Hilbert Space and State Vectors

For an entire chapter you solved differential equations. Wavefunctions were functions: things with graphs, nodes you could count, tails that decayed. That toolkit built every well and barrier in one dimension — and it is about to be outgrown. Spin has no wavefunction of position at all. Entanglement lives in no single coordinate. Even the humble statement "expand the initial state in energy eigenfunctions," which you used freely for the infinite well, quietly assumed a piece of mathematics nobody proved for you: that such expansions exist, converge, and conserve probability. This chapter — the formalism chapter — rebuilds quantum mechanics on its true foundation, and this opening article lays the cornerstone: the state of a quantum system is a vector in a Hilbert space. Not "is like a vector" — is one. Superposition, the physical heart of the theory, is nothing but vector addition. Normalization is having unit length. The wavefunctions of the last chapter turn out to be coordinates of these vectors in one particular basis, the way $(3, 4)$ are coordinates of an arrow — and spin states, photon polarizations, and qubits are vectors in smaller, friendlier spaces where the same geometry runs without any calculus at all. Three structural properties do all the work: complex linearity (states add, with complex coefficients that interfere), an inner product (states have overlaps, which become amplitudes), and completeness (infinite expansions converge to states genuinely in the space — the fine print that makes the previous chapter's Fourier games legal). Master this geometry and the next twenty articles — bras, operators, measurement — are annotations on one picture.

Key takeaways

A quantum state is a unit vector $|\psi\rangle$ in a complex Hilbert space $\mathcal H$ : a complex vector space with an inner product, complete in the induced norm.
Superposition is vector addition: $c_1|\psi_1\rangle + c_2|\psi_2\rangle$ is again a state (after normalization) — the complex coefficients carry interference.
The last chapter's wavefunctions are the special case $\mathcal H = L^2(\mathbb R)$ : square-integrable functions, with $\langle\phi|\psi\rangle = \int\phi^*\psi\,dx$ . Plane waves fail the entry test.
Physical states are rays: $|\psi\rangle$ and $e^{i\theta}|\psi\rangle$ describe identical physics — but relative phases inside a superposition are measurable.
Dimension is set by the physics: spin-½ lives in $\mathbb C^2$ (two components, no calculus); a particle in a well needs countably infinite dimensions (the ISW eigenbasis).
Completeness (Cauchy sequences converge in $\mathcal H$ ) is the license for infinite expansions $\sum_n c_n|n\rangle$ — with $\sum_n |c_n|^2 = 1$ as the conservation of probability.
Key idea: a quantum state is a unit ray in a complete complex inner-product space — linearity gives superposition, the inner product gives amplitudes, completeness makes infinite expansions rigorous.

The core insight

A quantum state is a unit ray in a complete complex inner-product space. Linearity gives superposition, the inner product gives probabilities, and completeness makes infinite expansions rigorous — these three properties are the entire structural content of "state" in quantum mechanics.

Where this lives today

The vocabulary of this article is the working language of a technology sector. A quantum computer's design document begins by declaring its Hilbert space: $n$ qubits inhabit $(\mathbb C^2)^{\otimes n}$ , dimension $2^n$ — that exponential vector space is the entire hardware thesis, and "we can prepare, protect, and steer unit vectors in it" is the engineering claim behind every roadmap. Quantum-chemistry codes budget accuracy by how they truncate the electronic Hilbert space; tensor-network methods live off the fact that physical states occupy a tiny structured corner of it. The phase distinction of equation (5) is an engineering spec: qubit gates are calibrated rotations of relative phase, global phase quotiented out of the definitions. And the completeness fine print still bites professionals — mathematical physicists spend real effort establishing that formally written states actually live in the space.

The physical picture

Start with the smallest quantum system there is, because it shows the geometry with nothing else in the room. A spin-½ particle's magnetic moment, measured along $z$ , gives one of two outcomes — call the states $|{\uparrow}\rangle$ and $|{\downarrow}\rangle$ . The complete state space is every complex combination $a|{\uparrow}\rangle + b|{\downarrow}\rangle$ with $|a|^2 + |b|^2 = 1$ : a two-dimensional complex vector space, $\mathbb C^2$ . No position, no wavefunction, no differential equation — and yet superposition, measurement statistics, and interference all run at full strength. The vector is the physics.

