Fourier Transforms and the Dirac Delta Function

Introduction

In our previous lesson, we established a cornerstone of quantum formalism: two observables can be measured simultaneously if and only if their corresponding Hermitian operators commute. This led us to the concept of a Complete Set of Commuting Observables (CSCO), which provides a unique set of labels—quantum numbers—for a system's state. Today, we turn our attention to the profound consequences of operators that do not commute, using the canonical example of position ( $x^$ ) and momentum ( $p^$ ).

This lesson will explore the deep relationship between the position and momentum representations of a quantum state. We will demonstrate how the Fourier transform acts as the bridge between these two descriptions. Along the way, we will encounter eigenfunctions that are not part of the standard Hilbert space $L^{2} (R)$ and introduce the Dirac delta function to handle their "orthonormality." To place these concepts on a firm mathematical footing, which will resonate with your background in both physics and mathematics, we will go beyond the standard undergraduate treatment and interpret these objects within the more powerful framework of the Rigged Hilbert Space (RHS), also known as a Gelfand triplet.

The Challenge of Continuous Spectra: Position and Momentum

Let's begin in the familiar position representation, where a state is described by a wave function $ψ (x)$ . The position operator $x^$ simply acts by multiplication, $x^ψ (x) = x ψ (x)$ , and the momentum operator is given by $p^= - i ℏ \frac{d}{d x}$ .

A central task in quantum mechanics is to find the eigenfunctions and eigenvalues of these operators. For the momentum operator, the eigenvalue equation is:

p^ψ_{p} (x) = - i ℏ \frac{d}{d x} ψ_{p} (x) = p ψ_{p} (x)

The solutions are the complex exponentials, or plane waves:

ψ_{p} (x) = C e^{i p x /ℏ}

where $C$ is a normalization constant. Here we encounter a fundamental problem. These eigenfunctions are not square-integrable over the real line:

\int_{- \infty}^{\infty} ∣ ψ_{p} (x) ∣^{2} d x = ∣ C ∣^{2} \int_{- \infty}^{\infty} 1 d x \to \infty

This means the momentum eigenfunctions do not belong to the Hilbert space $L^{2} (R)$ . This is a significant issue because the spectral theorem, as we have discussed it for discrete spectra, relies on eigenvectors that are members of the Hilbert space. How, then, can we represent a state in terms of its momentum components? The answer lies in generalizing the concept of a basis expansion from a discrete sum to a continuous integral.

From Fourier Series to Fourier Transform

The most natural way to understand this continuous expansion is to start with a system constrained to a finite space and then expand that space to infinity. The following video provides an excellent walkthrough of this idea.

To Understand the Fourier Transform, Start From Quantum Mechanics

The video "To Understand the Fourier Transform, Start From Quantum Mechanics" by Physics with Elliot masterfully builds the Fourier transform from the ground up, starting with a particle on a circle.

First, watch the segment on a particle on a circle. This shows how periodic boundary conditions lead to a discrete set of allowed momenta and how any wave function can be expanded as a Fourier series in terms of the corresponding momentum eigenfunctions. Next, see how this picture changes as the circle's radius goes to infinity. The segment on the infinite radius limit demonstrates how the discrete sum of the Fourier series becomes the integral of the Fourier transform, and the discrete momenta merge into a continuum. Finally, focus on the probabilistic interpretation. This part clarifies that the position-space wave function \psi(x) and the momentum-space wave function \hat{\psi}(k) are just two different representations of the same underlying quantum state, containing identical information.

This derivation shows that an arbitrary state $ψ (x)$ can be decomposed into a superposition of momentum eigenfunctions $e^{ik x}$ (where $p = ℏ k$ ), with the weighting function given by its Fourier transform, $ψ^(k)$ . The pair of equations connecting the position and momentum representations are:

Fourier Transform (Position to Momentum):

ψ^(k) = \frac{1}{2 π} \int_{- \infty}^{\infty} ψ (x) e^{- ik x} d x

Inverse Fourier Transform (Momentum to Position):

ψ (x) = \frac{1}{2 π} \int_{- \infty}^{\infty} ψ^(k) e^{ik x} d k

The Fourier transform provides a reversible map between the position-space representation of a quantum state, $\psi(x)$, and its momentum-space representation, $\hat{\psi}(k)$. The factor of $1/\sqrt{2\pi}$ can be distributed differently, but this symmetric convention is common.

