Ciro Santilli @cirosantilli 37

 Articles (11k) Discussions (25) Comments (63) Follows  Received likes Files

 New  Updated  Top  Announced  A-Z  Liked  Followed

Schrödinger picture  Updated 2025-07-11  +Created 1970-01-01

To better understand the discussion below, the best thing to do is to read it in parallel with the simplest possible example: Schrödinger picture example: quantum harmonic oscillator.

The state of a quantum system is a unit vector in a Hilbert space.

"Making a measurement" for an observable means applying a self-adjoint operator to the state, and after a measurement is done:

the state collapses to an eigenvector of the self adjoint operator
the result of the measurement is the eigenvalue of the self adjoint operator
the probability of a given result happening when the spectrum is discrete is proportional to the modulus of the projection on that eigenvector.
For continuous spectra such as that of the position operator in most systems, e.g. Schrödinger equation for a free one dimensional particle, the projection on each individual eigenvalue is zero, i.e. the probability of one absolutely exact position is zero. To get a non-zero result, measurement has to be done on a continuous range of eigenvectors (e.g. for position: "is the particle present between x=0 and x=1?"), and you have to integrate the probability over the projection on a continuous range of eigenvalues.
In such continuous cases, the probability collapses to an uniform distribution on the range after measurement.
The continuous position operator case is well illustrated at: Video "Visualization of Quantum Physics (Quantum Mechanics) by udiprod (2017)"

Those last two rules are also known as the Born rule.

Self adjoint operators are chosen because they have the following key properties:

their eigenvalues form an orthonormal basis
they are diagonalizable

Perhaps the easiest case to understand this for is that of spin, which has only a finite number of eigenvalues. Although it is a shame that fully understanding that requires a relativistic quantum theory such as the Dirac equation.

The next steps are to look at simple 1D bound states such as particle in a box and quantum harmonic oscillator.

This naturally generalizes to Schrödinger equation solution for the hydrogen atom.

The solution to the Schrödinger equation for a free one dimensional particle is a bit harder since the possible energies do not make up a countable set.

This formulation was apparently called more precisely Dirac-von Neumann axioms, but it because so dominant we just call it "the" formulation.

Quantum Field Theory lecture notes by David Tong (2007) mentions that:

if you were to write the wavefunction in quantum field theory, it would be a functional, that is a function of every possible configuration of the field $ϕ$ .

 Read the full article

TransferWise  Updated 2025-07-11  +Created 1970-01-01

 View more

This is a good company, first they truly helped reduce international transfer fees. They they continued to morph into a decent challenger bank.

Their Wise Interest account was amazing as of late 2023: wise.com/gb/interest/

Instant access with representative national interests and 0.29% fees.

Brick and mortar banks were way way behind in that regard!

E.g. October 2023, Wise was doing 4.87% interest after fees, while Barclay's best option was 1.16% above 5k pounds on the Rainy Day Saver (5% below). Ridiculous!

Update: On November 2023 unfortunately they more than doubled their fees from 0.19% to 0.46%:

but it still was a good option to keep cash in.

 Read the full article

Transversal time dilation  Updated 2025-07-11  +Created 1970-01-01

 View more

Light watch transverse to direction of motion. This case is interesting because it separates length contraction from time dilation completely.

Of course, as usual in special relativity, calling something "time dilation" leads us to mind boggling ideas of "symmetry breaking": if both frames have a light watch, how can both possibly observe the other to be time dilated?

And the answer to this, is the usual: in special relativity time and space are interwoven in a fucked up way, everything is just a spacetime event.

In this case, there are three spacetime events of interest: both clocks start at same position, your beam hits up at x=0, moving frame hits up at x>0.

Those two mentioned events are spacelike-separated events, and therefore even though they seem simultaneous to you, they are not going to be simultaneous to the moving observer!

If little clock one meter away from you tells you that at the time of some event (your light beam hit up) the moving light watch was only 50% up, this is just a number given by your one meter away watch!

