States and Representations

A quantum state is the predictive object associated with a preparation. For a pure state, the physical object is a ray in Hilbert space. A normalized state vector, a column of basis coefficients, a position-space wavefunction, and a momentum-space wavefunction can all represent that same ray.

This chapter develops those distinctions before probabilities, observables, and measurement rules are used in detail. Its central lesson is simple but far-reaching: a mathematical representative is not automatically the invariant physical object.

Helpful background. The Core Formalism overview supplies the roles of states, measurements, and dynamics; Bra–Ket Notation supplies the notation, and Inner-Product Conventions fixes conjugation and adjoints.

What This Chapter Owns

This chapter is the canonical home for

• the operational meaning of a quantum state;
• normalized state vectors as representatives of pure states;
• rays and the irrelevance of global phase;
• superposition, amplitudes, and observable relative phase;
• pure states as rays, rank-one density operators, and extremal states;
• bases, components, wavefunctions, and operator representations;
• position- and momentum-space representations;
• normalization of ordinary states and generalized continuum states;
• unitary changes of basis and representation-invariant quantities;
• projective Hilbert space as the space of physical pure states.

The chapter introduces mixed states only to distinguish them from coherent superpositions. The full theory of density operators, purity, ensembles, reduced states, and the Bloch-vector representation belongs to Density Operators. Detailed Hilbert-space analysis and generalized eigenvectors belong to Hilbert Spaces.

State, Vector, and Ray

A nonzero vector $\lvert\psi\rangle$ represents a pure state, but multiplying it by a nonzero complex number does not change the ray:

$$\lvert\psi\rangle \sim c\lvert\psi\rangle, \qquad c\in\mathbb C\setminus\{0\}.$$

Probability calculations are usually performed with a normalized representative,

$$\langle\psi\rvert\psi\rangle=1.$$

After normalization, the remaining redundancy is global phase:

$$\lvert\psi\rangle \sim e^{i\alpha}\lvert\psi\rangle.$$

For any observable $A$,

$$\langle e^{i\alpha}\psi\rvert A \lvert e^{i\alpha}\psi\rangle = \langle\psi\rvert A\lvert\psi\rangle.$$

Relative phase is different. In

$$\lvert\psi\rangle = a\lvert0\rangle + be^{i\phi}\lvert1\rangle,$$

changing $\phi$ can change interference probabilities in another basis. Global phase is redundant; relative phase is generally observable through interference.

Abstract State and Basis Components

For an orthonormal basis $\{\lvert n\rangle\}$,

$$\lvert\psi\rangle = \sum_n \psi_n\lvert n\rangle, \qquad \psi_n = \langle n\rvert\psi\rangle.$$

The coefficients $\psi_n$ depend on the basis. The state does not. Under a unitary change of basis, state components and operator matrices transform together so inner products, probabilities, traces, and expectation values remain invariant.

In a position representation,

$$\psi(x) = \langle x\rvert\psi\rangle.$$

In a momentum representation,

$$\widetilde\psi(p) = \langle p\rvert\psi\rangle.$$

These wavefunctions are not two physical states. They are coordinate descriptions of one state in two generalized bases.

Position and Momentum Representations

With the convention

$$\langle x\rvert p\rangle = \frac{1}{\sqrt{2\pi\hbar}} e^{ipx/\hbar},$$

the wavefunctions are related by

$$\widetilde\psi(p) = \frac{1}{\sqrt{2\pi\hbar}} \int_{-\infty}^{\infty} dx\, e^{-ipx/\hbar} \psi(x).$$

The inverse transform uses the opposite phase. Other Fourier normalizations are equally valid when applied consistently. The physical momentum probability in an interval is computed from $\lvert\widetilde\psi(p)\rvert^2dp$, not from a wave-number density without the appropriate Jacobian.

Normalization and Generalized States

For a normalizable position-space state,

$$\int dx\, \lvert\psi(x)\rvert^2 = 1.$$

Ideal position and momentum eigenkets are not normalizable Hilbert-space vectors. Their generalized normalization is distributional:

$$\langle x\rvert x'\rangle = \delta(x-x'),$$ $$\langle p\rvert p'\rangle = \delta(p-p').$$

Delta normalization does not mean the delta function has unit value at one point. It means the generalized basis acts correctly under integration. Box normalization and wave packets can provide regulated routes from idealized continuum states to ordinary normalized vectors.

