Conservation of Normalization

Three articles have been writing cheques against this one. Probability Density and Normalization told you to normalize once and trust the dynamics to keep the books; Born Statistical Interpretation noted that its entire reading of $|\Psi|^2$ collapses unless total probability stays exactly 1; Time-Dependent Schrödinger Equation promised the proof was two lines. Here the cheques are cashed. The claim is sharp and slightly miraculous: take any normalized state, let it evolve under the Schrödinger equation — spread, slosh, tunnel, interfere — and the integral $\int|\Psi|^2 dx$ remains 1 to infinitely many decimal places, forever, without anyone tuning anything. The proof is genuinely short, and every ingredient earns its keep. The time derivative of the total probability collapses, via the equation of motion, into a pure boundary term — the probability current of the previous article, evaluated at the ends of the universe — and normalizable states have nothing flowing there. Two hypotheses carry the whole weight: the potential must be real, and the state must vanish at infinity. Each is a door. Push on the first and you get complex "optical" potentials, the standard model of absorption and decay, where the norm dies exponentially on purpose and its half-life is physics you can measure. Push on the second and you rediscover why plane waves need special handling. Conservation of normalization is also your first sight of the deepest structural fact about quantum dynamics — evolution is unitary, a rigid rotation of the state — dressed here in its most concrete costume.

Key takeaways

The claim: $\dfrac{d}{dt}\displaystyle\int_{-\infty}^{\infty}|\Psi|^2\,dx = 0$ for every solution of the Schrödinger equation with real $V$ — normalize once, normalized forever.
Route 1 (direct): differentiate under the integral, use the TDSE twice; the integrand becomes $-\partial_x j$ and the integral collapses to boundary values of the current.
Route 2 (borrowed): integrate the continuity equation of Probability Current and the Continuity Equation over all space — one line.
Both hypotheses are load-bearing: $V$ real (else probability is created/destroyed locally) and $\Psi \to 0$ at $\pm\infty$ (else the boundary term survives).
Complex potentials are a feature: $V - i\Gamma/2$ makes the norm decay as $e^{-\Gamma t/\hbar}$ — the optical-potential model of absorption, with lifetime $\tau = \hbar/\Gamma$ .
The abstract face of the theorem is unitarity: $U(t) = e^{-i\hat Ht/\hbar}$ preserves norms because $\hat H$ is Hermitian — formalized later in the track.
Key idea: the Schrödinger equation with a real potential conserves total probability exactly — the time derivative of the norm is a boundary term, and normalizable states have empty boundaries.

The core insight

Because the Hamiltonian is Hermitian, the time derivative of the total probability reduces to a boundary term that vanishes for normalizable states; equivalently, time evolution $U(t) = e^{-i\hat Ht/\hbar}$ is unitary and preserves the norm.

Where this lives today

Unitarity is the quantum computer's constitution. Every gate in the 2026-era processors — superconducting, trapped-ion, neutral-atom — is a physically implemented $U = e^{-i\hat Ht/\hbar}$ , and "the norm is conserved" is why the machine's probabilities stay meaningful through a thousand-gate circuit. The industry's central enemy is framed in this article's terms: loss — a photon escaping a cavity, an atom tunnelling out of its tweezer — is norm leaking into untracked channels, modelled in every device simulation by exactly the complex-potential (more generally, Lindblad) bookkeeping of equation (3). Error-correction overheads are budgeted against measured $\tau = \hbar/\Gamma$ -style lifetimes: $T_1$ times on superconducting qubits, now pushing past milliseconds. The optical potential itself remains standard equipment: modern neutron-transport codes for reactor design still fit scattering data with complex $V - iW$ , and ab initio chemistry packages ship complex absorbing potentials (CAPs) both to terminate grids cleanly and — used deliberately — to compute resonance lifetimes of metastable anions, reading $\Gamma$ off the non-Hermitian eigenvalue exactly as equation (3) reads it off the norm.

The physical picture

Picture the whole real line as one sealed aquarium and $|\Psi|^2$ as the water. The previous article established the local plumbing law: water level at a point changes only by current flowing past it. Now ask about the total water. Add up the local law over the whole tank and everything interior cancels — every litre that leaves one region enters the next — so the total can change only through the tank's two far walls, at $x = \pm\infty$ . A normalizable state is one whose water (and whose current) dies away before reaching those walls. Nothing arrives at the boundary; nothing leaves; the total is eternal. That is the entire theorem, told as plumbing. What makes it worth an article is how much physics sits in the two hypotheses, and in what happens when each fails.