Figure

One geometry, two homes — the qubit's C² and the wavefunction's L² are the same structure

Now revisit the previous chapter with vector eyes. A wavefunction $\psi(x)$ assigns a complex number to each point $x$ — think of it as a column vector with a continuous index instead of a discrete one. Adding wavefunctions pointwise is vector addition; multiplying by a complex constant is scalar multiplication; and the integral $\int\phi^*(x)\psi(x)\,dx$ plays exactly the role the dot product plays for arrows: it measures overlap. The space of normalizable wavefunctions — those with $\int|\psi|^2dx < \infty$ — is called $L^2(\mathbb R)$ , and it is the infinite-dimensional sibling of the qubit's $\mathbb C^2$ . One set of axioms, every quantum system: the differences between a spin, a particle in a box, and a field mode are differences of dimension and labeling, not of structure. The chapter you just finished also supplied the natural coordinate systems. The infinite well's eigenfunctions $\{\psi_n\}$ are orthonormal — mutually perpendicular unit vectors — and complete: any state in the well can be written $\sum_n c_n\psi_n$ . That is not an analogy to expanding an arrow over $\hat x, \hat y, \hat z$ — it is the same operation with more axes. The genuinely new subtlety of infinite dimensions hides in the word "any": an infinite sum must converge, to something still inside the space. That demand — completeness — is the "Hilbert" in Hilbert space.

The three axioms, with the physics attached

Linearity. The state space is a complex vector space: for any two states and any complex numbers,

|\psi\rangle = c_1|\psi_1\rangle + c_2|\psi_2\rangle \ \in\ \mathcal H \qquad (1)

where: closure under superposition is the physical principle — double-slit interference, tunnelling doublets, chemistry's bonding orbitals all live in this one line — and the field being $\mathbb C$ rather than $\mathbb R$ is load-bearing: relative phases between components are what interfere, and the wave mechanics chapter showed a real-valued theory cannot even represent a moving free particle. (Notation: we adopt Dirac's ket $|\psi\rangle$ for "state vector" from here on; the full notation system is the next article's business.)

Inner product. Any two states have a complex overlap $\langle\phi|\psi\rangle$ , conjugate-symmetric ( $\langle\phi|\psi\rangle = \langle\psi|\phi\rangle^*$ ), linear in the second argument, and positive on the diagonal. In $L^2$ it is the familiar integral:

\langle\phi|\psi\rangle = \int_{-\infty}^{\infty}\phi^*(x)\,\psi(x)\,dx, \qquad \|\psi\| = \sqrt{\langle\psi|\psi\rangle} \qquad (2)

where: the induced norm $\|\psi\|$ is the vector's length, and a physical state must have $\|\psi\| = 1$ — that is normalization, now a geometric statement. The overlap's physical reading (probability amplitude) and its full axiom system, including the Cauchy–Schwarz inequality that keeps all amplitudes bounded, are developed two articles ahead in Inner Product and Norm of States; here the inner product's job is to define length and orthogonality so that bases make sense.

Completeness. Every Cauchy sequence of states — a sequence whose members become arbitrarily close to each other in norm — converges to a limit inside $\mathcal H$ . This is the axiom with no finite-dimensional content at all ( $\mathbb C^n$ has it automatically) and maximal infinite-dimensional consequence: it guarantees that the infinite expansions quantum mechanics runs on,

|\psi\rangle = \sum_{n=1}^{\infty} c_n |n\rangle, \qquad \sum_{n=1}^{\infty} |c_n|^2 = \|\psi\|^2 = 1 \qquad (3)

where: $\{|n\rangle\}$ is an orthonormal basis (the ISW eigenfunctions were your first), actually mean something: the partial sums form a Cauchy sequence (their mutual distances are controlled by the tail $\sum|c_n|^2$ , which vanishes), and completeness hands you the limit state. The equality on the right is Parseval's identity — Pythagoras in infinitely many dimensions — and reading $|c_n|^2$ as a probability, it is the conservation of total probability across a basis change. The fact that $L^2$ is complete is the Riesz–Fischer theorem (1907), one of the quiet mathematical pillars the theory stands on. (Which orthonormal sets are rich enough to expand everything — basis-hood itself — is the business of Complete Orthonormal Bases, later this chapter.)

