Time-Dependent Schrodinger Equation

Quantum mechanics has exactly one law of motion. Not one per system, not one per situation — one: $i\hbar\,\partial_t|\psi\rangle = \hat H|\psi\rangle$ . Every laser pulse steering a molecule, every qubit gate, every neutron interfering with itself obeys this single line. You met it in wave-mechanics clothes back in the foundations chapter — the Time-Dependent Schrödinger Equation of QM0 built it from de Broglie waves and drove a wavefunction $\Psi(x,t)$ through space. This article re-states the law where it actually lives: in Hilbert space, as a postulate about the state vector itself, with no mention of position until you choose a basis. The upgrade is not cosmetic. Read abstractly, the equation says the Hamiltonian generates the motion of the state — a rotation of a unit vector, steered by a Hermitian operator. Three structural theorems fall out in lines rather than pages: the norm never changes (probability is conserved by construction), the equation is linear (superpositions evolve component by component, coherences intact), and adding a constant to every energy changes nothing observable (energy has a gauge; only differences count). Each is proved here in the dagger notation, and each was earned the long way — by honest integrals — in the foundations chapter, so you will feel exactly what the formalism buys. What this article does not do is solve the equation: The Time-Evolution Operator integrates it into $\hat U(t)$ , Stationary States and Energy Eigenstates finds its fixed points, and Spectral Decomposition of Time Evolution turns solving into bookkeeping. Here we own the law itself — its anatomy, its guarantees, and the discipline of reading it representation-free.

Key takeaways

The dynamical postulate: $i\hbar\,\partial_t|\psi(t)\rangle = \hat H|\psi(t)\rangle$ . The state vector rotates in Hilbert space; the Hamiltonian is the generator of the rotation.
First order in time: the state now fixes the state forever (and backward). Quantum evolution between measurements is fully deterministic — no initial velocity, no second condition.
Hermiticity of $\hat H$ makes the evolution norm-preserving: $\tfrac{d}{dt}\langle\psi|\psi\rangle = 0$ in two lines. Probability conservation is built into the law, not added to it.
The equation is linear: if $|\psi_1(t)\rangle$ and $|\psi_2(t)\rangle$ are solutions, so is $\alpha|\psi_1\rangle + \beta|\psi_2\rangle$ , with the same $\alpha,\beta$ forever. Interference is a permanent feature, not an initial condition.
Energy is a gauge: replacing $\hat H \to \hat H + c\hat 1$ multiplies every state by the same invisible phase $e^{-ict/\hbar}$ . Only energy differences are physical.
The familiar PDE $i\hbar\,\partial_t\psi(x,t) = [-\tfrac{\hbar^2}{2m}\partial_x^2 + V(x)]\psi$ is this postulate written in the position basis — one representation among many.
Key idea: one first-order, linear, Hermitian-generated equation moves every quantum state; determinism, unitarity, and the irrelevance of absolute energy are theorems that fall out of its shape.

The core insight

$|\psi(t)\rangle$ is a vector in Hilbert space, and the TDSE says the Hamiltonian generates its rotation: $\tfrac{d}{dt}|\psi\rangle = -\tfrac{i}{\hbar}\hat H|\psi\rangle$ . Hermitian $\hat H$ ⇒ unitary, probability-conserving, deterministic evolution.

Where this lives today

Every quantum technology is an exercise in steering equation (1). A superconducting-qubit gate is a pulse $\hat H(t)$ engineered so that the induced rotation of $|\psi\rangle$ is exactly a chosen unitary — gate fidelity is measured against the TDSE's prediction, and calibration is the art of learning your actual $\hat H(t)$ . Quantum chemistry codes integrate the equation directly for molecules in laser fields, and attosecond spectroscopy now films the resulting charge sloshing on its natural 10–100 as timescale — cross-term beats of the kind Example 1 computed, in real molecules. The equation is also numerical physics' workhorse test bed: split-operator, Crank–Nicolson, and Krylov integrators are judged by how well they respect this article's theorems — an integrator that leaks norm is lying about probability, and modern geometric schemes conserve it to machine precision, a numerical homage to the dagger derivation.

