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Animation 18
Unitarity of position and wave-number translations
symmetry-ccr-unitarity.mp4
These are sparse samples decoded from the current MP4, not newly rendered illustrations. They can support checks of the sampled states and labels, but cannot establish continuous motion, timing, transitions, or the absence of problems between samples. Use the full MP4 when judging those properties.
1440 × 1040 · 30 fps · 17 s · 510 frames · 12 samples
MP4 SHA-256 b101e244c9651fa1e0979a1ac7450bd9234d09251a78efd750a888f8bfca636d
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Chapter context
Section: The Heisenberg Symmetry Group. Excerpts are verbatim; line numbers refer to the included chapter markdown.
Caption
Unitarity of position and wave-number translations
Image description
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
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.
Generation source
Main script --render writes the MP4; --check and --encoded-check write mathematical and decoded-video validation. The renderer is self-contained apart from external NumPy/Pillow/imageio-ffmpeg libraries.
Mapping evidence and limits
NAME and output path match. content/drafts/animations/symmetry-ccr-unitarity-validation.json records 1440 x 1040, 30 fps, 510 frames, 17 s; the matching encoded-validation.json records the same geometry and frame count and compares decoded frames with source renders.
Source SHA-256 values identify the exact downloadable bytes in this packet. The source mapping and recorded checks explain the likely generation pipeline; they do not prove that these exact source bytes produced the movie. A GitHub link pinned to a commit is provided only when the delivered source bytes exactly match that path at the build's Git HEAD.
- scripts/generate_symmetry_ccr_unitarity.py · GitHub at 974f9de35254
SHA-256 4fca981a5cf3731c12d09aaf28e4c5bbb8aafc528af6fd00a43f42f5eaddef8d
Existing generator checks (2 reports)
These are existing author-produced generator reports, copied without changes. Their checks were not rerun for this packet and are not independent certification. A report may describe an earlier generation run; inspect its contents before applying its claims to the current movie.
- symmetry-ccr-unitarity-validation.json
SHA-256 aaa39897c718ba34f0d3768436f21949a53ee3b77dfa0a30b01f4c99ecf463cd - symmetry-ccr-unitarity-encoded-validation.json
SHA-256 5e7d6deb56ef5d4bbb7920cabf4386df820e3d62c26ba4c784752a9d7bbcb88e
Decoded contact sheet
Extraction method: Twelve evenly spaced decoded frame indices, including first and last. Native-resolution JPEGs from the encoded MP4; timestamps read from FFmpeg showinfo. No source rerendering. Frame indices are zero-based. Sparse samples do not establish continuous motion or capture every transition.. Frame indices are zero-based.
Full-size sampled frames