Derivation APythagoras in Hilbert space — normalization survives expansion

We prove equation (3)'s right half: for a state expanded over an orthonormal set, the squared length is the sum of the squared coefficients. This is the workhorse identity behind "probabilities sum to one" in every basis, and the derivation is two lines of inner-product algebra — the point of presenting it is to watch the orthonormality do all the killing, and to see where infinite dimensionality genuinely needs the completeness axiom.

Step 1. Expand the norm using linearity of the inner product in each slot:

\|\psi\|^2 = \Bigl(\sum_m c_m|m\rangle\Bigr)^{\!\dagger}\Bigl(\sum_n c_n|n\rangle\Bigr) = \sum_{m,n} c_m^*\,c_n\,\langle m|n\rangle

Why the conjugate appears on $c_m$ : the inner product is antilinear in its first argument — pulling a constant out of the bra side conjugates it. This sign of sesquilinearity is developed fully in the inner-product article; here it is used once, correctly.

Step 2. Apply orthonormality $\langle m|n\rangle = \delta_{mn}$ : the double sum collapses to the diagonal,

\|\psi\|^2 = \sum_n |c_n|^2

Why this is Pythagoras: the components along mutually perpendicular unit axes contribute their squares independently — in two real dimensions this is the schoolroom theorem; here it holds for complex coefficients and countably many axes.

Step 3. Attach the physics. For a normalized state, $\sum_n|c_n|^2 = 1$ : the numbers $|c_n|^2$ form a probability distribution over the basis labels. This is the structural slot the Born rule will drop into (Born Rule and Measurement Probabilities, later this chapter) — the geometry is already probability-shaped before any measurement postulate is stated.

Step 4. Note the infinite-dimensional fine print. For infinitely many terms, Step 1's rearrangement requires the series to converge — which is exactly what completeness plus $\sum|c_n|^2 < \infty$ guarantees (the partial sums are Cauchy). In a merely "pre-Hilbert" space the sum could fail to be a state at all, and the identity would be about nothing. Verified: numerically for the tent state of Worked example 1 — 25 ISW coefficients give $\sum|c_n|^2 = 0.9999907$ , monotonically increasing partial sums (Cauchy behaviour), and the exact odd- $n$ coefficient law confirms the limit is 1; and exactly for the $\mathbb C^2$ case $(3/5)^2 + (4/5)^2 = 1$ .

Rays, phases, and what a "state" really is

One vector per state is almost right. Multiply $|\psi\rangle$ by a pure phase and every physical prediction — every $|\langle\phi|\psi\rangle|^2$ , every expectation value — is untouched, because the phase cancels against its conjugate. The honest statement is:

|\psi\rangle \ \sim\ e^{i\theta}|\psi\rangle \quad \text{(same physical state — a ray, not a vector)} \qquad (4)

where: a ray is the set of all unit vectors differing by a global phase. But keep the distinction razor-sharp: relative phase inside a superposition is physical. The states $\tfrac{1}{\sqrt2}(|{\uparrow}\rangle + |{\downarrow}\rangle)$ and $\tfrac{1}{\sqrt2}(|{\uparrow}\rangle + i|{\downarrow}\rangle)$ are genuinely different — measured along $x$ , the first gives "up" with certainty, the second a 50/50 coin flip. One phase you can never see; the other is the entire engine of interference. (You met this distinction dynamically in the wave chapter: the stationary states' global $e^{-iE t/\hbar}$ was invisible, while the relative phase between two energy components made densities slosh.)

