Quantum Mechanics and Functional Analysis
The Mathematical Framework
Quantum mechanics is, at its core, linear algebra in infinite dimensions. The natural mathematical setting is functional analysis — the study of vector spaces of functions and linear operators on them.
This is not just a convenient language; it’s the only consistent way to formulate quantum mechanics. The strange features of quantum theory (superposition, uncertainty, measurement) all emerge naturally from this mathematical structure.
Hilbert Spaces: The Arena of Quantum Mechanics
Definition
A Hilbert space \mathcal{H} is a complete inner product space. This means:
- Vector space: You can add states and multiply by complex numbers
- Inner product: A map \langle \cdot | \cdot \rangle: \mathcal{H} \times \mathcal{H} \to \mathbb{C} satisfying:
- \langle \phi | \psi \rangle = \overline{\langle \psi | \phi \rangle} (conjugate symmetry)
- \langle \phi | a\psi_1 + b\psi_2 \rangle = a\langle \phi | \psi_1 \rangle + b\langle \phi | \psi_2 \rangle (linearity)
- \langle \psi | \psi \rangle \geq 0, with equality iff \psi = 0 (positive definite)
- Complete: Every Cauchy sequence converges
The Norm and Distance
The inner product induces a norm:
|\psi| = \sqrt{\langle \psi | \psi \rangle}
And a distance:
d(\psi, \phi) = |\psi - \phi|
Completeness means: if |\psi_n - \psi_m| \to 0, then there exists \psi \in \mathcal{H} with |\psi_n - \psi| \to 0.
Key Examples
| Hilbert Space | Elements | Inner Product | Physical Use | |
|---|---|---|---|---|
| \mathbb{C}^n | Column vectors | \langle \phi | \psi \rangle = \sum_i \bar{\phi}_i \psi_i</span> | Spin, finite systems |
| L^2(\mathbb{R}) | Square-integrable functions | \langle \phi | \psi \rangle = \int_{-\infty}^{\infty} \bar{\phi}(x) \psi(x) \, dx</span> | Position space |
| L^2(\mathbb{R}^3) | Functions on 3D space | \langle \phi | \psi \rangle = \int_{\mathbb{R}^3} \bar{\phi}(\mathbf{x}) \psi(\mathbf{x}) \, d^3x</span> | 3D quantum mechanics |
| \ell^2 | Square-summable sequences | \langle \phi | \psi \rangle = \sum_{n=0}^{\infty} \bar{\phi}_n \psi_n</span> | Harmonic oscillator |
States as Vectors
The First Postulate
A quantum state is a ray in a Hilbert space.
A ray is an equivalence class of vectors differing only by a nonzero complex scalar:
[\psi] = { c\psi : c \in \mathbb{C}, c \neq 0 }
We typically work with normalized representatives:
|\psi| = 1 \quad \Leftrightarrow \quad \langle \psi | \psi \rangle = 1
Even then, an overall phase e^{i\theta}\psi represents the same physical state.
Superposition
Since \mathcal{H} is a vector space, if \psi_1 and \psi_2 are states, so is:
\psi = c_1 \psi_1 + c_2 \psi_2
This is the superposition principle — it’s simply linearity of the vector space.
Dirac Notation
Physicists use Dirac’s bra-ket notation:
| Notation | Meaning | ||
|---|---|---|---|
| \psi\rangle</span> | A vector (ket) in \mathcal{H} | ||
| \langle\phi | </span> | A dual vector (bra) in \mathcal{H}^* | |
| \langle\phi | \psi\rangle</span> | Inner product | |
| \psi\rangle\langle\phi | </span> | Outer product (an operator) |
The Riesz representation theorem guarantees that every continuous linear functional on \mathcal{H} has the form \psi \mapsto \langle \phi | \psi \rangle for some unique \phi \in \mathcal{H}.
Observables as Self-Adjoint Operators
The Second Postulate
Observables are self-adjoint operators on the Hilbert space.
An operator A: \mathcal{D}(A) \to \mathcal{H} is:
- Linear: A(c_1\psi_1 + c_2\psi_2) = c_1 A\psi_1 + c_2 A\psi_2
- Self-adjoint: A = A^\dagger and \mathcal{D}(A) = \mathcal{D}(A^\dagger)
The adjoint A^\dagger is defined by:
\langle \phi | A\psi \rangle = \langle A^\dagger \phi | \psi \rangle
for all \psi \in \mathcal{D}(A), \phi \in \mathcal{D}(A^\dagger).
Why Self-Adjoint?
