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I'm learning from Griffiths' book "Quantum Mechanics". In quantum mechanics, a particle has a variety of accessible energy states. Then is the energy of the particle always conserved? Energy states are decided randomly, this makes me think energy would not be conserved. I got some alternative ideas:

  1. Average value of energy is conserved. Energy is statistically conserved, but there are fluctuations.
  2. Few particles effect each other, so the energy loss of one particle is compensated by another particle. Therefore energy is always conserved and there is no fluctuation.
Thomas Fritsch
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Then is the energy of the particle is conserved always?

In QM system energy conservation law applies to wavefunction, so that given Hamiltonian operator and particle wavefunction at a time $t=0$, particle wavefunction at a time $t$ can be calculated as : $$ {\displaystyle\left|\psi (t)\right\rangle =e^{-iHt/\hbar }\left|\psi (0)\right\rangle } $$ However, if measurement on particle is performed, then energy conservation law at wavefunction level breaks

Agnius Vasiliauskas
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