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Mathematics for Physics

104 Conservation of Energy Transition Law

An energy-conservation rule that is used to match photon energy to an atomic energy difference when an atom jumps between two levels.

The transition energy rule. When an atom jumps from energy \(E_{i}\) to energy \(E_{f}\), it emits or absorbs a photon whose energy equals the gap. This principle is used to compute the frequency of the spectral line.

The transition energy rule is

\[ E_{\mathrm{photon}} = \Delta E_{\mathrm{atom}} = \lvert E_{f} - E_{i}\rvert \]

where

  • \(E_{\mathrm{photon}}\) is the energy of the photon.
  • \(\Delta E_{\mathrm{atom}}\) is the change in atomic energy.
  • \(E_{i}\) and \(E_{f}\) are the initial and final atomic energies.

Absorption versus emission. Absorption raises the atom and the photon vanishes. Emission lowers the atom and a photon is created. This principle is used to decide the direction of the jump from whether light is taken in or given out.

The Bohr frequency condition. Combined with \(E=hf\), the same rule fixes the frequency of the line. This principle is used to convert a level diagram into a spectrum.

The Bohr frequency condition is

\[ hf = \lvert E_{f} - E_{i}\rvert \]

where

  • \(h\) is Planck’s constant.
  • \(f\) is the photon frequency.
  • \(E_{i}\) and \(E_{f}\) are the atomic energies.