# symmetry-ccr-unitarity.mp4

Chapter section: The Heisenberg Symmetry Group

Caption (verbatim):

Unitarity of position and wave-number translations

Image description (verbatim):

Four panes show two wave functions translating in x or k while their overlap is preserved, with schematic state-space projections for a real, positive overlap

## Before the animation

Chapter lines 869–869:

This invariance, as we've seen, is called unitarity. It preserves the distinguishability of states under symmetry transformations and makes those transformations reversible. When time evolution is itself unitary, the evolution of the state is deterministic and reversible. Translations in position and wave number preserve the inner product:

Chapter lines 871–883:

```math
\begin{gathered}
\text{Translation by }a\text{ in }x\\[0.5em]
\langle T_x(a)\psi,T_x(a)\chi\rangle\\
=\langle\psi,\chi\rangle
\end{gathered}
\qquad
\begin{gathered}
\text{Translation by }b\text{ in }k\\[0.5em]
\langle T_k(b)\psi,T_k(b)\chi\rangle\\
=\langle\psi,\chi\rangle
\end{gathered}
```

## After the animation

Chapter lines 891–891:

In addition to position and wave number, waves have a third independent way of changing. A wave's **phase**, $\phi$, refers to where it is in its cyclic pattern. For example, a phase shift of $2\pi$, or one full "cycle," returns the wave to its exact initial state. We need to be a bit careful here. For a pure mode, shifting its position is indistinguishable from shifting its phase, somewhat in the way the turning of a barbershop sign appears as though its stripes are moving up and down. We might, then, be tempted to think there is no difference between position and phase shifts. But the single mode is an idealization. In the general case, in which the wave function is a packet composed of modes, position translation shifts the entire function. Phase translation shifts each mode by the same fraction of its cycle, changing the function while leaving its magnitude envelope unchanged.