The physical picture

Forget space for a moment. The state is a unit vector $|\psi\rangle$ in Hilbert space — the arrow installed by Hilbert Space and State Vectors. Dynamics says this arrow moves, and the TDSE says how: at every instant the Hamiltonian reads the current arrow and dictates its velocity, $d|\psi\rangle/dt = -\tfrac{i}{\hbar}\hat H|\psi\rangle$ . The factor $-i/\hbar$ does real work here: because the velocity is $i$ times a Hermitian action, the motion is a pure rotation — the arrow turns, and its length, the total probability, never changes. A classical phase-space point flows; a quantum state vector precesses.

Figure

One law, two faces — the abstract rotation and its position-basis shadow

The wave-mechanics equation you already own is the right panel: project the rotating arrow onto position eigenkets and the components $\psi(x,t) = \langle x|\psi(t)\rangle$ inherit a PDE. The physics is in the left panel — everything the PDE does, spreading packets and beating densities alike, is the shadow of one rigid rotation. The second thing to see is what "first order in time" means. Newton's law is second order: launching a trajectory takes position and velocity. The TDSE asks for the state at one instant, nothing else — the equation computes the velocity from the state. One photograph fixes the whole film, forward and backward. That determinism coexists peacefully with quantum randomness because the state is not an outcome: outcomes come from measurement, a different postulate entirely (Measurement and Wavefunction Collapse owns that story).

Figure

Newton needs two dials — the TDSE needs one photograph

The law in ket language

Here is the postulate, stated where it lives. For a closed system with Hamiltonian operator $\hat H$ — the observable of total energy, Hermitian per Hermitian Operators and Observables — the state ket obeys

i\hbar\,\frac{\partial}{\partial t}\,|\psi(t)\rangle \;=\; \hat H\,|\psi(t)\rangle \qquad (1)

where: $|\psi(t)\rangle$ — the state vector at time $t$ , a unit ket in Hilbert space; $\hat H$ — the Hamiltonian, possibly itself time-dependent; $\hbar = 1.055\times10^{-34}\ \mathrm{J\,s}$ — the conversion between energy and angular frequency. Read the units: $\hat H|\psi\rangle$ has energy times state; dividing by $i\hbar$ makes a rate. An energy $E$ corresponds to a phase angular velocity $E/\hbar$ — energy is rotation rate, which is why $\hbar$ appears exactly once. Equation (1) is a postulate — entry four in the axiom list of The Postulates of Quantum Mechanics — not a theorem. The de Broglie-wave construction of the QM0 Time-Dependent Schrödinger Equation motivates it; nothing derives it. Its confirmation is a century of spectroscopy. To recover wave mechanics, take components in the position basis. The bridge machinery — how $\langle x|\hat H|\psi\rangle$ becomes a differential operator acting on $\psi(x,t)$ — is owned by Position Representation; the result is the familiar

i\hbar\,\frac{\partial}{\partial t}\,\psi(x,t) \;=\; \left[-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x,t)\right]\psi(x,t) \qquad (2)

where: $\psi(x,t) = \langle x|\psi(t)\rangle$ — the position-basis components of the rotating ket. Equation (2) is not a second law; it is equation (1) wearing one particular basis. The momentum-basis version of the same law is a first-order equation for $\varphi(p,t)$ (Momentum Representation does that translation). Same arrow, different shadows. One more reading of (1) before the theorems: being first order and linear, it guarantees a map — the time-evolution operator $\hat U(t,t_0)$ — carrying initial states to final states. Constructing $\hat U$ , proving it unitary, and solving it is the business of The Time-Evolution Operator, the next article. Here we extract what follows from the equation's shape alone.

