The time-dependent equation in Dirac notation

Newton's second law is second order: launching a projectile takes a position and a velocity. The classical wave equation is second order too — a string needs its shape and its velocity profile. The Schrödinger equation is first order: d|Ψ⟩/dt is already determined by |Ψ⟩ itself, so a single ket at a single instant fixes the entire history, forward and backward. There is no independent 'velocity of the state' to prescribe and no freedom left once |Ψ(0)⟩ is given. That makes quantum evolution between measurements perfectly deterministic — given the ket at noon, the equation names the ket at any other time. Reversibility is built in as well: for a real Hamiltonian such as p̂²/2m + V(x̂), complex-conjugate the position-space equation and send t → −t and you hold an equally valid solution, which is why any irreversibility in physics must be smuggled in from outside this law.

Delete the i and watch what breaks: dΨ/dt = −(1/ħ)ĤΨ is a diffusion-type equation, and expanding in energy eigenvectors makes each component scale as the real exponential e^{(−Eₙt/ħ)}. The norm is no longer conserved and, after a long time, only the lowest-energy component survives, because every excited fraction dies relative to it. Restore the i and the same expansion gives e^{(−iEₙt/ħ)}: each component keeps its length and only turns its phase, at angular rate Eₙ/ħ. Rotation instead of decay is the entire difference between quantum evolution and heat flow, and it hangs on one factor. Numerical physicists exploit the broken version on purpose: propagate in 'imaginary time' t → −iτ, renormalising as you go, and any starting guess with a nonzero ground-state overlap relaxes onto the ground state — a standard SciPy exercise, and a reminder of exactly what the i was doing in the equation you meant to solve.

Differentiate ⟨Ψ|Ψ⟩ and use the equation twice. The ket side gives d|Ψ⟩/dt = −(i/ħ)Ĥ|Ψ⟩; daggering the whole equation gives d⟨Ψ|/dt = +(i/ħ)⟨Ψ|Ĥ†. Adding the two contributions, d⟨Ψ|Ψ⟩/dt = (i/ħ)⟨Ψ|(Ĥ† − Ĥ)|Ψ⟩, which vanishes exactly when Ĥ† = Ĥ. A self-adjoint Hamiltonian therefore conserves total probability with no side condition: normalise once, normalised forever. Both hypotheses earn their keep. In the position representation the dagger move is an integration by parts, so 'self-adjoint' includes the boundary terms — a wavefunction that fails to fall off fast enough can leak norm through the edge of the calculation. And the failure mode is a tool: add a complex absorbing potential −iW(x) near the grid boundary and the norm decays deliberately, swallowing outgoing flux before a periodic box wraps it back around.

Dirac notation earns its keep the moment you need numbers. Apply ⟨k| to the equation and insert Σⱼ|j⟩⟨j| = 𝟙: iħ ċₖ = Σⱼ Hₖⱼ cⱼ, with cₖ = ⟨k|Ψ⟩ and Hₖⱼ = ⟨k|Ĥ|j⟩. One abstract law has become N coupled first-order ODEs — precisely what scipy.integrate.solveᵢᵥₚ eats. The smallest honest case is ammonia's two inversion states, nitrogen above or below the H₃ plane, with Ĥ = E₀𝟙 − Δσ̂ₓ: iħċ₁ = E₀c₁ − Δc₂ and iħċ₂ = −Δc₁ + E₀c₂. Starting in |1⟩, the populations trade as P₂ = sin²(Δt/ħ), and with Δ = 4.9 × 10⁻⁵ eV that oscillation runs at 2Δ/h ≈ 24 GHz — the microwave line the ammonia maser rides. The same two-line reduction, with the basis grown from 2 kets to a spatial grid, is how every wave-packet movie in this course is computed.

The equation contains |Ψ⟩ to the first power and nothing else, so if |Ψ₁(t)⟩ and |Ψ₂(t)⟩ both solve it, α|Ψ₁(t)⟩ + β|Ψ₂(t)⟩ solves it with the same constants α and β at every time. Superposition is not an initial-condition trick that the dynamics erodes; it is preserved exactly, which is why interference survives evolution and a two-slit pattern can be computed one path at a time and summed at the screen. Linearity together with unitarity also forbids state-space chaos: unitarity keeps the inner product of two nearby kets constant, so no trajectory in Hilbert space peels away from a neighbour exponentially — quantum 'chaos' has to hide in the spectrum of Ĥ instead. A practical corollary: adding E₀𝟙 to Ĥ multiplies every solution by the one global phase e^{(−iE₀t/ħ)}, so only energy differences ever reach an experiment, and the zero of potential energy is yours to place.

Every symbol carries an assumption. The Ĥ of the previous topic is p̂²/2m + V(x̂): non-relativistic, since the equation is first order in time but second order in space, and no Lorentz transformation preserves that mismatch — an electron near c needs the Dirac equation. The state space is L²(ℝ), so spin is absent until you tensor on ℂ² and add the Pauli term. The system must be closed: couple it to an environment you do not track and the ket description itself fails, replaced by a density matrix whose master equation, of Lindblad form in the Markovian limit, carries precisely the non-unitary terms this law forbids. And nothing in this smooth, deterministic flow ever selects a measurement outcome — the Born rule and projection of Unit 3 are separate postulates. Whenever a problem seems to need the state to 'jump', the jump is being imported from that other postulate, not from anything iħ d|Ψ⟩/dt = Ĥ|Ψ⟩ can do.