Schrödinger evolution and stationary states

The time-dependent Schrödinger equation is first order in time. Given an initial state and a suitable self-adjoint Hamiltonian, it determines the later state. For a single nonrelativistic particle without electromagnetic vector potentials, $H=p^2/(2m)+V(x,t)$. This model assumes the regime where relativistic corrections and particle creation are negligible. It does not claim that every microscopic system is literally a single wave moving along one spatial line.

The factor $i$ is central. With a self-adjoint Hamiltonian, the evolution is unitary and preserves the state's norm, so total probability stays one. For a time-independent Hamiltonian the evolution operator is $U(t)=e^{-iHt/\hbar}$. A time-dependent Hamiltonian generally requires time ordering; inserting an ordinary exponential of its integral can fail when Hamiltonians at different times do not commute. The simple spectral method below deliberately assumes time independence.

An energy eigenstate of a time-independent Hamiltonian evolves by one overall phase, $e^{-iE_nt/\hbar}$. The state vector therefore changes in our chosen mathematical convention, but its physical ray does not. All time-independent measurement probabilities are constant. Such a state is called stationary. A stationary probability density should not be interpreted as an electron sitting motionless at one definite location.

A coherent superposition of different energies evolves differently. Each energy coefficient acquires its own phase, so relative phases change at frequencies determined by energy differences. The energy probabilities themselves remain fixed for a time-independent Hamiltonian, yet position or another incompatible observable can show time dependence through interference. Adding a constant to every energy changes only a common phase and leaves closed-system probabilities unchanged.

Consider $H=\operatorname{diag}(0,\Delta)$, with a positive energy gap $\Delta$. Prepare $|+\rangle=(|0\rangle+|1\rangle)/\sqrt{2}$. The zero/one probabilities remain one half, but an $X$-basis measurement recombines the components. Its plus probability is $\cos^2(\Delta t/(2\hbar))$. A system can therefore have conserved energy statistics and changing statistics for another observable at the same time.

The oscillation period is $2\pi\hbar/\Delta$, equivalently $h/\Delta$. This gives a useful units check: energy divided by $\hbar$ is angular frequency. Experiments often include decoherence and uncertain timing, which reduce observed contrast. Our calculation isolates ideal unitary evolution; it does not model those environmental processes or a drive. Keeping that boundary explicit helps diagnose whether a disagreement comes from arithmetic or from a missing physical ingredient.