Theorem one: the norm never moves

The foundations chapter proved probability conservation the honest way: Conservation of Normalization differentiated $\int|\Psi|^2\,dx$ under the integral and watched boundary terms die — hypotheses you could touch. The formalism buys the same theorem in two lines, and the two lines expose exactly which property of $\hat H$ carries the load.

Derivation ANorm conservation in two lines — the dagger route

We prove $\tfrac{d}{dt}\langle\psi(t)|\psi(t)\rangle = 0$ for any state obeying equation (1) with Hermitian $\hat H$ — the abstract sequel to the integral proof of Conservation of Normalization, which this argument neither repeats nor replaces (the integral route showed where probability flows locally; this route shows why the total is rigid). The strategy: differentiate the bracket with the product rule, feed in the TDSE and its dagger, and watch the two terms cancel. The single point of failure to watch: the step $\hat H^\dagger = \hat H$ . Everything else is bookkeeping.

Step 1. Dagger the equation of motion. Taking the adjoint of (1) — the dual-correspondence rules of Hermitian Conjugate and Adjoint Operators — gives the bra equation $-i\hbar\,\partial_t\langle\psi| = \langle\psi|\hat H^\dagger$ . Why the sign flips: the adjoint conjugates scalars, and $(i\hbar)^* = -i\hbar$ . The bra rotates oppositely to the ket — it must, if their pairing is to stand still.

Step 2. Differentiate the norm with the product rule:

\frac{d}{dt}\langle\psi|\psi\rangle = \left(\partial_t\langle\psi|\right)|\psi\rangle + \langle\psi|\left(\partial_t|\psi\rangle\right) = -\frac{1}{i\hbar}\langle\psi|\hat H^\dagger|\psi\rangle + \frac{1}{i\hbar}\langle\psi|\hat H|\psi\rangle

Step 3. Impose Hermiticity. With $\hat H^\dagger = \hat H$ the two terms are equal and opposite:

\frac{d}{dt}\langle\psi|\psi\rangle = \frac{1}{i\hbar}\left[\langle\psi|\hat H|\psi\rangle - \langle\psi|\hat H|\psi\rangle\right] = 0

Why this is the whole story: the cancellation used nothing about $\hat H$ except its Hermiticity — not time-independence, not any particular potential. Even a driven system with $\hat H(t)$ conserves the norm at every instant, provided $\hat H(t)$ stays Hermitian.

Step 4. Read the converse as a design rule. Give the Hamiltonian an anti-Hermitian part, $\hat H = \hat H_0 - i\Gamma/2$ , and Step 3 leaves a remainder: the norm decays as $e^{-\Gamma t/\hbar}$ . That is not a bug but a modelling tool — the optical potential of Conservation of Normalization, which owns the lifetime dictionary $\tau = \hbar/\Gamma$ . Verified: norm constant to $10^{-13}$ over a time grid for a random 6-level Hermitian $\hat H$ ; with $\hat H - 0.1i\hat 1$ the norm decays to $e^{-0.2} = 0.819$ exactly as predicted; the same script confirms $|\langle\phi(t)|\psi(t)\rangle|$ is constant — inner products, not just norms, are preserved.

The last verified line deserves its sentence: the same two-line argument applied to two different solutions shows $\tfrac{d}{dt}\langle\phi(t)|\psi(t)\rangle = 0$ . Overlaps between states are rigid. Two states that start distinguishable stay exactly as distinguishable forever — unitary evolution neither creates nor destroys information. The operator-level statement of this fact, $\hat U^\dagger\hat U = \hat 1$ with both directions of the equivalence, is the norm-preservation theorem of Unitary Operators and Transformations; the dynamical version, with $\hat U$ built explicitly from $\hat H$ , belongs to The Time-Evolution Operator.