Figure

Total probability can change only at the ends of the universe

The second picture to install is the failure mode — because it is deliberately engineered every day. Make the potential complex, $V(x) - i\Gamma/2$ , and the plumbing law gains a drain: probability now disappears locally, at rate $\Gamma/\hbar$ per unit of probability present, without travelling anywhere. The tank empties exponentially. Far from being a bug, this is the standard model of a particle that can be absorbed — a neutron captured by a nucleus, a molecule ionized by a detector — where "the particle stopped being described by this wavefunction" is exactly the physics you want the norm to report.

Figure

Real potential: sealed tank · Complex potential: tank with a drain

The theorem, twice

Two proofs, one theorem. The first is self-contained — Griffiths' classic §1.4 computation — and shows exactly where each hypothesis bites. The second borrows the continuity equation wholesale and is over in a breath; its brevity is the payoff of last article's work.

Derivation ARoute 1 — the direct proof, with every hypothesis visible

We compute $\frac{d}{dt}\int|\Psi|^2dx$ head-on: differentiate under the integral, trade every time derivative for space derivatives via the Schrödinger equation, and watch the integrand become a total derivative whose integral is a boundary term. The two assumptions — real potential, normalizable state — each get one moment where the proof would die without them; the goal of walking this route rather than quoting the result is to catch both moments red-handed.

Step 1. Differentiate under the integral. $\dfrac{dN}{dt} = \displaystyle\int_{-\infty}^{\infty}\frac{\partial}{\partial t}|\Psi|^2\,dx = \int_{-\infty}^{\infty}\left(\Psi^*\,\partial_t\Psi + \Psi\,\partial_t\Psi^*\right)dx$ . Why: the integration limits are fixed, so the derivative passes inside; the product rule splits the density's change between the two factors.

Step 2. Substitute the equation of motion — and its honest conjugate. From the TDSE, $\partial_t\Psi = \frac{i\hbar}{2m}\partial_x^2\Psi - \frac{i}{\hbar}V\Psi$ ; conjugating, $\partial_t\Psi^* = -\frac{i\hbar}{2m}\partial_x^2\Psi^* + \frac{i}{\hbar}V^*\Psi^*$ . Why: note the $V^*$ — kept deliberately, because this is hypothesis number one's moment. The potential terms in Step 1 combine to $\frac{i}{\hbar}(V^* - V)|\Psi|^2$ , which vanishes if and only if $V$ is real. A complex potential injects or drains probability at every point where $\mathrm{Im}\,V \neq 0$ .

Step 3. Collapse the kinetic terms into a total derivative. With real $V$ : $\dfrac{dN}{dt} = \dfrac{i\hbar}{2m}\displaystyle\int\left(\Psi^*\partial_x^2\Psi - \Psi\,\partial_x^2\Psi^*\right)dx = \dfrac{i\hbar}{2m}\int \partial_x\!\left(\Psi^*\partial_x\Psi - \Psi\,\partial_x\Psi^*\right)dx$ . Why: the same Wronskian identity that built the current last article — the cross terms $\partial_x\Psi^*\partial_x\Psi$ cancel in the difference. The integrand is now recognizably $-\partial_x j$ times constants.

Step 4. Evaluate the boundary term — hypothesis two's moment.

\frac{dN}{dt} = \frac{i\hbar}{2m}\left[\Psi^*\frac{\partial\Psi}{\partial x} - \Psi\frac{\partial\Psi^*}{\partial x}\right]_{-\infty}^{\infty} = 0 \qquad (1)

Why: a normalizable $\Psi$ must vanish at $\pm\infty$ (faster than $1/\sqrt{|x|}$ , and physical states' derivatives stay bounded), so the bracket dies at both ends. Here is where plane waves escape the theorem: $|e^{ikx}|$ does not vanish anywhere, the bracket survives, and the naive argument says nothing — the loophole flagged since Bound versus Scattering States.