Pure States and Density Operators

A normalized vector defines the rank-one density operator

$$\rho_\psi = \lvert\psi\rangle \langle\psi\rvert.$$

It is invariant under global phase and obeys

$$\rho_\psi^2=\rho_\psi, \qquad \operatorname{Tr}\rho_\psi=1.$$

This representation makes the ray structure explicit: every normalized vector on the same ray produces the same projector. A pure global state can nevertheless have a mixed reduced state when it is entangled with another subsystem. “Pure” and “unentangled” are not synonyms.

Chapter Page Map

Question	Canonical page	Main distinction
What predictive role does a state play?	Quantum States	preparation and probability assignment
How is a pure state represented in Hilbert space?	State Vectors	vector representative versus physical state
Why is global phase unobservable?	Rays and Global Phase	vector versus ray
Why can relative phase affect outcomes?	Superposition and Relative Phase	coherent superposition versus mixture
What makes a state pure?	Pure States	ray, rank-one projector, extremal state
How do abstract objects acquire components?	Bases and Representations	object versus coordinates
What is a wavefunction?	Wavefunctions as Representations	position amplitude versus abstract state
How is the same state represented in momentum space?	Momentum-Space Representation	Fourier-related generalized bases
Which normalization applies?	Normalization	unit norm, delta normalization, box regulation
What changes under a new basis?	Change of Basis	components change, invariants do not
What is the geometry of pure-state space?	Projective Hilbert Space	Hilbert vectors modulo complex scale

State Vectors, Rays and Global Phase, and Superposition and Relative Phase form the state-vector starting sequence. The remaining articles extend that foundation to representation changes, normalization, purity, and state-space geometry.

Suggested Routes

First systematic pass

Read State Vectors, Rays and Global Phase, and Superposition and Relative Phase in order. Quantum States gives the broader preparation-and-prediction perspective; Pure States connects rays to rank-one projectors and extremal states.

Representation and wave-mechanics route

Read Bases and Representations, then Wavefunctions as Representations, Momentum-Space Representation, Normalization, and Change of Basis. Continue to Wave Mechanics and Model Systems for differential equations and solvable Hamiltonians.

Quantum-information and geometry route

For mixtures and reduced states, complete State Vectors, Projectors, Probability Amplitudes, the Born Rule, and Expectation Values before Density Operators. This supplies the pure-state and trace-rule capabilities needed for mixtures and reduced states.

The geometry branch reads Pure States, Change of Basis, and Projective Hilbert Space. For a qubit, projective state space becomes the Bloch sphere; continue to Geometric Quantum Mechanics Overview for distances, phases, and symplectic structure.

Mathematical-care route

Pair normalization and continuous representations with Generalized Eigenvectors, Rigged Hilbert Spaces: First Look, and Position and Momentum Representations.

Representation-Invariant Checks

Under a unitary basis change, verify that the following are unchanged:

$$\langle\phi\rvert\psi\rangle, \qquad \langle\psi\rvert A\lvert\psi\rangle, \qquad \operatorname{Tr}A, \qquad \det A$$

when the determinant is defined in the finite-dimensional setting. Eigenvalues and transition probabilities are likewise invariant. Individual vector components and matrix entries are not.

For numerical state representations, also check normalization, phase conventions, grid measure, Fourier scaling, and convergence with basis or grid size.

Common Mistakes

• Calling a vector a unique physical pure state. Nonzero scalar multiples lie on the same ray.
• Removing a relative phase as though it were global. Relative phase can change interference.
• Equating a coherent superposition with a classical mixture. Their off-diagonal coherences and measurement statistics differ.
• Treating a wavefunction as basis independent. It is a component function in a chosen generalized basis.
• Changing state components without changing operator matrices. A basis change must transform all representatives consistently.
• Normalizing a generalized eigenstate to one. Ideal continuum eigenstates use delta normalization or a regulator.
• Confusing pure with unentangled. Purity refers to the state on the Hilbert space being described.
• Assuming finite truncation preserves projective geometry and operator relations exactly. Convergence and truncation effects require checks.

Cross-Links

• Formalism Overview
• Observables and Operators
• Born Rule
• Density Operators
• Bra–Ket Notation
• Hilbert Spaces
• Wave Mechanics and Model Systems
• Representation Translation Table

References

• I. Bengtsson and K. Życzkowski, Geometry of Quantum States, 2nd ed., Cambridge University Press, 2017, doi:10.1017/9781139207010.
• P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford University Press, 1958.
• A. Peres, Quantum Theory: Concepts and Methods, Kluwer, 1995.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955.