Theorem two: energy has a gauge

Where is the zero of energy? Chemistry measures from dissociation, solid-state from the Fermi level, particle physics from the vacuum. If absolute energy mattered, these communities would disagree about dynamics. They do not, and equation (1) says why.

Go deeper · Derivation B · 1 minShifting every energy shifts nothing observable

Derivation BShifting every energy shifts nothing observable

We prove that replacing $\hat H \to \hat H + c\,\hat 1$ (a constant $c$ added to every energy — a relabeling of the zero) changes every solution by the same global phase, and therefore changes no prediction. The strategy: guess the repaired solution, substitute, and confirm; then audit observability through the Born rule. Watch how the identity operator's blandness — it commutes with everything — is what makes the repair global.

Step 1. Let $|\psi(t)\rangle$ solve (1) with $\hat H$ . Define the candidate $|\tilde\psi(t)\rangle = e^{-ict/\hbar}|\psi(t)\rangle$ and substitute into the shifted equation:

i\hbar\,\partial_t|\tilde\psi\rangle = i\hbar\left(-\tfrac{ic}{\hbar}\right)e^{-ict/\hbar}|\psi\rangle + e^{-ict/\hbar}\,i\hbar\,\partial_t|\psi\rangle = e^{-ict/\hbar}\left[c + \hat H\right]|\psi\rangle = (\hat H + c\hat 1)\,|\tilde\psi\rangle

Why the phase commutes through: $e^{-ict/\hbar}$ is a number, and $\hat H$ is linear — the scalar slides past the operator untouched. The shifted Hamiltonian's solutions are the old solutions wearing a uniformly rotating phase.

Step 2. Audit observability. Any Born probability compares the state against a fixed eigenket: $P = |\langle a|\tilde\psi(t)\rangle|^2 = |e^{-ict/\hbar}|^2\,|\langle a|\psi(t)\rangle|^2 = |\langle a|\psi(t)\rangle|^2$ — untouched, since $|e^{i\theta}| = 1$ . Expectation values sandwich the state on both sides, so the phase meets its conjugate and cancels. Nothing measurable notices $c$ . Why "global" is the operative word: the phase multiplies the entire state, and physical states are rays — the ray card of Hilbert Space and State Vectors. A relative phase between components of a superposition is a different animal entirely and is very much observable.

Step 3. Extract the corollary that runs all of spectroscopy: in any superposition, component $n$ acquires phase $e^{-i(E_n + c)t/\hbar}$ , and every cross term — every beat, every observable oscillation — depends only on $(E_m + c) - (E_n + c) = E_m - E_n$ . Bohr frequencies survive the gauge; absolute levels do not. The systematic development of this cross-term machinery is Spectral Decomposition of Time Evolution. Verified: numerically, $\hat U_{H+c}(t) = e^{-ict/\hbar}\,\hat U_H(t)$ to $10^{-13}$ on a random 6-level system, and every $|amplitude|^2$ agrees to $10^{-14}$ ; symbolically, $e^{-i(E+c)t/\hbar} = e^{-ict/\hbar}e^{-iEt/\hbar}$ exactly.

Theorem three: superpositions evolve as superpositions

The third structural fact is so easy to prove that its consequences are routinely underrated.

Go deeper · Derivation C · 1 minLinearity — the solution set is a vector space

Derivation CLinearity — the solution set is a vector space

We show that if $|\psi_1(t)\rangle$ and $|\psi_2(t)\rangle$ each solve equation (1), then so does $|\chi(t)\rangle = \alpha|\psi_1(t)\rangle + \beta|\psi_2(t)\rangle$ for any fixed complex $\alpha, \beta$ — and that the coefficients never drift. The point of the derivation is less the algebra than the reading: quantum interference is not a delicate initial condition that dynamics erodes; it is preserved by the law itself. Watch that $\alpha$ and $\beta$ enter nowhere in the equation — only in the state.