Step 5. State the result. $N(t) = N(0)$ : normalize at $t = 0$ and the constant $A$ chosen in Probability Density and Normalization stays correct at every later time, through any dynamics a real potential can generate. Why: this is the license behind "normalize once, keep forever" — and behind the Born rule's coherence, since probabilities that summed to 1 must keep doing so. *Verified: sympy reproduces Steps 1–4 symbolically, including the $(V^* - V)$ remainder for complex $V$ and the boundary-term form (1); Griffiths & Schroeter §1.4, Eqs. 1.20–1.28.*

Go deeper · Derivation B · 1 minRoute 2 — one line via continuity, and the unitarity reading

Derivation BRoute 2 — one line via continuity, and the unitarity reading

If the local law is already on the shelf, global conservation is an integral away. This card runs the borrowed proof, then translates the theorem into the language it will wear for the rest of the track: time evolution as a norm-preserving — unitary — transformation, the quantum analogue of a rigid rotation. Nothing new is proven in the translation; the value is seeing one fact in its second costume before the Formalism chapter makes that costume standard issue.

Step 1. Integrate the continuity equation over all space. From Probability Current and the Continuity Equation: $\partial_t\rho + \partial_x j = 0$ , so $\dfrac{dN}{dt} = \displaystyle\int\partial_t\rho\,dx = -\int\partial_x j\,dx = j(-\infty) - j(+\infty) = 0$ . Why: the interior plumbing telescopes away; only the walls speak, and for normalizable states the walls are dry. One line, as promised — the entire cost was paid last article.

Step 2. Recast as an inner-product statement. $N = \int\Psi^*\Psi\,dx$ is the squared length of the state, in the inner product the Formalism chapter will formalize on Hilbert space. The theorem says: Schrödinger evolution never changes lengths. Why: transformations that preserve all lengths (and, by the polarization identity, all inner products) are called unitary — the complex-space cousins of rotations.

Step 3. Name the generator. For time-independent $\hat H$ the evolution operator is $U(t) = e^{-i\hat Ht/\hbar}$ , and unitarity $U^\dagger U = \mathbb{1}$ holds precisely because $\hat H$ is Hermitian — the anti-Hermitian exponent $-i\hat Ht/\hbar$ exponentiates to an isometry, exactly as an imaginary exponent $e^{i\theta}$ has unit modulus. Why: Hermiticity of the Hamiltonian, reality of the potential, and conservation of probability are one fact in three vocabularies. The operator-level machinery — adjoints, unitary operators, the exponential map — is built properly in the Formalism chapter (Unitary Operators and Transformations); this article supplies the concrete integral it will abstract. *Verified: Step 1 is the divergence-theorem argument checked symbolically; the $U(t)$ norm-preservation is verified on a two-eigenstate superposition ( $|c_1|^2 + |c_2|^2$ constant) in the article's verify script; Sakurai §2.1.*

Breaking it on purpose: the optical potential

Now open the door that Step 2 guarded. Give the potential a constant negative imaginary part,

V(x) \;\longrightarrow\; V(x) - \frac{i\,\Gamma}{2}, \qquad \Gamma > 0 \qquad (2)

where: $\Gamma$ — a positive constant with units of energy, the absorption width. Rerunning Step 2 of the direct proof, the potential terms no longer cancel; they leave exactly $\frac{i}{\hbar}(V^* - V)|\Psi|^2 = -\frac{\Gamma}{\hbar}|\Psi|^2$ , so

\frac{dN}{dt} = -\frac{\Gamma}{\hbar}\,N \quad\Longrightarrow\quad N(t) = e^{-\Gamma t/\hbar}, \qquad \tau = \frac{\hbar}{\Gamma} \qquad (3)

where: $\tau$ — the lifetime, the time for the survival probability to fall to $1/e$ . Read $N(t)$ as the probability that the particle has not yet been absorbed: the "violation" is bookkeeping for a process — capture, ionization, detection — that removes the particle from the channel this wavefunction describes. Nuclear physicists have modelled neutron absorption this way since the 1950s (the "optical model," because the nucleus acts like a cloudy refracting medium), and the same trick, dressed as a complex absorbing potential, sits at the edge of every modern quantum-chemistry grid to soak up outgoing flux before it reflects off the simulation box. The relation $\tau = \hbar/\Gamma$ is also your first sight of the energy–time reciprocity that governs all decaying states: wide resonances die fast, narrow ones linger — a theme that returns quantitatively with the track's scattering chapter.

