Fourier analysis reference

The physical content of the Fourier transform in one sentence: decompose a function into a superposition of waves of different frequencies. In quantum mechanics, “frequency” is momentum ($p=\hbar k$), so the Fourier transform is precisely the change-of-basis formula between the position and momentum representations.

Where the main text needs this: 2.10 The free particle uses it to build wave packets; 3.3 Hilbert space interprets it as an expansion in a continuous basis; 7.6 Fermi’s golden rule and the Chapter 11 path integral lean heavily on Gaussian integrals; appendix C.2 The split-operator method uses the FFT to hop back and forth between the two representations.

Transform conventions

The same “Fourier transform” differs between books by a constant or a sign in the exponent, so let us nail down this book’s convention first.

Mathematical convention (symmetric version, $k$ is the wavenumber):

$$\tilde f(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} f(x)\,\ee^{-\ii kx}\,\dd x, \qquad f(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \tilde f(k)\,\ee^{\ii kx}\,\dd k\tag{A.2.1}$$

Quantum-mechanical convention ($p=\hbar k$, momentum wavefunction written $\phi(p)$):

$$\phi(p)=\frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{\infty}\psi(x)\,\ee^{-\ii px/\hbar}\,\dd x, \qquad \psi(x)=\frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{\infty}\phi(p)\,\ee^{\ii px/\hbar}\,\dd p\tag{A.2.2}$$

The prefactor $1/\sqrt{2\pi\hbar}$ is there so that both sides are normalised simultaneously (see Parseval below). In Dirac notation this is nothing but inserting a complete set of momentum eigenstates: $\phi(p)=\braket{p}{\psi}$, with $\braket{x}{p}=\ee^{\ii px/\hbar}/\sqrt{2\pi\hbar}$.

The δ function

The Dirac δ function is not an ordinary function but a “value picker”: whenever it appears inside an integral, it pulls out the value of the integrand at the position of the spike:

$$\int_{-\infty}^{\infty} f(x)\,\delta(x-x_0)\,\dd x=f(x_0)\tag{A.2.3}$$

Picture it as a limiting spike whose width goes to zero while its area stays 1. Its most important representation is a superposition of plane waves:

$$\delta(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\ee^{\ii kx}\,\dd k\tag{A.2.4}$$

— plane waves of every wavenumber added with equal weight, cancelling everywhere and piling up in phase only at $x=0$. This formula is the origin of the continuous-basis orthonormality relations $\braket{x}{x'}=\delta(x-x')$ and $\braket{p}{p'}=\delta(p-p')$.

Frequently used properties:

Property	Formula
Scaling	$\delta(ax)=\delta(x)/\lvert a\rvert$
Composite argument	$\delta(g(x))=\sum_i \delta(x-x_i)/\lvert g'(x_i)\rvert$, where the $x_i$ are the simple roots of $g$
Derivative	$\int f(x)\,\delta'(x-x_0)\,\dd x=-f'(x_0)$
Step function	$\dd\Theta/\dd x=\delta(x)$

Table of standard transform pairs

Listed in the mathematical (symmetric) convention. To switch to the ħ convention, set $k\to p/\hbar$ and multiply the result by $1/\sqrt{\hbar}$.

$f(x)$	$\tilde f(k)$	Remarks
$\ee^{-x^2/2\sigma^2}$	$\sigma\,\ee^{-\sigma^2k^2/2}$	Gaussian stays Gaussian; widths are reciprocal, $\sigma_x\sigma_k=1$
$\ee^{ik_0x}$	$\sqrt{2\pi}\,\delta(k-k_0)$	Plane wave = a single momentum
$\delta(x-x_0)$	$\ee^{-\ii kx_0}/\sqrt{2\pi}$	A position eigenstate contains every momentum
Square pulse (equal to 1 for $\lvert x\rvert<a$)	$\sqrt{2/\pi}\,\dfrac{\sin ka}{k}$	sinc-shaped; the mathematical prototype of single-slit diffraction
$\ee^{-a\lvert x\rvert}$	$\sqrt{2/\pi}\,\dfrac{a}{a^2+k^2}$	Lorentzian; exponential decay ⇔ power-law tails
$f(x-x_0)$	$\ee^{-\ii kx_0}\tilde f(k)$	Translation ⇔ attaching a phase
$f'(x)$	$\ii k\tilde f(k)$	Differentiation ⇔ multiplying by $\ii k$; the algorithmic basis of C.2

The first row deserves a second look: the narrower a Gaussian wave packet (small $\sigma$), the wider its momentum distribution — this is the uncertainty principle showing up at the Fourier level, and the Gaussian saturates the bound exactly, $\sigma_x\sigma_p=\hbar/2$.

