# symmetry-complex-phase-modes.mp4

Chapter section: The Heisenberg Symmetry Group

Caption (verbatim):

Visualizing Phase Change

Image description (verbatim):

Nine complex modes and their exact sum rotate through five phase turns while their magnitude envelopes remain fixed

## Before 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.

## After the animation

Chapter lines 899–899:

We can also discover and define phase directly from our symmetry group's commutation relations, which gives us a useful algebraic packaging of the group structure. Let's ask the question:

Chapter lines 901–903:

```math
[\hat X, \hat K] = \; ?
```