Limiting cases

Case	Result	Meaning
$V$ real, $\Psi$ normalizable	$dN/dt = 0$ exactly	the theorem: normalize once, forever
plane wave $e^{ikx}$	boundary term survives; proof silent	non-normalizable idealizations need box normalization or packets
$V - i\Gamma/2$ , constant $\Gamma$	$N = e^{-\Gamma t/\hbar}$ , $\tau = \hbar/\Gamma$	the optical potential: absorption as engineered non-Hermiticity
$\mathrm{Im}\,V = -\Gamma(x)/2$ localized	norm drains only where $\Gamma(x) \neq 0$	complex absorbing boundaries in numerical grids
superposition $\sum c_n\psi_n e^{-iE_nt/\hbar}$	$\sum\lvert c_n\rvert^2 = 1$ at all $t$	unitarity in the energy basis — weights frozen, per Stationary States

Common trap“The wavefunction spreads, so the probability must dilute”

Spreading redistributes; it never dilutes. A free wavepacket that doubles its width halves its typical density — $|\Psi|^2$ at the centre drops — but the integral of $|\Psi|^2$ over all space is untouched: the same unit of probability now covers more ground more thinly, like a fixed shot of ink stirred into more water. Conflating density with total probability produces phantom paradoxes ("the particle is disappearing as it spreads!") and real calculational errors — re-normalizing at each time step, which wrecks relative phases and with them every interference prediction. The norm needs setting exactly once. The legitimate exceptions are the engineered ones: complex potentials drain the norm by design, and there the decay of $N(t)$ — not its constancy — is the measurable physics. Know which regime you are in before you touch that normalization constant.

Worked examples

1 · Auditing conservation on a beating superposition (confidence builder)

Take the equal two-state box superposition from Stationary States and Separation of Variables: $\Psi = \frac{1}{\sqrt2}\left(\psi_1 e^{-iE_1t/\hbar} + \psi_2 e^{-iE_2t/\hbar}\right)$ , with $\psi_{1,2}$ the orthonormal ISW eigenfunctions. Its density visibly sloshes at $\omega_{21}$ . Verify by direct integration that the total norm is nonetheless constant.

Step 1. Write the norm: $N(t) = \frac12\int\psi_1^2\,dx + \frac12\int\psi_2^2\,dx + \mathrm{Re}\!\left[e^{i(E_1 - E_2)t/\hbar}\int\psi_1\psi_2\,dx\right]$ .

Step 2. The first two integrals are 1 each (normalized eigenfunctions), contributing $\frac12 + \frac12 = 1$ .

Step 3. The cross term carries the only time dependence — and it is multiplied by $\int\psi_1\psi_2\,dx = 0$ , by orthogonality of distinct eigenfunctions. So $N(t) = \boxed{1}$ at every instant, exactly.

Step 4. Read the anatomy: the local density genuinely oscillates ( $\rho$ has a $\psi_1\psi_2\cos\omega_{21}t$ term — probability really flows from side to side), but the flow integrates to zero because the spatial pattern of the beat, $\psi_1\psi_2$ , has zero net area. Local motion, global stasis: the continuity picture in one example. This is also unitarity in the energy basis — the weights $\left(\frac12, \frac12\right)$ never budge.

2 · A neutron in a cloudy nucleus (exam level)

In the optical model, a low-energy neutron inside a heavy nucleus feels $V - iW$ with absorption strength $W \approx 3.0\ \mathrm{MeV}$ (so $\Gamma = 2W = 6.0\ \mathrm{MeV}$ ). Find the neutron's mean survival time inside the nucleus, and compare it with the $\sim10^{-22}\ \mathrm{s}$ it takes the neutron to cross a nucleus. What does the comparison say physically?

Step 1. Lifetime from equation (3): $\tau = \dfrac{\hbar}{\Gamma} = \dfrac{1.055\times10^{-34}}{6.0\times10^{6}\times1.602\times10^{-19}} = \boxed{1.1\times10^{-22}\ \mathrm{s}}$ .

Step 2. Crossing time: a $\sim10\ \mathrm{MeV}$ neutron moves at $v \approx 4\times10^{7}\ \mathrm{m/s}$ ; across a nucleus of diameter $\sim10\ \mathrm{fm}$ , $t_{\text{cross}} = 10^{-14}\,\mathrm{m}/4\times10^{7}\,\mathrm{m/s} \approx 2.5\times10^{-22}\ \mathrm{s}$ .