\text{global phase: unobservable} \qquad\qquad \text{relative phase: interference} \qquad (5)

where: the practical rule of thumb — a phase multiplying the whole ket drops out of every modulus; a phase multiplying one term of a sum survives inside cross terms. The reality-choice freedom for 1D bound states (Non-Degeneracy of One-Dimensional Bound States) was exactly a use of ray freedom: fixing the unobservable phase to make the function real.

Go deeper · Derivation B · 1 minGlobal phase drops out of everything — relative phase survives in the cross term

Derivation BGlobal phase drops out of everything — relative phase survives in the cross term

We prove the ray statement (4) honestly: multiplying a state by $e^{i\theta}$ changes no prediction of quantum mechanics, while a phase on one branch of a superposition does. The strategy is to run the two candidate phases through the only two objects predictions are made of — squared overlaps and expectation values — and watch where conjugation cancels the phase and where it cannot. This one card is the licence for every "choose the phase conveniently" move in the subject, including the reality choice for 1D bound states.

Step 1. Squared overlaps. For any fixed $|\phi\rangle$ : $|\langle\phi|e^{i\theta}\psi\rangle|^2 = |e^{i\theta}|^2\,|\langle\phi|\psi\rangle|^2 = |\langle\phi|\psi\rangle|^2$ , since $|e^{i\theta}| = 1$ . Why the phase escaped the bra unharmed: linearity in the ket pulled it straight out; the modulus then executed it.

Step 2. Expectation values. $\langle e^{i\theta}\psi|\hat A|e^{i\theta}\psi\rangle = e^{-i\theta}e^{i\theta}\langle\psi|\hat A|\psi\rangle = \langle\psi|\hat A|\psi\rangle$ — the bra contributes the conjugate phase, and the pair annihilates. Every moment of every observable is untouched; $|\psi\rangle$ and $e^{i\theta}|\psi\rangle$ are experimentally identical.

Step 3. Now phase one branch only. Take $|\psi_\varphi\rangle = \tfrac{1}{\sqrt2}(|1\rangle + e^{i\varphi}|2\rangle)$ with $\langle 1|2\rangle = 0$ , and probe with $|\phi\rangle = \tfrac{1}{\sqrt2}(|1\rangle + |2\rangle)$ :

|\langle\phi|\psi_\varphi\rangle|^2 = \tfrac14\,|1 + e^{i\varphi}|^2 = \tfrac{1 + \cos\varphi}{2}

Why it survived: the phase now sits between two terms of a sum, and the modulus-squared's cross term $2\,\mathrm{Re}(e^{i\varphi})$ preserves it. Relative phase = interference term, by direct computation.

Step 4. Read the dichotomy. $\varphi$ interpolates the probability continuously from 1 (in phase) through $\tfrac12$ ( $\varphi = \pi/2$ ) to 0 (anti-phase) — maximally physical; $\theta$ never appears anywhere — pure gauge. The vector-to-physics map is many-to-one exactly along global phases: states are rays. Verified: numerically on explicit $\mathbb C^2$ vectors — elementwise $|e^{i\theta}\psi|^2 = |\psi|^2$ at $\theta = 0.77$ , and the probe probabilities $1.00$ vs $0.50$ for $\varphi = 0$ vs $\varphi = \pi/2$ , matching $(1+\cos\varphi)/2$ .

And who is allowed into the space at all? For $L^2(\mathbb R)$ , the entry test is $\int|\psi|^2 < \infty$ . The Gaussian passes. The tent function passes. Discontinuous, spiky, physically grotesque functions pass — $L^2$ is roomier than the set of "nice" wavefunctions, and the admissibility conditions of the wave chapter carve out the physical subset. Who fails is more instructive: the plane wave $e^{ikx}$ , the backbone of scattering theory, has infinite norm — it is not in the space (Free Particle and Plane Waves met this head-on). The formalism's mature answer — delta-normalized continuum states living just outside $\mathcal H$ , made rigorous in a "rigged" extension — is scheduled for Continuous Spectra and Dirac Delta Normalization and Rigged Hilbert Space at this chapter's far end. Until then: plane waves are limits of legitimate states (wave packets), and every honest calculation can be phrased with the packets.

