# symmetry-action-phase-desktop.mp4

Chapter section: From Wave Mechanics to Quantum Mechanics

Caption (verbatim):

No separate caption in the chapter.

Image description (verbatim):

Three wave-number shells share the same proper-time interval while phase cycles and action accumulates at corresponding rates

## Before the animation

Chapter lines 1297–1297:

The smaller $\hbar$ is, the more phase cycles a given action difference produces. For candidate paths, we use the wave mode appropriate to each segment and add the accumulated phases, following the procedure described earlier. For two candidate paths:

Chapter lines 1299–1301:

```math
\Delta\phi=\frac{\Delta S}{\hbar}.
```

## After the animation

Chapter lines 1307–1307:

But this is exactly our condition for the path sum to be dominated by stationary paths. For everyday bodies, $\hbar$ is tiny compared with action differences so that their motion is effectively deterministic. Are we saying that the laws of motion we learn in high school physics are an approximation? Yes, a very good approximation.

Chapter lines 1309–1309:

We can now express our $H_3$ commutation relation in units of action: