# symmetry-ccr-x-k-translations-symmetric.mp4

Chapter section: The Heisenberg Symmetry Group

Caption (verbatim):

Wave Packet translated in position and momentum space

Image description (verbatim):

Separate translations in position and wave number, each shown in its own representation

## Before the animation

Chapter lines 839–839:

We can understand Fourier analysis in terms of the symmetry group that acts on wave functions, the Heisenberg group. This group's actions preserve the overlaps between states, and therefore their distinguishability. They do not in general preserve the behavior of those states as they evolve. In this sense, they are symmetries of state space, whether or not they are also dynamical symmetries of a particular system. This group not only underlies Fourier analysis, but in defining similarity and distinguishability of wave functions, supplies an essential ingredient for a logically viable notion of state.

Chapter lines 841–841:

We can translate a wave function either in $x$-space or in $k$-space. While shifting the wave number isn't a translation in the familiar physical space we live in, from a mathematical perspective, $k$-space is the dual, or equivalent up to role reversal, of $x$-space. 

## After the animation

Chapter lines 849–849:

The overlap between two complex wave functions is their **inner product**:

Chapter lines 851–855:

```math
\langle\psi_1,\psi_2\rangle
=
\int_{-\infty}^{\infty}\psi_1^*(x)\psi_2(x)\,dx.
```