---
tag: QPH28QH036
kind: concept
name: Bargmann transform
msc: [44A15, 30H20, "46E22"]
specializes: [YCK709QB06]
sources:
  - url: https://en.wikipedia.org/wiki/Segal%E2%80%93Bargmann_space
---

## Summary

The Bargmann transform is a unitary map from $L^2(\R)$ onto the Segal–Bargmann space, the Hilbert space of [[entire functions|0G4F0XB9X2]] $F$ on $\C$ with $\int_\C \abs{F(z)}^2 e^{-\abs{z}^2}\, dA(z) < \infty$.
It sends the [[Hermite functions|52VS07H6GK]] to the normalized monomials $z^n / \sqrt{n!}$.
It turns the creation and annihilation operators of the harmonic oscillator into multiplication by $z$ and differentiation in $z$.
This makes the oscillator very simple.
It is closely related to the short-time Fourier transform with a Gaussian window.

## Explanations

### L1 human

A function on the real line is turned into a holomorphic function on the complex plane.
The real part of $z$ plays the role of position and the imaginary part the role of momentum, so the transform shows a quantum state in phase space.
The Gaussian $e^{-x^2/2}$, the ground state of the harmonic oscillator, becomes the constant function $1$.
Higher energy states become the powers $z^n$, up to a factor.