Parseval’s theorem and the convolution theorem

Parseval’s (Plancherel’s) theorem: the “total amount” is unchanged by the transform,

$$\int_{-\infty}^{\infty}\lvert f(x)\rvert^2\,\dd x=\int_{-\infty}^{\infty}\lvert\tilde f(k)\rvert^2\,\dd k\tag{A.2.5}$$

Physical translation: if $\psi(x)$ is normalised, then $\phi(p)$ is automatically normalised too — total probability is 1 in the position representation and still 1 after switching to momentum. That is why $\lvert\phi(p)\rvert^2$ can be directly interpreted as the probability density for momentum.

Convolution theorem: the convolution of two functions, $(f*g)(x)=\int f(y)\,g(x-y)\,\dd y$ (shift $g$ everywhere and stack the copies weighted by $f$), becomes a product after transforming:

$$\widetilde{f*g}(k)=\sqrt{2\pi}\,\tilde f(k)\,\tilde g(k)\tag{A.2.6}$$

The use: convolutions are hard, products are easy, so “transform — multiply — transform back” is the standard play. Numerically this is exactly how FFT convolution works, and it is what the C.2 split-operator method does at every step.

Gaussian integrals

The most frequently occurring family of integrals in quantum mechanics. The basic formula ($\operatorname{Re}a>0$):

$$\int_{-\infty}^{\infty}\ee^{-ax^2}\,\dd x=\sqrt{\frac{\pi}{a}}\tag{A.2.7}$$

Gaussian integral with a linear term: completing the squarebasic~5 minexpand

The target:

$$I=\int_{-\infty}^{\infty}\ee^{-ax^2+bx}\,\dd x\tag{A.2.8}$$

Complete the square. Write the exponent as a perfect square:

$$-ax^2+bx=-a\left(x-\frac{b}{2a}\right)^2+\frac{b^2}{4a}\tag{A.2.9}$$

Shift the integration variable $u=x-b/2a$ (the limits are unchanged) and pull out the constant factor:

$$I=\ee^{b^2/4a}\int_{-\infty}^{\infty}\ee^{-au^2}\,\dd u =\sqrt{\frac{\pi}{a}}\;\ee^{b^2/4a}\tag{A.2.10}$$

This one formula handles most situations: $b$ may be complex (for instance $b=\ii k$ — this is exactly where “the Fourier transform of a Gaussian is a Gaussian” comes from), and $a$ may carry an imaginary part (in the path integral $a\propto\ii$, i.e. the Fresnel integral; the result has the same form, understood as convergent in the limiting sense).

Versions with polynomial factors are generated by “differentiating with respect to the parameter”:

$$\int x^2\,\ee^{-ax^2}\dd x=-\frac{\partial}{\partial a}\int\ee^{-ax^2}\dd x =\frac{1}{2a}\sqrt{\frac{\pi}{a}}\tag{A.2.11}$$

Odd powers give $\int x^{2n+1}\ee^{-ax^2}\dd x=0$ (an odd function integrated over a symmetric interval vanishes).

∎Key formulas

• Momentum representation (ħ convention)

  $$\phi(p)=\frac{1}{\sqrt{2\pi\hbar}}\int\psi(x)\,\ee^{-\ii px/\hbar}\,\dd x$$

  This is ⟨p|ψ⟩ — inserting the complete momentum basis

• Plane-wave form of the δ function

  $$\delta(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\ee^{\ii kx}\,\dd k$$

  Origin of continuous-basis orthonormalisation

• Parseval's theorem

  $$\int|\psi(x)|^2\dd x=\int|\phi(p)|^2\dd p$$

  Position-normalised ⇔ momentum-normalised; only then does |φ(p)|² have a probability reading

• Convolution theorem

  $$\widetilde{f*g}=\sqrt{2\pi}\,\tilde f\,\tilde g$$

  When a convolution is hard, multiply in Fourier space instead

• Gaussian integral

  $$\int_{-\infty}^{\infty}\ee^{-ax^2+bx}\dd x=\sqrt{\frac{\pi}{a}}\,\ee^{b^2/4a}$$

  Complete the square + shift; b may be complex — the basic brick of the path integral

?Self-check3 questions

1. 1.

   A Gaussian wave packet becomes narrower in position space. What happens to its momentum distribution?

   • It broadens — Fourier widths are reciprocal
   • It narrows too — the two move together
   • It stays the same — normalisation pins the distribution down
   • It depends on the central momentum of the packet

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2. 2.

   What is the direct physical meaning of Parseval's theorem in quantum mechanics?

   • A wavefunction normalised in the position representation is automatically normalised in the momentum representation
   • Energy is conserved under the Fourier transform
   • The product of position and momentum uncertainties has a lower bound
   • The group velocity of a wave packet equals the particle velocity

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3. 3.

   What is ∫ f(x) δ(2x) dx?

   • f(0)/2
   • f(0)
   • 2f(0)
   • 0

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