# symmetry-amplitude-probability.mp4

Chapter section: From Wave Mechanics to Quantum Mechanics

Caption (verbatim):

From intensity to position measurement probability density

Image description (verbatim):

A complex amplitude and its squared magnitude predict the distribution of repeated position measurements at two wavelengths

## Before the animation

Chapter lines 1231–1235:

```math
\rho(x)=|\psi(x)|^2,
\qquad
\int_{-\infty}^{\infty}\rho(x)\,dx=1.
```

Chapter lines 1237–1237:

We could write the same relationship in the wave-number representation. In either case, the interpretation is as follows. A state is a superposition of the eigenstates in the chosen basis. An ideal measurement leaves the state in a specific eigenstate of the measured operator. The value of the measurement is the eigenvalue of that state. The squared magnitudes of the coefficients multiplying those eigenstates determine the outcome probabilities. They must sum to one for discrete outcomes. For continuous quantities such as position and wave number, they give probability densities that integrate to one. In this view, we may say, loosely, and only if we are so inclined, that the thing that "is" is a wave packet, and the wave-number distribution we measure is given by the squared magnitudes of its Fourier components.

## After the animation

Chapter lines 1245–1245:

To connect our wave description to mechanics, we need to relate phase to action.

Chapter lines 1247–1247:

Action, the quantity extremized by a physically valid path, can be constructed from the structure of **spacetime**, as articulated in the theory of special relativity, which will be the topic of our next chapter. Crudely speaking, because the quantity to be extremized must be agreed upon by all observers, it is natural that it should be an invariant of symmetry actions on spacetime. This leads to the result that the action is, in free motion, for massive bodies, proportional to an invariant built from translations — the time elapsed along a path as measured in a body's rest frame — times a dual invariant built from time and space translation generators. The former quantity is called **proper time** while the latter is the body's **mass**.