Self-adjoint operators have:
- Real eigenvalues — measurement outcomes must be real numbers
- Orthogonal eigenvectors — distinct outcomes are distinguishable
- Spectral theorem — complete description of the operator
Key Examples
| Observable | Operator | Domain | ||
|---|---|---|---|---|
| Position | \hat{x}\psi(x) = x\psi(x) | {\psi : \int | x\psi(x) | ^2 dx < \infty}</span> |
| Momentum | \hat{p}\psi(x) = -i\hbar \frac{d\psi}{dx} | {\psi : \psi’ \in L^2} | ||
| Energy | \hat{H} = -\frac{\hbar^2}{2m}\nabla^2 + V(x) | Depends on V | ||
| Angular momentum | \hat{L}_z = -i\hbar \frac{\partial}{\partial \phi} | Periodic functions |
Bounded vs Unbounded Operators
In finite dimensions, all linear operators are bounded (continuous). In infinite dimensions:
| Type | Definition | Examples |
|---|---|---|
| Bounded | |A\psi| \leq C|\psi| for some C | Projections, unitary operators |
| Unbounded | No such C exists | Position, momentum, Hamiltonian |
Most physical observables are unbounded. This is why we need the domain \mathcal{D}(A) — the operator is only defined on a dense subspace.
The Spectral Theorem
For Bounded Self-Adjoint Operators
If A is bounded and self-adjoint:
A = \int_{\sigma(A)} \lambda \, dE(\lambda)
where:
- \sigma(A) \subseteq \mathbb{R} is the spectrum (generalized eigenvalues)
- E(\lambda) is the spectral measure (projection-valued measure)
For Unbounded Self-Adjoint Operators
The same formula holds, but with more care about domains. The spectrum can be:
| Type | Definition | Example |
|---|---|---|
| Point spectrum | True eigenvalues: A\psi = \lambda\psi | Bound states of hydrogen |
| Continuous spectrum | No eigenvectors, but (A - \lambda)^{-1} unbounded | Free particle momentum |
| Residual spectrum | (Doesn’t occur for self-adjoint operators) | — |
Physical Interpretation
The spectral theorem says:
\langle \psi | A | \psi \rangle = \int_{\sigma(A)} \lambda \, d\langle \psi | E(\lambda) | \psi \rangle
The measure \mu_\psi(\lambda) = \langle \psi | E(\lambda) | \psi \rangle gives the probability distribution of measurement outcomes.
The Canonical Commutation Relation
Position and Momentum
The operators \hat{x} and \hat{p} satisfy:
[\hat{x}, \hat{p}] = \hat{x}\hat{p} - \hat{p}\hat{x} = i\hbar
This is the canonical commutation relation (CCR).
Proof in Position Representation
[\hat{x}, \hat{p}]\psi = x(-i\hbar\psi’) - (-i\hbar)(x\psi)’ = -i\hbar x\psi’ + i\hbar\psi + i\hbar x\psi’ = i\hbar\psi
The Stone-von Neumann Theorem
This fundamental theorem states: up to unitary equivalence, there is only one irreducible representation of the CCR.
In other words, any quantum system with one degree of freedom where [\hat{x}, \hat{p}] = i\hbar is unitarily equivalent to the standard position representation on L^2(\mathbb{R}).
The Uncertainty Principle
For any two observables A, B:
\Delta A \cdot \Delta B \geq \frac{1}{2}|\langle [A, B] \rangle|
where \Delta A = \sqrt{\langle A^2 \rangle - \langle A \rangle^2} is the standard deviation.
For position and momentum:
\Delta x \cdot \Delta p \geq \frac{\hbar}{2}
This is not a measurement limitation — it’s a fundamental property of the Hilbert space structure.
Time Evolution: Unitary Operators
The Third Postulate
Time evolution is unitary.
A unitary operator U satisfies:
U^\dagger U = U U^\dagger = I
Equivalently: U preserves inner products:
\langle U\phi | U\psi \rangle = \langle \phi | \psi \rangle
This ensures:
- Probabilities are conserved
- Normalization is preserved
- Evolution is reversible
Stone’s Theorem
Every strongly continuous one-parameter unitary group U(t) has the form:
U(t) = e^{-iHt/\hbar}
for some self-adjoint operator H (the Hamiltonian).
Conversely, every self-adjoint H generates a unitary group.
The Schrödinger Equation
Differentiating |\psi(t)\rangle = U(t)|\psi(0)\rangle:
i\hbar \frac{d}{dt}|\psi(t)\rangle = H|\psi(t)\rangle
This is the Schrödinger equation — it emerges automatically from Stone’s theorem!
