Fourier Analysis and Distributions

Fourier analysis builds a function from waves. A broad range of wavelengths can combine into a narrow pulse; a nearly single wavelength produces a long wave train. In quantum mechanics, the same calculation changes a position wavefunction into a momentum wavefunction.

Distributions extend this language to idealized point sources and plane waves. Start with Fourier transforms and packets; return to the distribution pages when a delta function or a singular integral enters your calculation.

Enter this chapter

• For a first calculation: review Complex Exponentials if needed, then read Fourier Transform and work through its Gaussian example. Wave Packets shows what the transform means for localization and motion.
• For a particle in a box or on a ring: start with Fourier Series. The boundary conditions select a discrete set of waves.
• For continuous momentum states: read Delta Function, then Distributions. Their defining rule is to test a singular expression inside an integral.
• For singular response and scattering integrals: continue to Distributional Derivatives and Principal Value Distributions.

Keep the transform and inverse together

For position $x$ and physical momentum $p$, we use

$$\phi(p)=\frac{1}{\sqrt{2\pi\hbar}} \int e^{-ipx/\hbar}\psi(x)\,dx,$$ $$\psi(x)=\frac{1}{\sqrt{2\pi\hbar}} \int e^{ipx/\hbar}\phi(p)\,dp.$$

The signs are opposite, and both prefactors are the same. The exponent is dimensionless. Plancherel’s theorem ensures that $\int|\psi|^2dx=\int|\phi|^2dp$: these are two descriptions of the same normalized state.

Mathematical tables often use wave number $k=p/\hbar$, sometimes with different prefactors. Fourier Transform Tables for QM explains the conversion; Fourier Transform Conventions gives a compact reminder.

Find the calculation you need

Page	Central question
Periodic Functions and Fourier Series	How are periodic functions expanded in discrete orthogonal modes?
Fourier Transform	How is a position-space function decomposed into momentum components?
Inverse Fourier Transform	Under which convention and hypotheses is the original function reconstructed?
Plancherel and Parseval Theorems	Why are norms and inner products preserved?
Convolution	When does a sliding integral become multiplication after transformation?
Wave Packets	How do normalizable superpositions balance spatial and momentum localization?
Position and Momentum Representations	How do states and operators look in momentum space?
Delta Function	How is ideal localization defined under an integral?
Distributions	Which test-function framework makes generalized functions precise?
Distributional Derivatives	How do derivatives encode jumps and point sources?
Principal Value Distributions	How are singular integrals and $i0$ prescriptions interpreted?
Poisson Summation Formula	How are real-space and reciprocal-space lattice sums related?
Fourier Transform Tables for QM	How are transform pairs translated safely into the local convention?

Check your understanding

Suppose you move a packet to the right without changing its shape: $\psi_a(x)=\psi(x-a)$. Does its momentum probability distribution change?

Solution

Set $u=x-a$ in the forward transform. This gives $\phi_a(p)=e^{-ipa/\hbar}\phi(p)$. The extra factor has modulus one, so $|\phi_a(p)|^2=|\phi(p)|^2$. The displacement is stored in relative phases of momentum components, not in their probabilities.

For sampled functions on a computer, continue to the Fast Fourier Transform. A finite grid introduces spacing, window-size, and periodicity assumptions in addition to the continuous formulas above.

References

• Folland, Gerald B. Fourier Analysis and Its Applications. American Mathematical Society, 2009. Fourier series, transforms, and convergence.
• Lighthill, M. J. Introduction to Fourier Analysis and Generalised Functions. Cambridge University Press, 1958. A concise introduction to distributions through Fourier calculations.
• Stein, Elias M., and Rami Shakarchi. Fourier Analysis: An Introduction. Princeton University Press, 2003. Proofs and examples for the mathematical theory.