In Dirac notation, we identify $ψ (x) = ⟨ x ∣ ψ ⟩$ and, up to a normalization factor, $ψ^(p) = ⟨ p ∣ ψ ⟩$ . The Fourier transform is thus the mathematical tool for changing from the position basis to the momentum basis.

The Dirac Delta Function

We now need a way to express the orthonormality of these continuous basis states. In the discrete case, we had $⟨ ψ_{n} ∣ ψ_{m} ⟩ = δ_{nm}$ . What is the continuous analogue for $⟨ x ∣ x^{'} ⟩$ or $⟨ p ∣ p^{'} ⟩$ ?

Let's use the completeness relation for the position basis, $I = \int ∣ x ⟩ ⟨ x ∣ d x$ , and insert it into the representation of $ψ (x^{'})$ :

ψ (x^{'}) = ⟨ x^{'} ∣ ψ ⟩ = ⟨ x^{'} ∣ (\int ∣ x ⟩ ⟨ x ∣ d x) ∣ ψ ⟩ = \int ⟨ x^{'} ∣ x ⟩ ⟨ x ∣ ψ ⟩ d x = \int ⟨ x^{'} ∣ x ⟩ ψ (x) d x

For this equation to hold for any function $ψ (x)$ , the object $⟨ x^{'} ∣ x ⟩$ must have the property that it is zero everywhere except at $x = x^{'}$ , and its integral "picks out" the value $ψ (x^{'})$ . This object is the Dirac delta function, $δ (x - x^{'})$ .

The Dirac delta function $\delta(x)$ is defined by its sifting property: $\int f(x)\delta(x-a)dx = f(a)$. It can be heuristically visualized as the limit of a sequence of functions, like a Gaussian with a vanishing width and constant area.

While the "infinitely tall, infinitesimally narrow spike" is a useful picture, it's not mathematically rigorous. The defining characteristic of the Dirac delta is its behavior under an integral. The following video provides a more careful treatment, which is essential for a deeper understanding.

Ch 5: What are Dirac deltas and wavefunction inner products? | Maths of Quantum Mechanics

The video "What are Dirac deltas and wavefunction inner products?" from the Quantum Sense channel provides a nuanced look at this crucial mathematical tool.

Watch the introduction to see why the delta function is needed to extract coefficients from a continuous expansion. Pay close attention to the discussion of the big spike interpretation and its limitations. The video presents an example of a delta function representation (related to the Fourier transform) that does not vanish away from the origin, highlighting why the sifting property is the proper definition. Finally, see how this leads to the continuous orthonormal condition, \langle x|x' \rangle = \delta(x-x'), and the familiar integral form for inner products in a continuous basis.

With the Dirac delta, we can write the orthonormality and completeness relations for both bases:

Orthonormality: $⟨ x ∣ x^{'} ⟩ = δ (x - x^{'})$ and $⟨ p ∣ p^{'} ⟩ = δ (p - p^{'})$
Completeness: $\int ∣ x ⟩ ⟨ x ∣ d x = I$ and $\int ∣ p ⟩ ⟨ p ∣ d p = I$
Transformation: The change of basis matrix element is $⟨ x ∣ p ⟩ = \frac{1}{2 π ℏ} e^{i p x /ℏ}$ , which is just the (normalized) momentum eigenfunction in the position basis.

The Rigged Hilbert Space: A Rigorous Framework

We have now established a working formalism, but it rests on two mathematically delicate objects: non-normalizable eigenfunctions and the Dirac delta "function." Your studies in QFT have likely accustomed you to the physicist's pragmatic use of such tools. However, a rigorous foundation exists in the form of the Rigged Hilbert Space (RHS), or Gelfand triplet.

The core idea is to replace the single Hilbert space $H$ with a triplet of spaces:

Φ \subset H \subset Φ^{\times}

$H$ is the standard Hilbert space of square-integrable functions, $L^{2} (R)$ .
$Φ$ is a smaller, dense subspace of "well-behaved" functions. For quantum mechanics, this is typically the Schwartz space, $S (R)$ , which consists of infinitely differentiable functions that decrease, along with all their derivatives, faster than any inverse power of $x$ . These functions represent physically reasonable, well-localized states.
$Φ^{\times}$ is the anti-dual space to $Φ$ . It is the space of all continuous anti-linear functionals on $Φ$ . A functional is a map that takes a function from $Φ$ and returns a complex number. This dual space is larger than $H$ .

