# symmetry-collision-shared-scale.mp4

Chapter section: From Wave Mechanics to Quantum Mechanics

Caption (verbatim):

A shared conversion factor makes phase and action select the same collision

Image description (verbatim):

Varying a candidate collision point leaves phase and action stationary at different points for unequal conversion factors, and at the same point for a shared factor

## Before the animation

Chapter lines 1276–1281:

```math
\begin{aligned}
\delta\phi_1&=\epsilon,\qquad\delta\phi_2=-\epsilon,\\
\delta S&=a_1\delta\phi_1+a_2\delta\phi_2=(a_1-a_2)\epsilon.
\end{aligned}
```

Chapter lines 1283–1283:

If the collision changes a particle’s speed or direction, we can shift the candidate collision point so that its phase changes to first order. Choose such a shift, so \(\epsilon\ne0\). For the action to be stationary at the same point, $a_1=a_2$. The shared conversion factor is $\hbar$.

## After the animation

Chapter lines 1291–1291:

We can measure its numerical value in the lab by comparing mass, as measured through collisions, to wavelength, as measured through interference. We then find:

Chapter lines 1293–1295:

```math
\frac{S}{\phi}:=\hbar\approx 1.055\times10^{-34}\,\mathrm{J\,s}.
```