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Mathematics for Physics

66 Conservation of Energy Transition Law

The energy-conservation rule that when an atom transitions between two energy levels, it must emit or absorb a photon whose energy equals the difference between those states, that is used to match photon energy to an atomic energy difference.

(\(E_{\text{photon}} = \Delta E_{\text{atom}}\)).

definition [d] (Conservation of Energy Transition Law) From Knight: according to Einstein, a photon of frequency \(f\) has energy \(E_{\mathrm{photon}} = hf\). If an atom jumps from an initial state with energy \(E_{i}\) to a final state with energy \(E_{f}\), energy will be conserved if the atom emits or absorbs a photon with

  • \(E_{\mathrm{photon}} = \Delta E_{\mathrm{atom}} = |E_{f} - E_{i}|\) .

When an atom is excited to a higher energy level by absorbing a photon, the photon vanishes. Thus energy conservation requires

  • \(E_{\mathrm{photon}} = \Delta E_{\mathrm{atom}}\) .

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.
  • \(f\) is the photon frequency.
  • \(h\) is Planck’s constant.

66.1 Elementary Example

66.1.1 Simple

An atom drops from \(E_{i} = -1.5\,\mathrm{eV}\) to \(E_{f} = -3.4\,\mathrm{eV}\).

\[ \Delta E_{\mathrm{atom}} = 1.9\,\mathrm{eV} \]

\[ E_{\mathrm{photon}} = 1.9\,\mathrm{eV} \]

where

  • the emitted photon carries exactly \(\Delta E_{\mathrm{atom}}\).

66.1.2 General

Three downward jumps each emit a photon equal to the level gap.

\[ \Delta E_{1} = |E_{3}-E_{2}|,\quad \Delta E_{2} = |E_{4}-E_{2}|,\quad \Delta E_{3} = |E_{5}-E_{2}| \]

\[ E_{\mathrm{photon},k} = \Delta E_{k} \]

where

  • each \(E_{\mathrm{photon},k}\) matches the corresponding atomic gap.