The wave equation contains the entire state of a particle.
From mathematical formulation of quantum mechanics remember that the wave equation is a vector in Hilbert space.
And a single vector can be represented in many different ways in different basis, and two of those ways happen to be the position and the momentum representations.
More importantly, position and momentum are first and foremost operators associated with observables: the position operator and the momentum operator. And both of their eigenvalue sets form a basis of the Hilbert space according to the spectral theorem.
When you represent a wave equation as a function, you have to say what the variable of the function means. And depending on weather you say "it means position" or "it means momentum", the position and momentum operators will be written differently.
This is well shown at: Video "Visualization of Quantum Physics (Quantum Mechanics) by udiprod (2017)".
Furthermore, the position and momentum representations are equivalent: one is the Fourier transform of the other: position and momentum space. Remember that notably we can always take the Fourier transform of a function in due to Carleson's theorem.
Then the uncertainty principle follows immediately from a general property of the Fourier transform: en.wikipedia.org/w/index.php?title=Fourier_transform&oldid=961707157#Uncertainty_principle
In precise terms, the uncertainty principle talks about the standard deviation of two measures.
We can visualize the uncertainty principle more intuitively by thinking of a wave function that is a real flat top bump function with a flat top in 1D. We can then change the width of the support, but when we do that, the top goes higher to keep probability equal to 1. The momentum is 0 everywhere, except in the edges of the support. Then:
- to localize the wave in space at position 0 to reduce the space uncertainty, we have to reduce the support. However, doing so makes the momentum variation on the edges more and more important, as the slope will go up and down faster (higher top, and less x space for descent), leading to a larger variance (note that average momentum is still 0, due to to symmetry of the bump function)
- to localize the momentum as much as possible at 0, we can make the support wider and wider. This makes the bumps at the edges smaller and smaller. However, this also obviously delocalises the wave function more and more, increasing the variance of x
Bibliography:
- www.youtube.com/watch?v=bIIjIZBKgtI&list=PL54DF0652B30D99A4&index=59 "K2. Heisenberg Uncertainty Relation" by doctorphys (2011)
- physics.stackexchange.com/questions/132111/uncertainty-principle-intuition Uncertainty Principle Intuition on Physics Stack Exchange
This is good. But it misses some key operations, so much so that makes Ciro not want to learn/use it daily.
Topological Quantum Computer by Professor John Preskill
. Source. Topological Quantum Computation by Jason Alicea (2021)
Source. FBI vs Un-Defuseable Bomb by Qxir (2021)
Source. Tells the story of Harvey's Resort Hotel bombing (1980)Mahayana adds a bunch of stuff on top of the Pali Canon. Most of it appears to be random mysticism. Maybe there is something good in it... maybe.
Ciro Santilli believes that molecular biology technologies will be a large part of the next big things as shown at: Section "Molecular biology technologies".
Bibliography:
- www.youtube.com/watch?v=mS563_Teges&list=PLQbPquAyEw4dQ3zOLrdS1eF_KJJbUUyBx Biophysical Techniques Course 2022 by the MRC Laboratory of Molecular Biology. Holy crap that playlist is a tour de force of molecular biology techniques in 2022!
Ciro's Edict #5 Completely rebrand Cirodown to OurBigBook Updated 2025-07-01 +Created 1970-01-01
The name cirodown should not appear anywhere now, except with very few exceptions, e.g.:
- github.com/cirosantilli/cirodown to github.com/ourbigbook/ourbigbook
- file extension from
.ciroto.bigb - the Node.js NPM package was renamed from
cirodowntoourbibook. - all in-code instances
I have also squatted
OurBigBook on all major social media handles for near future usage, e.g.: twitter.com/ourbigbook and so on.I was going to do this sooner or later, it was inevitable, but the timing was partly triggered due to noticing that English speakers (and likely many other nationalities) are not able to easily read/hear/pronounce "Ciro".
Ciro's Edict #5 Link to an image or video in another file that has an
\x on title from another file Updated 2025-07-01 +Created 1970-01-01Issue report at: github.com/ourbigbook/ourbigbook/issues/198 Suppose you had:
programming-language.ciro
= Programming language
\Image[https://raw.githubusercontent.com/cirosantilli/media/master/python-logo.jpg]
{title=The \x[python-programming-language] logo}
== Python
{c}
{disambiguate=programming-language}= Logos I like
\x[image-the-python-logo]Now, when rendering
\x[image-the-python-logo], we would need to fetch two IDs:image-the-python-logofor theTheandlogopartpython-programming-languageitself, to know that\x[python-programming-language]should render asPython
But after group all SQL queries together was done, there was no way to know that rendering
image-the-python-logo would imply also fetching python-programming-language.This was solved by adding a new database entry type,
REFS_TABLE_X_TITLE_TITLE to the existing References table, which tracks dependencies between IDs.The best open source implementation as of 2020 seems to be: Mozilla rr.
This is not a label that Ciro Santilli likes to give lightly. But maybe sometimes, it is inevitable.
Bibliography:
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