Infinite-Dimensional Hilbert Spaces

A wavefunction has infinitely many possible components, but it still has lengths, angles, and orthogonal projections. A Hilbert space keeps this familiar geometry while making limits of approximations meaningful. The new care is mostly about which limit is taken and which vectors an operator can act on.

Helpful background. Review inner products and orthonormal bases. You should be able to expand a finite-dimensional vector and calculate its norm.

Enter this chapter

Start with Hilbert Spaces and $L^2$ Spaces. Then read Completeness and Orthonormal Bases. These three pages explain why an infinite expansion is a limit in norm, why changing a wavefunction at one point does not change its $L^2$ vector, and why boundary conditions need a separate discussion.

For a first course in wave mechanics, continue to Position and Momentum Representations. You can return to the more technical operator-domain questions when they arise in a calculation.

One example to keep in mind

The sine functions $e_n(x)=\sqrt{2/L}\sin(n\pi x/L)$ form a complete orthonormal basis of $L^2(0,L)$. They can expand the constant function even though every finite sine sum vanishes at the endpoints. There is no contradiction: $L^2$ convergence controls an integral of the squared error, not endpoint values.

By contrast, applying the Dirichlet Hamiltonian $-\hbar^2d^2/(2m\,dx^2)$ requires differentiability and zero boundary values in its operator domain. An arbitrary $L^2$ vector need not satisfy these requirements. Domains of Operators separates membership in the state space from permission to apply an unbounded operator.

Choose a route

• Understand observables: Bounded Operators → Unbounded Operators → Domains → Adjoints → Self-Adjointness.
• Understand spectra and scattering notation: Spectral Theorem → Continuous Spectra → Generalized Eigenvectors. The rigged Hilbert space introduction is an optional next step.
• Understand mixed states: Trace-Class and Hilbert–Schmidt Operators explains which infinite-dimensional operators have a trace; then read Density Operators.

Check your understanding

A calculation shows $\lVert\psi_N-\psi\rVert\to0$. Does it follow that $\lVert A\psi_N-A\psi\rVert\to0$?

Solution

Yes for bounded $A$, because the second norm is at most $\lVert A\rVert\lVert\psi_N-\psi\rVert$. Not in general for unbounded $A$: the vectors must first lie in its domain, and Hilbert-norm convergence alone need not control their images. For example, $Ae_n=ne_n$ and $\psi_n=e_n/n$ give $\psi_n\to0$ but $\lVert A\psi_n\rVert=1$.

Page map

Page	Central question
Hilbert Spaces	Why must an inner-product state space be complete?
$L^2$ Spaces	What does square integrability mean, and why are wavefunctions equivalence classes?
Completeness and Orthonormal Bases	In which sense does an infinite basis expansion converge?
Separable Hilbert Spaces	Why can a countable basis describe standard infinite-dimensional state spaces?
Bounded Operators	Which operators are continuous and defined safely on the full space?
Unbounded Operators	Why do position, momentum, and Hamiltonians require additional care?
Domains of Operators	How do regularity and boundary conditions become part of an operator?
Adjoint Operators	How does the inner product determine an adjoint and its domain?
Symmetric versus Self-Adjoint Operators	Why is symmetry weaker than self-adjointness?
Spectral Theorem, Practical Version	How do projection-valued measures unify discrete and continuous spectra?
Continuous Spectra	What replaces a normalizable eigenbasis in continuous spectral sectors?
Position and Momentum Representations	How does one state acquire position- and momentum-space wavefunctions?
Generalized Eigenvectors	How should delta-normalized kets be read and used?
Rigged Hilbert Spaces, First Look	Which larger space gives generalized kets a disciplined home?
Trace-Class and Hilbert-Schmidt Operators	Which operator classes make traces and density operators controlled?

References

• J. B. Conway, A Course in Functional Analysis, 2nd ed., Springer, 1990.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics, Volume I: Functional Analysis, revised and enlarged ed., Academic Press, 1980.
• B. Simon, Trace Ideals and Their Applications, 2nd ed., American Mathematical Society, 2005.
• G. Teschl, Mathematical Methods in Quantum Mechanics, 2nd ed., American Mathematical Society, 2014.
• J. Weidmann, Linear Operators in Hilbert Spaces, Springer, 1980.