Step 1. Substitute and use linearity of both $\partial_t$ and $\hat H$ :

i\hbar\,\partial_t|\chi\rangle = \alpha\,i\hbar\,\partial_t|\psi_1\rangle + \beta\,i\hbar\,\partial_t|\psi_2\rangle = \alpha\hat H|\psi_1\rangle + \beta\hat H|\psi_2\rangle = \hat H|\chi\rangle

Why there is content here: the same one-line computation fails for any nonlinear equation — add two solutions of Navier–Stokes and you do not get a solution. The quantum law is exactly linear, as far as every experiment has pushed.

Step 2. Note what stays fixed. The coefficients $\alpha, \beta$ are the same at every time: evolution moves the components, never the recipe. The relative phase and magnitude of the branches — the data interference patterns are made of — ride along undamaged, the dynamical half of why the two-slit pattern of Double-Slit Experiment survives the trip to the screen.

Step 3. Mark the theorem's boundary honestly. Linearity holds for the closed-system law (1). Measurement — the update of Measurement and Wavefunction Collapse — is not of this form: no linear deterministic map sends every superposition onto a single eigenstate. That tension is the measurement problem, and it lives in the entanglement chapter, not in this equation. Verified: $\hat U(\alpha\psi_1 + \beta\psi_2) = \alpha\,\hat U\psi_1 + \beta\,\hat U\psi_2$ to $10^{-13}$ numerically for random complex coefficients.

Limiting cases

case	what equation (1) becomes	meaning
$\hat H$ time-independent	$\lvert\psi(t)\rangle = e^{-i\hat H(t-t_0)/\hbar}\lvert\psi(t_0)\rangle$	one exponential solves everything; built and proven in The Time-Evolution Operator
eigenstate input $\hat H\lvert E\rangle = E\lvert E\rangle$	$\lvert\psi(t)\rangle = e^{-iEt/\hbar}\lvert E\rangle$	phase-only motion — the fixed points, owned by Stationary States and Energy Eigenstates
$\hat H(t)$ commuting at different times	$\lvert\psi(t)\rangle = e^{-\frac{i}{\hbar}\int_{t_0}^{t}\hat H(t')dt'}\lvert\psi(t_0)\rangle$	the exponent integrates but never needs reordering
$\hat H(t)$ non-commuting, $[\hat H(t_1),\hat H(t_2)]\neq 0$	$\hat U = \mathcal{T}\exp[-\frac{i}{\hbar}\int \hat H\,dt']$	naive exponentiation fails; the repair is Time-Ordered Exponential and the Dyson Series
$\hbar \to 0$ against classical actions	phases spin infinitely fast	stationary-phase dominance — the semiclassical bridge, teased at Bohr–Sommerfeld and Semiclassical Quantization in 1D
anti-Hermitian part $-i\Gamma/2$	norm decays $e^{-\Gamma t/\hbar}$	open-system modelling; the optical potential of Conservation of Normalization

Common trapThe exponential solution always works — just integrate the Hamiltonian

Two errors hide in that sentence. First: $|\psi(t)\rangle = e^{-i\hat Ht/\hbar}|\psi(0)\rangle$ presumes $\hat H$ time-independent. If $\hat H(t)$ varies and fails to commute with itself at different times — a driven qubit, a pulsed laser — the innocent $\exp[-\tfrac{i}{\hbar}\int\hat H\,dt']$ is simply wrong: slices of the evolution refuse to be reordered inside one exponent. The honest object is the time-ordered exponential of Time-Ordered Exponential and the Dyson Series. Second: conflating the abstract equation with its position-space PDE. Equation (2) is one representation of (1); a spin-½ has a perfectly good TDSE — a $2\times2$ matrix equation — and no wavefunction of $x$ at all. Solve the equation where the system lives.