Figure

Truncation is observable-dependent — the tent state's norm converges fast, its energy slowly

Limiting cases

case	result	meaning
dimension 2	$\mathcal H = \mathbb C^2$ : spin-½, polarization, qubit	full quantum structure, zero calculus — the cleanest lab for the axioms
dimension $n$	$\mathbb C^n$ : $n$ -level atoms, spin- $j$ ( $n = 2j{+}1$ )	completeness automatic; everything is matrix algebra
countably infinite	$L^2$ of a bound problem (ISW basis)	completeness is a real theorem (Riesz–Fischer); expansions need convergence
plane waves, $\delta(x)$	not in $\mathcal H$	idealized limits; delta-normalization and rigged spaces handle them later
global phase $e^{i\theta}\|\psi\rangle$	same physical state (ray)	gauge freedom; fix it however convenient
relative phase in a sum	different physical state	interference — the measurable content of complex linearity
composite systems	$\mathcal H_1 \otimes \mathcal H_2$ (tensor product)	dimension multiplies; entanglement lives here — later chapters

Common trapThe state space is the set of all wavefunctions, and any two that differ are physically different

Two corrections in one. First, membership: $\mathcal H$ contains only normalizable objects — plane waves and delta spikes, however useful, are not states; treating them as such without the (later) delta-normalization scaffolding produces divergent nonsense like $\langle k|k\rangle = \infty$ taken literally. Second, distinctness: vectors differing by a global phase are the same state — the map from vectors to physics is many-to-one (rays). The two errors pull in opposite directions: one admits too much, the other counts too much. The membership test is $\|\psi\| < \infty$ ; the identity test is "equal up to one overall phase." Everything else — including a phase on one branch of a superposition — is physically distinct, and interference experiments will tell you so.

Worked examples

Example 1 — a plucked box: expansion, probabilities, convergence (confidence builder). A particle in a box (width $L = 1$ , ISW basis $\psi_n = \sqrt2\sin n\pi x$ ) is prepared in the "plucked string" tent state: $\psi \propto x$ up to the midpoint, then $\propto 1 - x$ . Expand it, read the probabilities, and check Pythagoras.

Step 1. Normalize. $\int_0^1\text{tent}^2 = 1/12$ , so the normalized amplitude is $A = \sqrt{12}$ .

Step 2. Coefficients. $c_n = \langle n|\psi\rangle = \sqrt{24}\int_0^1 \text{tent}\cdot\sin(n\pi x)\,dx = \dfrac{4\sqrt6}{n^2\pi^2}\times(-1)^{(n-1)/2}$ for odd $n$ , and $c_n = 0$ for even $n$ — the tent is symmetric about the centre, so odd-parity basis states drop out (the parity article's selection logic, working inside an expansion).

Step 3. Probabilities. $|c_1|^2 = 96/\pi^4 = \boxed{98.55\%}$ : the pluck is overwhelmingly the ground state. $|c_3|^2 = |c_1|^2/81 = 1.22\%$ , $|c_5|^2 = 0.16\%$ … An energy measurement almost always returns $E_1$ .

Step 4. Pythagoras. Numerically, 25 terms give $\sum|c_n|^2 = 0.99999$ , climbing monotonically toward 1 — the Cauchy convergence that completeness promises. But note the sting in the tail: the energy expectation $\sum|c_n|^2 n^2 E_1 = 12/\pi^2\,E_1 \approx 1.216\,E_1$ converges much more slowly (25 terms: $1.197$ ) — the kink at the tent's peak feeds high- $n$ components whose $n^2$ weighting resists truncation. Norms forgive kinks; energies do not. Convergence in what sense is a real question in $\mathcal H$ , not pedantry.

Example 2 — a qubit, no calculus anywhere (exam level). A qubit is prepared with amplitude $\tfrac{1+i}{2}$ on $|0\rangle$ and unknown amplitude $c$ on $|1\rangle$ . (a) Fix $|c|$ by normalization. (b) Give the probabilities of each outcome in the computational basis. (c) Exhibit two legal choices of $c$ that are physically different states, and one that is the same state in disguise.

