist in specific, discrete orbits with quantized energy levels (line spectrum).
Atomic Stability	Accelerating electrons should continuously radiate energy and spiral into the nucleus, making atoms unstable.	Electrons in stationary orbits do not radiate energy, ensuring atomic stability.
Energy Emission/Absorption	Atoms should emit/absorb light continuously across all frequencies.	Atoms emit/absorb photons only when electrons transition between specific energy levels, leading to discrete spectral lines.
Angular Momentum	Angular momentum of electron can take any value.	Angular momentum is quantized, $L = n\frac{h}{2\pi}$.

The classical model of the atom, based on Rutherford's planetary model and classical electromagnetism, predicted that electrons orbiting the nucleus should continuously radiate energy and eventually collapse into the nucleus, rendering atoms unstable. It also failed to explain the observed discrete line spectra of elements. In stark contrast, Bohr's model introduced the revolutionary concept of quantized energy levels and angular momentum, proposing that electrons exist in stable, non-radiating 'stationary states.' This explained atomic stability and the characteristic line spectra, marking a pivotal shift towards quantum understanding.