The key insight is this: The kets $∣ x ⟩$ and $∣ p ⟩$ are not elements of $H$ , but they are well-defined elements of the larger space $Φ^{\times}$ . They are not vectors in the Hilbert space, but functionals that act on the "nice" vectors in $Φ$ .

The position "eigenket" $∣ x_{0} ⟩$ is the functional that evaluates a function at $x_{0}$ : $⟨ ψ ∣ x_{0} ⟩ = ψ (x_{0})$ .
The momentum "eigenket" $∣ p_{0} ⟩$ is the functional that performs a Fourier transform: $⟨ ψ ∣ p_{0} ⟩ = \int ψ^{*} (x) e^{i p_{0} x /ℏ} d x$ .

These are called generalized eigenvectors, and the RHS framework provides a generalized spectral theorem (the Gelfand-Maurin theorem) that guarantees self-adjoint operators like $x^$ and $p^$ have a complete set of these generalized eigenvectors.

The following readings provide a formal development of these ideas. Given your mathematical inclinations, they should offer a satisfying level of rigor.

[PDF] Rigged Hilbert Space Theory for Hermitian and Quasi-Hermitian ...

This document provides a detailed mathematical introduction to Rigged Hilbert Spaces, using the momentum operator as its primary example.

Begin with the introduction to the Dirac Formulation of Quantum Mechanics on page 7. Focus on the discussion starting from the problem with continuous spectra, which explicitly points out why momentum eigenstates are not in the Hilbert space. Next, read the Introduction to the Mathematics of Rigged Hilbert Spaces on pages 8-9. This short section gives a clear, high-level summary of the Gelfand triplet \Phi \subset H \subset \Phi^\times and its purpose. For the formal details, proceed to Section 2.5.2, Nuclear Rigged Hilbert Spaces, bras, kets and its Application to Momentum, starting on page 20. Read from the beginning of the section to the end of Example 2.19 to see the formal construction using the Schwartz space S(\mathbb{R}). Finally, read the Nuclear Spectral Theorem section on pages 23-24. This theorem (Theorem 2.23 and Example 2.24) is the central result, guaranteeing the existence of a complete set of generalized eigenvectors and explicitly showing how it applies to the momentum operator via the Fourier transform.

To broaden your perspective, the next reading places the RHS in the wider context of group theory and special functions, directly addressing the position and momentum bases.

[PDF] Groups, Special Functions and Rigged Hilbert Spaces

This review article contextualizes the RHS and provides a clear summary of its application to the position and momentum representations.

Read Section 2, Rigged Hilbert Spaces (p. 2-3). This provides another clear, concise definition of the Gelfand triplet and its properties. Continue on page 4 to the discussion of the Gelfand–Maurin theorem. This section explains how the RHS allows for a rigorous spectral decomposition for operators with continuous spectra. Finally, read Section 5.1, Continuous and Discrete Bases and RHS (p. 15-16). This section is directly relevant, as it explicitly constructs the position and momentum continuous bases within the RHS framework using the Schwartz space, and then derives the relations between them, including the resolution of identity and the Fourier transform.

Conclusion

In this lesson, we have navigated the crucial relationship between the position and momentum representations, moving from intuitive pictures to a mathematically rigorous foundation.

Key Takeaways:

The eigenfunctions of operators with continuous spectra, like position and momentum, are not elements of the standard Hilbert space $L^{2} (R)$ as they are not square-integrable.
The position-space wave function $ψ (x) = ⟨ x ∣ ψ ⟩$ and momentum-space wave function $ψ ~ (p) = ⟨ p ∣ ψ ⟩$ are related by the Fourier transform, which serves as the unitary transformation between the two continuous bases.
The "orthonormality" of these continuous basis states is expressed using the Dirac delta function, such that $⟨ x ∣ x^{'} ⟩ = δ (x - x^{'})$ .
The Rigged Hilbert Space (or Gelfand triplet) $Φ \subset H \subset Φ^{\times}$ provides the rigorous mathematical framework for these ideas. In this picture, the kets $∣ x ⟩$ and $∣ p ⟩$ are interpreted as generalized eigenvectors that exist in the dual space $Φ^{\times}$ , acting as functionals on the space of "nice" physical states $Φ$ .

Having solidified our understanding of single-particle state representations, we are now prepared to tackle systems with multiple particles. In our next lesson, we will introduce the tensor product, the mathematical tool used to construct the state space of composite systems. This will lay the essential groundwork for understanding one of the most uniquely quantum phenomena: entanglement.

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