Schrodinger Equation Time Evolution Estimator

Each energy eigenstate evolves with its own phase factor, φ_n(x)e^(−iE_nt/ħ), a direct and exact consequence of the time-dependent Schrödinger equation. A superposition of two eigenstates therefore develops a relative phase between its two components that grows steadily with time, producing an oscillating (time-dependent) probability density even though each individual eigenstate alone would be perfectly static.

For a superposition of two energy eigenstates E1 and E2, the relative phase between them repeats every T = h / |E2 − E1|, the beat period. This is an exact result, not an approximation, it comes directly from the two phase factors e^(−iE1t/ħ) and e^(−iE2t/ħ) realigning after exactly one period.

For an equal-weight superposition of two eigenstates, the probability of measuring the system back in its original configuration (its state at t=0) is P(t) = (1 + cos(ωt)) / 2, where ω = ΔE/ħ. This oscillates between 1 (fully returned) and 0 (fully 'flipped' to the other component) once per beat period, an exact, closed-form result.

A single eigenstate's probability density is |φ_n(x)e^(−iE_nt/ħ)|² = |φ_n(x)|², since the phase factor has magnitude 1 and cancels out completely. This is why eigenstates are called 'stationary states', their measurable properties never change with time, no matter how long you wait, only genuine superpositions of different energies produce observable time evolution.

At t = T/2, the relative phase between the two components reaches exactly π, and for an equal-weight superposition the survival probability drops to precisely 0, the system has completely 'flipped' away from its initial configuration. At t = T, the phase returns to a full 2π and the system exactly reproduces its state at t=0.

The period T = h/ΔE is inversely proportional to the energy gap ΔE, so systems with widely separated energy levels (like tightly bound atomic electrons) oscillate on extremely fast, femtosecond or even attosecond timescales, while systems with closely spaced levels oscillate far more slowly. This inverse relationship is exactly analogous to E = hf for a single photon.

This is exact. Given that φ1 and φ2 are true energy eigenstates (exact solutions of the time-independent Schrödinger equation), their time evolution under the full time-dependent Schrödinger equation is known in closed form with no approximation at all, only the eigenstates themselves might come from an approximate method in a harder problem.

This mechanism underlies quantum beats seen in atomic and molecular spectroscopy, Rabi-like oscillations in simplified two-level systems, the periodic 'sloshing' of a particle-in-a-box wave packet built from two eigenstates, and the femtosecond-to-attosecond electronic oscillations studied in ultrafast laser spectroscopy of atoms and molecules.

Yes, a superposition of many eigenstates evolves as a sum of many phase factors, each with its own frequency, and the resulting motion is generally more complex than simple sinusoidal oscillation, it can show partial and full quantum revivals at times related by the different pairwise beat periods. This calculator focuses on the simplest, fully solvable two-state case.

Not in this idealized picture. Because both components are exact energy eigenstates of an isolated system, the oscillation continues forever with the same amplitude, this is unitary, reversible quantum evolution. In real physical systems, interactions with an environment (decoherence) typically damp such oscillations over time, but that additional physics is beyond this closed two-state calculation.