Worked examples

1 · Auditing rigidity on a two-level superposition (confidence builder)

A system has energy eigenkets $|1\rangle, |2\rangle$ with $E_1 = 1.0\ \mathrm{eV}$ , $E_2 = 2.5\ \mathrm{eV}$ . It starts in $|\psi(0)\rangle = 0.6|1\rangle + 0.8|2\rangle$ . Write $|\psi(t)\rangle$ , and verify by hand that the norm and both level probabilities are constant, while the relative phase evolves.

Step 1. Each eigencomponent rotates at its own rate (the fixed-point solution from the limiting-cases table): $|\psi(t)\rangle = 0.6\,e^{-iE_1t/\hbar}|1\rangle + 0.8\,e^{-iE_2t/\hbar}|2\rangle$ .

Step 2. Norm: $\langle\psi|\psi\rangle = |0.6\,e^{-iE_1t/\hbar}|^2 + |0.8\,e^{-iE_2t/\hbar}|^2 = 0.36 + 0.64 = \boxed{1}$ — each clock has unit modulus, so time cancels term by term, exactly as the dagger derivation promised.

Step 3. Level probabilities: $P(E_1) = |\langle 1|\psi(t)\rangle|^2 = \boxed{0.36}$ , $P(E_2) = \boxed{0.64}$ , at every $t$ . Evolution never transfers weight between energy levels — energy statistics are rigid (the general theorem lives in Stationary States and Energy Eigenstates).

Step 4. What does move: the relative phase, winding at $\omega_{21} = \dfrac{E_2 - E_1}{\hbar} = \dfrac{1.5\ \mathrm{eV}}{\hbar} = 2.28\times10^{15}\ \mathrm{rad/s}$ . Any observable connecting the two levels oscillates at exactly this Bohr frequency — the cross-term machinery of Spectral Decomposition of Time Evolution.

2 · The double-well qubit clock (exam level)

A particle in a symmetric double well tunnels between the left and right wells. The two lowest eigenstates are the symmetric and antisymmetric combinations, split by $E_2 - E_1 = 2\Delta = 4.0\ \mathrm{\mu eV}$ . The particle starts fully in the left well, $|L\rangle = \tfrac{1}{\sqrt2}(|+\rangle + |-\rangle)$ . Find the period of the left-right oscillation, the probability of finding it on the left at $t = 0.10\ \mathrm{ns}$ , and the first instant of even odds.

Step 1. Attach the clocks. With $\hat H|\pm\rangle = E_\pm|\pm\rangle$ : $|\psi(t)\rangle = \tfrac{1}{\sqrt2}\left(e^{-iE_+t/\hbar}|+\rangle + e^{-iE_-t/\hbar}|-\rangle\right)$ , and the left-well probability is $P_L(t) = |\langle L|\psi(t)\rangle|^2 = \cos^2\!\left(\dfrac{\Delta t}{\hbar}\right)$ with $2\Delta = E_- - E_+$ the splitting.

Step 2. Period of $P_L$ : the cosine-squared repeats when $\Delta t/\hbar = \pi$ , i.e. $T = \dfrac{\pi\hbar}{\Delta} = \dfrac{h}{2\Delta} = \dfrac{4.136\times10^{-15}\ \mathrm{eV\,s}}{4.0\times10^{-6}\ \mathrm{eV}} = \boxed{1.03\ \mathrm{ns}}$ . A four-micro-electron-volt splitting — a hundred-millionth of a chemical bond — is a nanosecond metronome.

Step 3. At $t = 0.10\ \mathrm{ns}$ : $P_L = \cos^2\!\left(\pi\cdot\dfrac{0.10}{1.034}\right) = \cos^2(0.304\ \mathrm{rad}) = \boxed{0.91}$ . Early on, the particle has barely begun to leak — the cosine is flat at its crest.

Step 4. Even odds when $\cos^2 = \tfrac12$ : first at $t = T/4 = \boxed{0.26\ \mathrm{ns}}$ . There the state is an equal coherent mixture of left and right — not a coin flip but a superposition, and the distinction is measurable through interference (the doctrine of Density Operator and Mixed States).