Step 1. Normalization: $\left|\tfrac{1+i}{2}\right|^2 = \tfrac{1+1}{4} = \tfrac12$ , so $|c|^2 = \tfrac12$ : $|c| = \boxed{1/\sqrt2}$ , phase free.

Step 2. Probabilities: $P(0) = P(1) = \boxed{1/2}$ — set by moduli alone; no choice of phase for $c$ changes them.

Step 3. The phases. Choose $c = \tfrac{1}{\sqrt2}$ versus $c = \tfrac{i}{\sqrt2}$ : same $P(0), P(1)$ , but these are different rays — a measurement in the rotated basis $\tfrac{1}{\sqrt2}(|0\rangle \pm |1\rangle)$ distinguishes them with certainty margin (their overlaps with $|+\rangle$ differ: relative phase is physical). By contrast, $c = \tfrac{1}{\sqrt2}$ with the entire state multiplied by $e^{i\pi/4}$ — equivalently amplitudes $\bigl(\tfrac{1+i}{2}e^{i\pi/4}, \tfrac{e^{i\pi/4}}{\sqrt2}\bigr)$ — is the same state: every basis, every observable, identical statistics. Ray freedom in action: one phase is a relabeling, the other is physics.

The man who proved the space complete

Origins · 1 minFrigyes Riesz and how this was discovered

Frigyes Riesz

1880–1956 · Budapest, Szeged

Of this article's three axioms, two were classical geometry generalized; the third — completeness — had to be proved about the space physics would need, and Riesz is the mathematician who proved it. In 1907, simultaneously with Ernst Fischer, he established what is now the Riesz–Fischer theorem: the square-integrable functions form a complete space — every Cauchy sequence of wavefunctions converges to a wavefunction — and, strikingly, that $L^2$ is isomorphic to the space of square-summable sequences. That last clause is quietly the mathematical heart of quantum mechanics' great 1926 reconciliation: Schrödinger's wavefunctions and Heisenberg's infinite vectors are the same Hilbert space in two coordinate systems, and the isomorphism physicists invoked was Riesz's, nineteen years old and waiting. His 1907 representation theorem — every continuous linear functional comes from an inner product with a unique vector — is the licence for the bra–ket duality built in the very next article. Working from Szeged, far from the Göttingen mainstream, Riesz went on to co-found functional analysis as a discipline (his 1952 textbook with his student Szőkefalvi-Nagy trained a generation); the hard analytic floor under this chapter — that expansions converge, that duals exist, that the geometry is honest — is largely his pouring.

History

1900

— Hilbert's Paris problems include the axiomatization of physics — a request his own mathematics would later answer

1904

— Hilbert's integral-equation papers begin the geometry of infinitely many variables

1907

— Riesz and Fischer prove L² is complete — the theorem that makes wavefunction expansions converge

1925

— Heisenberg, Born, and Jordan's matrix mechanics: states as infinite vectors, in Hilbert's own town

1926

— Schrödinger's wave mechanics; the two theories' equivalence begs for a common structure

1927

— von Neumann axiomatizes quantum states as vectors in what he names "Hilbert space"

1932

— von Neumann's Mathematische Grundlagen completes the framework this chapter teaches

Going further

References. Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., Ch. 1 · Shankar, Principles of Quantum Mechanics, 2nd ed., Ch. 1 · Griffiths & Schroeter, Introduction to Quantum Mechanics, 3rd ed., Ch. 3 & Appendix A · Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, Vol. I, Ch. II · von Neumann, Mathematical Foundations of Quantum Mechanics (1932; Princeton ed. 2018).

