States and Representations

A quantum state lets you predict measurement probabilities. A ket, a column vector, and a wavefunction can describe the same state: they are different ways of expressing its amplitudes. This chapter teaches you to move between these descriptions without mistaking a change of coordinates for a change of preparation.

Start with state vectors if you know complex vectors and inner products. The linear-algebra introduction provides those tools if you need them.

Helpful background. Minimal language distinguishes states, measurements, and dynamics; bra–ket notation explains bras, kets, inner products, and outer products; inner-product conventions fixes conjugation, norms, adjoints, and transition amplitudes.

One state, different amplitudes

For an orthonormal basis $\{\lvert e_j\rangle\}$, a normalized state has the expansion

$$\lvert\psi\rangle=\sum_j c_j\lvert e_j\rangle, \qquad c_j=\langle e_j\rvert\psi\rangle, \qquad \sum_j\lvert c_j\rvert^2=1.$$

The coefficients depend on the basis. Multiplying every coefficient by the same phase $e^{i\alpha}$ leaves the pure physical state unchanged. Changing their relative phases can change predictions.

For example, $\lvert+\rangle=(\lvert0\rangle+\lvert1\rangle)/\sqrt2$ and $\lvert-\rangle=(\lvert0\rangle-\lvert1\rangle)/\sqrt2$ both give equal probabilities for outcomes $0$ and $1$. Yet they are orthogonal: a measurement in the $\{\lvert+\rangle,\lvert-\rangle\}$ basis distinguishes them perfectly. Their minus sign is a relative phase, not an overall phase.

For motion in space, the position wavefunction $\psi(x)$ replaces a discrete list of amplitudes. Its normalization uses an integral, and the corresponding momentum wavefunction is a Fourier transform. Begin with ordinary square-integrable packets before treating ideal plane waves, whose delta normalization is a different mathematical construction.

Reading sequence

Question	Article	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
How do physical procedures, objects, and representations differ?	Mathematical Objects and Physical Meaning	modeling a preparation versus rewriting its description
What is the geometry of pure-state space?	Projective Hilbert Space	Hilbert vectors modulo complex scale
Which matrix arguments extend to wavefunctions?	Finite and Infinite-Dimensional Quantum Mechanics	shared physical rules versus domain and convergence requirements

Choose your next calculation

For a first pass, read quantum states, state vectors, global phase, and relative phase in that order. Then use bases and representations to calculate the same probability in two bases.

For spatial motion, continue through wavefunctions, momentum representation, and normalization before entering wave mechanics. Projective Hilbert space develops the geometry of rays after the elementary calculations are familiar.

Use the objects-and-meaning comparison to separate basis changes from changes of preparation. The finite–infinite comparison then checks the domains, spectra, and convergence assumptions that become essential beyond finite matrices.

Try it. Are $\lvert+\rangle$ and $i\lvert+\rangle$ different pure states? Are $\lvert+\rangle$ and $\lvert-\rangle$?

Solution

The first pair differs by a global phase and represents one ray. The second pair represents orthogonal rays; the measurement described above separates them with certainty.

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.