Step 5. Register check: the Bohr frequency $\omega_{21} = 2\Delta/\hbar = 6.1\times10^{9}\ \mathrm{rad/s}$ is microwave-band tunnelling — the same physics as ammonia's maser doublet (its own $\Delta = 49.4\ \mathrm{\mu eV}$ , worked at Time-Independent Schrödinger Equation, runs about twenty-five times faster).

Origins · 1 minVladimir Fock and how this was discovered

Vladimir Fock

1898–1974 · Leningrad

Fock took the time-dependent equation seriously as an equation — while others solved for spectra, he asked what its solutions do when the Hamiltonian itself moves. With Max Born in 1928 he proved the adiabatic theorem: drag $\hat H(t)$ slowly and a system in the $n$ -th eigenstate stays there, accumulating precisely the phases the equation dictates — the result behind every slow-tuning protocol from molecular physics to quantum annealing, and the soil from which Berry's geometric phase later grew. He worked out the TDSE's behavior under gauge transformations (why $\hat H \to \hat H + c$ and its electromagnetic cousins are invisible); his name is also on Hartree–Fock and on Fock space, the many-particle arena built downstream of this chapter. In Stalin's Leningrad he defended the equation against ideological attack — arrested twice, released twice, unbowed — insisting quantum mechanics needed no philosophical repair, only careful reading.

History

1926

— Schrödinger publishes the time-dependent wave equation, capping the four-paper annus mirabilis

1927

— Ehrenfest connects the new equation's averages to Newton's law

1928

— Born and Fock prove the adiabatic theorem for slowly driven Hamiltonians

1932

— von Neumann casts dynamics as a one-parameter unitary group on Hilbert space

1948

— Feynman reformulates the same law as a sum over histories

1984

— Berry isolates the geometric phase hiding in adiabatic solutions

Going further

Builds on — Hilbert Space and State Vectors · Hermitian Operators and Observables · Born Rule and Measurement Probabilities · Time-Dependent Schrödinger Equation (the QM0 wave-mechanics twin: its de Broglie construction, plane-wave check, and first-order-determinism card are recalled here, never re-derived) · Conservation of Normalization (the integral-route proof this article's dagger route abstracts).

Leads to — The Time-Evolution Operator (integrating the law into $\hat U$ ) · Stationary States and Energy Eigenstates (its fixed points) · Spectral Decomposition of Time Evolution (the general solution as bookkeeping) · Time-Ordered Exponential and the Dyson Series (driven, non-commuting $\hat H(t)$ ) · The Schrodinger Picture and The Heisenberg Picture (who carries the time) · Ehrenfest's Theorem in the dynamics register (the commutator engine for $\tfrac{d}{dt}\langle\hat A\rangle$ , QM3's own version).

References — Griffiths, Introduction to Quantum Mechanics, 3rd ed., Ch. 1–2 · Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., Ch. 2 · Shankar, Principles of Quantum Mechanics, 2nd ed., Ch. 4 · Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, Vol. I, Ch. III · Townsend, A Modern Approach to Quantum Mechanics, 2nd ed., Ch. 4.

Provenance — Equations are standard and cross-checked against Griffiths 3rd ed. Ch. 1–2, Sakurai & Napolitano 2nd ed. Ch. 2, Shankar 2nd ed. Ch. 4, Cohen-Tannoudji Vol. I Ch. III, and Townsend 2nd ed. Ch. 4. A verifier should confirm: (1) the Hermiticity-implies-unitarity derivation; (2) the separation-of-variables reduction to the time-independent Schrödinger equation; (3) the Ehrenfest derivation including $[H,x]=-i\hbar p/m$ and $[H,p]=i\hbar V'(x)$ from $[x,p]=i\hbar$ ; (4) the numerical value $\hbar=1.054571817\times10^{-34}\,\mathrm{J\,s}$ (CODATA). No nonstandard numbers are asserted. (Lane note: items 2 and 3 are owned upstream — separation by Time-Independent Schrödinger Equation and Stationary States and Separation of Variables, the Ehrenfest commutators by the dynamics-chapter Ehrenfest's Theorem — cited here rather than reproduced.)