Provenance. Equations and axioms are standard Hilbert-space structure verifiable against Sakurai & Napolitano Ch.1, Shankar Ch.1, Griffiths Ch.3 & Appendix A, and Cohen-Tannoudji Vol.1 Ch.II. A verifier should recompute: the antilinearity derivation from conjugate symmetry; the Pythagoras/normalization sum $\sum|c_n|^2=1$ ; the Cauchy–Schwarz proof via the projection-subtracted vector; and the numerical check $(3/5)^2+(4/5)^2=1$ . For this edition, verification was executed (sympy/numpy): Gaussian in / plane-wave out of $L^2$ ; the tent-state expansion (exact $c_n = 4\sqrt6/n^2\pi^2$ odd- $n$ law, $|c_1|^2 = 96/\pi^4 = 98.55\%$ , Parseval to $10^{-5}$ in 25 terms, monotone Cauchy partial sums, and the slow-converging $\langle H\rangle = 12/\pi^2\,E_1$ with its 25-term shortfall exhibited); global-vs-relative phase on explicit $\mathbb C^2$ states; and both worked examples. Per this chapter's ownership map, the Cauchy–Schwarz derivation itself is executed in Inner Product and Norm of States (two articles ahead); this article states only norm and orthogonality.

Builds on: Linear algebra (complex vector spaces) · Complex numbers · The Postulates of Quantum Mechanics

Leads to: Dirac Bra-Ket Notation · Inner Product and Norm of States · Linear Operators on Hilbert Space · Complete Orthonormal Bases · Born Rule and Measurement Probabilities

Is the Hilbert space something physically real, or just bookkeeping?

Physicists split on the metaphysics; the operational content is not in dispute. The structure — linearity, inner product, completeness — encodes experimentally enforced facts: superposition and interference exist, probabilities come from squared overlaps and sum to one, fine expansions converge. Whether the vector "exists" or merely catalogues predictions is an interpretation question (this track's foundations chapter). What you must treat as real in practice: the ray structure and relative phases, because experiments measure them.

Why must the scalars be complex? Real superpositions look like they'd work fine.

Interference with a tunable phase is the experimental reason: a relative factor $e^{i\varphi}$ interpolates continuously between constructive and destructive, and two-path experiments trace out exactly that continuum. Structurally, time evolution needs it: $e^{-iEt/\hbar}$ phases are what make stationary states stationary while superpositions move, and the wave chapter showed a real wavefunction carries zero probability current — a real-only theory has no travelling states. (Real and quaternionic quantum mechanics have been studied seriously; locality and composition arguments keep landing on $\mathbb C$ as the physical choice.)

Do I need to check completeness whenever I use a basis expansion?

No — you inherit it, via Riesz–Fischer ( $L^2$ is complete) and the spectral theory that gives well-behaved Hamiltonians orthonormal eigenbases. What deserves your attention is the rate and sense of convergence: Example 1's tent state has a norm expansion converging beautifully and an energy expansion converging painfully, because the kink stuffs the high- $n$ tail. Truncation error depends on which observable you ask about — that is where the mathematics touches your homework.

If the wavefunction is just coordinates of a vector, what basis was Chapter 1 secretly using?

The position "basis": $\psi(x)$ is the component of $|\psi\rangle$ along the idealized position state at $x$ — written $\langle x|\psi\rangle$ once bra-ket notation arrives. The scare quotes are earned: position states are delta-normalized continuum objects outside $\mathcal H$ , and making that basis honest is the delta-normalization/rigged-space program later this chapter. The safe version available now: any orthonormal discrete basis, with $\psi$ 's coordinates being the $c_n$ .

What plays the role of Hilbert space for two particles, or a particle with spin?

Tensor products. Two particles: $L^2 \otimes L^2$ ; a spinful electron: $L^2 \otimes \mathbb C^2$ (a two-component wavefunction). Dimensions multiply — which is why many-body spaces are so catastrophically large, and why entanglement (states of the product space that are not products of states) exists at all. The construction gets its own treatment in the identical-particles and entanglement chapters; this article's axioms hold verbatim in every factor and in the product.

Is every unit vector in the Hilbert space a physically preparable state?

The formalism imposes no restriction; nature adds asterisks. Superselection rules forbid certain superpositions (different electric charges); energy constraints make most of a many-body space unreachable in the universe's lifetime; decoherence renders some superpositions (alive+dead cats) unstable on absurd timescales. The formalism is deliberately permissive — dynamics and environment carve out the accessible corner, a theme the entanglement chapter takes up in earnest.