Step 3. Compare: $\tau \sim t_{\text{cross}}$ — the survival probability falls to $e^{-1}$ in roughly one crossing. The nucleus is strongly absorbing but not black: a neutron has order-unity odds of being captured per transit, which is exactly why the optical model (grey, cloudy, partially transparent) fits neutron-scattering data better than either a transparent well or a perfectly black disk.

Step 4. Sanity checks: units — MeV × the eV-to-joule factor cancels against $\hbar$ in J·s to give seconds ✓; scaling — doubling the absorption $W$ halves $\tau$ , more cloud, faster capture ✓; and at $W \to 0$ the real-potential theorem resumes: $\tau \to \infty$ , the sealed tank again ✓.

Origins · 1 minEmmy Noether and how this was discovered

Emmy Noether

1882–1935 · Erlangen, Göttingen, Bryn Mawr

Conservation of probability is not an accident of the Schrödinger equation — it is a symmetry wearing a theorem, and the woman who proved that correspondence is the reason we know it. Noether's 1918 theorem, produced in Göttingen while the university still refused her a paid position (Hilbert famously asked the faculty whether they were "a university or a bathing society"), established that every continuous symmetry of a physical law begets a conservation law. The quantum instance: the Schrödinger equation is unchanged when $\Psi$ is multiplied by a constant phase $e^{i\theta}$ — nothing observable can depend on the global phase — and the conserved quantity that symmetry purchases is exactly $\int|\Psi|^2dx$ , this article's norm. Promote the phase symmetry from global to local and Noether's machinery generates electric charge conservation and, ultimately, the gauge structure of the Standard Model. Exiled from Germany in 1933 with the other Jewish faculty, she taught her final two years at Bryn Mawr; Einstein's obituary called her the most significant creative mathematical genius yet produced since the higher education of women began. Every "the norm is conserved" in this chapter is a small monument to her.

History

1918

— Noether's theorem: continuous symmetry ⇒ conservation law; global phase invariance will be probability's symmetry.

1926

— Born's probability reading makes ∫|Ψ|² = 1 a physical necessity; Schrödinger's equation is checked to preserve it.

1927

— Unitarity of quantum evolution is articulated in the Hilbert-space formulations of the new mechanics.

1932

— Von Neumann's Mathematical Foundations makes norm-preserving (unitary) evolution an axiom of the theory.

1954

— Feshbach, Porter and Weisskopf's optical model uses complex potentials to fit neutron absorption — conservation broken by design.

Going further

Texts. Griffiths & Schroeter, Introduction to Quantum Mechanics, 3rd ed., §1.4 (Normalization) and §1.5 (Momentum) · Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., §2.1 (Time Evolution, unitarity) · Shankar, Principles of Quantum Mechanics, 2nd ed., Ch. 4 (Probability current) · Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, Vol. 1, Ch. III (Probability conservation) · Townsend, A Modern Approach to Quantum Mechanics, 2nd ed., Ch. 6.

Provenance. Equations and the normalization-conservation proof verified against Griffiths & Schroeter §1.4 (Eqs. 1.20–1.28), including the probability current. Unitarity argument checked against Sakurai & Napolitano §2.1. Continuity-equation route follows Shankar Ch. 4 and Cohen-Tannoudji Vol. 1 Ch. III; the flux relation $R + T = 1$ mentioned by the base is owned in this track by Transmission and Reflection Coefficients. A verifier should recompute: (1) the cancellation of the $V$ terms requires $V \in \mathbb{R}$ , with remainder $\frac{i}{\hbar}(V^* - V)|\Psi|^2$ ; (2) the boundary-term form of $dN/dt$ ; (3) the optical-potential decay $N = e^{-\Gamma t/\hbar}$ and the numbers $\Gamma = 6.0\ \mathrm{MeV} \Rightarrow \tau = 1.1\times10^{-22}\ \mathrm{s}$ , crossing time $\approx 2.5\times10^{-22}\ \mathrm{s}$ ; (4) the two-state norm audit via orthogonality. Constant: $\hbar = 1.055\times10^{-34}\ \mathrm{J\,s}$ (CODATA).

