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Mathematics for Physics

103 Conservation of Energy

A principle that is used to keep the total energy of an isolated quantum system constant in time, where an isolated system exchanges no energy with its surroundings.

Time-independent energy eigenvalues. If the Hamiltonian does not depend on time, the energy eigenvalues are constant and a stationary state keeps a definite energy. This principle is used to assign a fixed \(E\) to each energy eigenstate.

Conservation of \(\langle H\rangle\). The expectation value of \(\hat{H}\) is constant when \(\partial\hat{H}/\partial t=0\). This principle is used to treat \(\langle H\rangle\) as the conserved energy of a general state.

The conservation of mean energy is

\[ \dfrac{\partial\hat{H}}{\partial t} = 0 \implies \dfrac{d\langle H\rangle}{dt} = 0 \]

where

  • \(\hat{H}\) is the Hamiltonian.
  • \(t\) is time.

Energy conservation in a radiative transition. In a radiative jump the atom plus the photon conserve energy: \(hf=\lvert E_{i}-E_{f}\rvert\). This principle is used to match spectral lines to level differences.

The Bohr frequency condition is

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

where

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

103.1 References

  1. Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — time-independent Hamiltonian and conserved energy.
  2. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — photon energy and atomic jumps.