1.4: Changes in time

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Schroedinger’s equation \(\hat{H} \phi = i \hbar \partial \phi / \partial t\) shows us that the Hamiltonian (energy operator) is related to the change in wavefunction in time. A system prepared in an eigenstate of the Hamiltonian has time-invariant probability density. A system prepared in an eigenstate of a non-commuting operator has a probability density which varies in time. It is this time independence (conservation law) which makes eigenstates of the energy operator so useful.

When we measure some property of a system, the act of making the measurement collapses the system into an eigenstate of the appropriate operator. All memory of the previous state of the system is lost in this collapse, except in the special case when the state is degenerate, as we’ll see later. The system then evolves according to its Hamiltonian.