Prerequisites. The Wavefunction · Born Statistical Interpretation · Probability Density and Normalization · Time-Dependent Schrödinger Equation · Probability Current and the Continuity Equation. Leads to. Stationary States and Separation of Variables · Time-Independent Schrödinger Equation · the unitary-evolution formalism of the Formalism chapter.

Is conservation of normalization the same thing as conservation of energy?

Different theorems from different symmetries, easily conflated because both say "something stays constant." Norm conservation follows from global phase symmetry (Noether's quantum instance) and holds for any real potential, even a time-dependent one — drive a system with a laser and probabilities still sum to 1 while the energy distribution pumps freely. Energy conservation follows from time-translation symmetry and requires $\hat H$ independent of $t$ . A driven atom conserves norm but not energy; no standard quantum system does the reverse. Keeping the two symmetries distinct will pay off repeatedly in the Dynamics chapter.

Does measurement violate the conservation proved here?

It sidesteps it. The theorem governs evolution under the Schrödinger equation; the textbook measurement update — collapse to an eigenstate, then renormalize — is not Schrödinger evolution, and the explicit renormalization step is the bookkeeping scar it leaves. Whether collapse is a genuine second dynamical process or an effective description of entanglement with the apparatus is the measurement problem, treated honestly in the Formalism chapter and the track's final chapter. What is uncontested: between measurements, the norm is untouchable.

Why exactly does a complex potential model absorption rather than just being nonsense?

Because the drained norm has an operational meaning: $N(t)$ is the probability that the particle is still in the channel your wavefunction describes. A neutron that gets captured, or an electron that ionizes away, has not vanished from the universe — it has left the subspace you chose to model. The imaginary potential is the effective, coarse-grained trace of coupling to those unmodelled channels (one can derive optical potentials honestly by "integrating out" the absorption channels — Feshbach's projection formalism). The rigorous fully-conserving description puts the lost probability in explicit other channels or an environment; the density-operator machinery for that lives at the track's later register.

The proof needs Ψ → 0 at infinity. Do all normalizable functions actually behave that well?

Almost — and the fine print is worth one honest look. Square-integrability alone permits pathologies (functions with ever-narrower spikes that never settle to zero), but such functions cannot solve the Schrödinger equation with a physical potential: solutions of a second-order ODE with finite energy are smooth enough that $\Psi \to 0$ and $\partial_x\Psi$ stays bounded, which is all the boundary term needs. This is why textbooks say "physically admissible" rather than merely "normalizable" — the full checklist of good behaviour is cataloged later in this chapter by Admissibility Conditions on Wavefunctions.

Is there a local version of this theorem, or only the global integral statement?

The local version is the continuity equation itself — the previous article — and it is strictly stronger: probability is conserved pointwise, in the sense that no region's probability changes except by flux through its boundary. The global theorem is the weakest consequence (integrate over everything). The hierarchy mirrors electromagnetism: local charge conservation $\partial_t\rho + \nabla\cdot\mathbf j = 0$ implies, but is not implied by, "total charge is constant." Whenever you can, cite the local law; the global one is what survives at the level of measured totals.

If evolution is a 'rigid rotation', what plays the role of the axis and angle?

Pleasingly literal answers exist. Decompose the state over energy eigenstates: evolution multiplies component $n$ by the phase $e^{-iE_nt/\hbar}$ — so the "axes" are the energy eigendirections (they are only spun within their own phase, never mixed), and the "angles" are $E_nt/\hbar$ , each growing at its own rate. A unitary is exactly a rotation that may turn each complex axis at a different speed. This picture also explains why norms are safe: rotations move the state around the unit sphere of Hilbert space but never off it. The Formalism chapter (Unitary Operators and Transformations) does this with full machinery.

Do nonlinear modifications of quantum mechanics break norm conservation?

Generically yes, and that is one of the sharpest experimental handles on them. Add a term $\epsilon|\Psi|^2\Psi$ to the Schrödinger equation and, depending on its form, either the norm drifts or — if you engineer the nonlinearity to preserve norms — superluminal signalling tends to sneak in elsewhere (Gisin's theorem-style arguments). Interferometry and spectroscopy bound such terms at extraordinary levels. Meanwhile the effective nonlinear equations physicists use daily (Gross–Pitaevskii for condensates, nonlinear optics) are mean-field descriptions living inside ordinary linear QM — their "norm" bookkeeping is particle number, conserved by construction. The linear, unitary core has survived every test to date.