Is the time-dependent Schrödinger equation derived from anything deeper?

No. It is a postulate — the dynamics axiom of the theory. The de Broglie-wave construction of the QM0 twin article is a motivation, not a derivation, and it says so honestly. What can be derived are its structural consequences (unitarity, linearity, the energy gauge) and, in the other direction, constraints that make it look inevitable: demand a first-order, deterministic, probability-conserving evolution and Stone's theorem essentially forces the form $i\hbar\,\partial_t|\psi\rangle = \hat H|\psi\rangle$ with Hermitian $\hat H$ — that argument lives in The Time-Evolution Operator.

If the equation is deterministic, where does quantum randomness come from?

From measurement, which is a different postulate. Between measurements the state vector evolves as rigidly as a planet: $|\psi(0)\rangle$ fixes $|\psi(t)\rangle$ exactly. Randomness enters only when the Born rule converts the state into outcome probabilities, and the update after an outcome — collapse — is not generated by equation (1) at all (Measurement and Wavefunction Collapse owns the mechanics, and the linearity derivation here shows why no Hermitian generator could produce it).

Why must the time derivative come with an i? What breaks without it?

Drop the $i$ and the equation becomes $\hbar\,\partial_t|\psi\rangle = -\hat H|\psi\rangle$ — a diffusion equation whose solutions decay toward the lowest eigenstate instead of oscillating: norms die, interference washes out. The $i$ converts the Hermitian generator into a rotation — phases wind, moduli hold. (Numericists exploit the broken version deliberately: imaginary-time evolution relaxes trial states onto ground states precisely because it violates unitarity.)

Does norm conservation still hold when the Hamiltonian depends on time?

Yes — the dagger proof never used time-independence, only Hermiticity, and it holds instant by instant. What time-dependence breaks is the solution formula: $e^{-i\hat Ht/\hbar}$ presumes one fixed $\hat H$ . A driven system still evolves unitarily, but its unitary is the time-ordered exponential (Time-Ordered Exponential and the Dyson Series), and energy is no longer conserved because the drive can pump it in and out.

What exactly is the relation between the ket equation and the wavefunction PDE I learned first?

Strictly: take the inner product of equation (1) with the position bra $\langle x|$ , and let Position Representation translate $\langle x|\hat H|\psi\rangle$ into $[-\tfrac{\hbar^2}{2m}\partial_x^2 + V(x)]\psi(x,t)$ . The PDE is the component form of the ket equation in one particular basis — no more fundamental than the momentum-space version or the two-component matrix equation of a spin. Systems without a position degree of freedom (spin, photon polarization) have a TDSE and no Schrödinger PDE.

Can I always split a hard Hamiltonian as H₀ plus a small piece and treat the small piece separately?

That instinct is exactly where the subject goes next, but the naive split fails when the pieces do not commute: $e^{-i(\hat H_0 + \hat V)t/\hbar} \neq e^{-i\hat H_0t/\hbar}e^{-i\hat Vt/\hbar}$ in general. Making the split work honestly is the content of the interaction picture and time-dependent perturbation theory, built from The Interaction (Dirac) Picture onward — with the Dyson series as the engine.

Is there a version of this equation for mixed states or open systems?

For a closed system in ensemble packaging, equation (1) induces the von Neumann equation $i\hbar\,\partial_t\hat\rho = [\hat H, \hat\rho]$ (Density Operator and Mixed States introduces $\hat\rho$ ). Genuinely open systems — coupled to an untracked environment — need Lindblad generators that add controlled dissipation; the optical potential of the norm derivation's Step 4 is that family's crudest member.