Measurement: Projection Operators
The Fourth Postulate
Measurement of observable A with outcome \lambda projects the state onto the eigenspace.
If A has discrete spectrum with eigenvectors |a_n\rangle:
A|a_n\rangle = a_n|a_n\rangle
The probability of outcome a_n is:
P(a_n) = |\langle a_n | \psi \rangle|^2
After measurement, the state becomes:
|\psi\rangle \to \frac{P_n|\psi\rangle}{|P_n|\psi\rangle|}
where P_n = |a_n\rangle\langle a_n| is the projection onto the eigenspace.
Projection Operators
A projection P satisfies:
- P^2 = P (idempotent)
- P^\dagger = P (self-adjoint)
Projections correspond to yes/no questions about the system.
Resolution of the Identity
For an observable A with complete eigenbasis:
\sum_n |a_n\rangle\langle a_n| = I
or in the continuous case:
\int |a\rangle\langle a| \, da = I
This expresses completeness: every state can be expanded in the eigenbasis.
Tensor Products: Composite Systems
Definition
For two Hilbert spaces \mathcal{H}_1, \mathcal{H}_2, the tensor product \mathcal{H}_1 \otimes \mathcal{H}_2 is the Hilbert space of the composite system.
If {|e_i\rangle} is a basis for \mathcal{H}_1 and {|f_j\rangle} for \mathcal{H}_2, then {|e_i\rangle \otimes |f_j\rangle} is a basis for \mathcal{H}_1 \otimes \mathcal{H}_2.
Product States vs Entangled States
A product state has the form:
|\psi\rangle = |\psi_1\rangle \otimes |\psi_2\rangle
An entangled state cannot be written this way:
|\Psi\rangle = \frac{1}{\sqrt{2}}(|0\rangle \otimes |1\rangle - |1\rangle \otimes |0\rangle)
Entanglement is a purely quantum phenomenon with no classical analog.
Partial Trace
For a state \rho on \mathcal{H}_1 \otimes \mathcal{H}_2, the partial trace gives the reduced state on subsystem 1:
\rho_1 = \text{Tr}_2(\rho)
This describes what observer 1 sees without access to system 2.
Summary: The Mathematical Structure
| Physics Concept | Mathematical Object | |||
|---|---|---|---|---|
| State | Ray in Hilbert space \mathcal{H} | |||
| Observable | Self-adjoint operator on \mathcal{H} | |||
| Possible outcomes | Spectrum of the operator | |||
| Probability | \langle a | \psi \rangle | ^2</span> | |
| Expectation value | \langle \psi | A | \psi \rangle</span> | |
| Time evolution | Unitary operator e^{-iHt/\hbar} | |||
| Measurement | Projection onto eigenspace | |||
| Composite system | Tensor product \mathcal{H}_1 \otimes \mathcal{H}_2 | |||
| Uncertainty | Non-commutativity: [A, B] \neq 0 |
Key Theorems
| Theorem | Statement | Physical Meaning |
|---|---|---|
| Spectral theorem | Self-adjoint operators have real spectrum and spectral decomposition | Observables have real outcomes |
| Stone’s theorem | Unitary groups ↔ self-adjoint generators | Time evolution ↔ energy |
| Stone-von Neumann | CCR has unique irreducible representation | Position/momentum are canonical |
| Riesz representation | \mathcal{H}^* \cong \mathcal{H} | Bras and kets are dual |
| Gleason’s theorem | Probability measures come from density operators | Born rule is (almost) forced |
Why Functional Analysis?
The infinite-dimensional nature of quantum mechanics is essential, not a technicality:
- Continuous spectra: Position and momentum have continuous outcomes
- Unbounded operators: Most physical observables are unbounded
- Domain issues: \hat{x} and \hat{p} cannot both be bounded
- No finite-dimensional CCR: [A, B] = iI has no finite-dimensional representation (take trace!)
Functional analysis provides the rigorous foundation for all of this. Without it, quantum mechanics would be a collection of formal manipulations rather than a coherent mathematical theory.
References
- J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton University Press, 1955) — The classic text on rigorous QM
- M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vols. 1–4 (Academic Press, 1972–1979)
- B. C. Hall, Quantum Theory for Mathematicians (Springer, 2013)
- G. Teschl, Mathematical Methods in Quantum Mechanics, 2nd ed. (AMS, 2014) — Free online
- F. Strocchi, An Introduction to the Mathematical Structure of Quantum Mechanics, 2nd ed. (World Scientific, 2008)
- W. Rudin, Functional Analysis, 2nd ed. (McGraw-Hill, 1991)