 Read the full article

Common Crawl  Updated 2025-07-11  +Created 1970-01-01

 View more

commoncrawl.org/

Amazing project, that basically makes a more searchable Wayback Machine.

A bit hard to use their data though, partly due to size, but also lack of free to use querrying mechanisms, and how obtuse Amazon S3 is to use.

Notably, aws-cli with an account is the only reliable way, everything else is way too broken, e.g. trying the to check the an index index.commoncrawl.org/CC-MAIN-2023-06/ very often 500s.

But still, their projct is amazing.

The only out-of-the-box search they seem to have is: urlsearch.commoncrawl.org/ for domains/URLs. It is good, but there could be so much more... notably IPs.

Also could should document the data shape a bit better.

Sample sizes can be found at: commoncrawl.org/2023/04/mar-apr-2023-crawl-archive-now-available/

To explore the data, after login:

aws s3 ls s3://commoncrawl/crawl-data/CC-MAIN-2013-20/

Copy the toplevel directory only:

aws s3 cp s3://commoncrawl/crawl-data/CC-MAIN-2013-20/ . --recursive --exclude "*/*"

Copy some wet/wat files:

aws s3 cp s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368696381249/wat/CC-MAIN-20130516092621-00000-ip-10-60-113-184.ec2.internal.warc.wat.gz .
aws s3 sync s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368696381249/wet/CC-MAIN-20130516092621-00000-ip-10-60-113-184.ec2.internal.warc.wet.gz .

Directory structrure:

cc-index.paths.gz (1K)
cc-index-table.paths.gz (1K)

segment.paths.gz (1.7K) Sample lines:

crawl-data/CC-MAIN-2013-20/segments/1368696381249/
crawl-data/CC-MAIN-2013-20/segments/1368696381630/

index.html (2.3K)

wat.paths.gz (98K) Sample lines:

crawl-data/CC-MAIN-2013-20/segments/1368696381249/wat/CC-MAIN-20130516092621-00000-ip-10-60-113-184.ec2.internal.warc.wat.gz
crawl-data/CC-MAIN-2013-20/segments/1368696381249/wat/CC-MAIN-20130516092621-00001-ip-10-60-113-184.ec2.internal.warc.wat.gz

wet.paths.gz (98K) Sample lines:

crawl-data/CC-MAIN-2013-20/segments/1368696381249/wet/CC-MAIN-20130516092621-00000-ip-10-60-113-184.ec2.internal.warc.wet.gz
crawl-data/CC-MAIN-2013-20/segments/1368696381249/wet/CC-MAIN-20130516092621-00001-ip-10-60-113-184.ec2.internal.warc.wet.gz

warc.paths.gz (99K)

crawl-data/CC-MAIN-2013-20/segments/1368696381249/warc/CC-MAIN-20130516092621-00000-ip-10-60-113-184.ec2.internal.warc.gz
crawl-data/CC-MAIN-2013-20/segments/1368696381249/warc/CC-MAIN-20130516092621-00001-ip-10-60-113-184.ec2.internal.warc.gz

segments: directgory with actual data

1368696381249: one of many segments, any meaning of name?

CC-MAIN-20130516092621-00000-ip-10-60-113-184.ec2.internal.warc.wet.gz (142M, 334M unzipped)

A tiny bit of metadata, and then plaintext content from the website, e.g. the second one:

WARC/1.0
WARC-Type: conversion
WARC-Target-URI: http://004eeb5.netsolhost.com/stephensilver.htm
WARC-Date: 2013-05-18T08:11:02Z
WARC-Record-ID: <urn:uuid:773b31ba-ddc6-47a5-ae24-d08141b9944d>
WARC-Refers-To: <urn:uuid:4b1bdbff-4926-4ced-86f6-072f5bb3837a>
WARC-Block-Digest: sha1:LQFSCR2LIJQYMPTXRHWU7HAPQTVSYS3A
Content-Type: text/plain
Content-Length: 12046

Stephen Silver is a journalist and editor who specializes in the areas of politics, pop culture, film and sports. He works as an editor with the North American Publishing Co. and as a film critic with The Trend, a local newspaper in the Philadelphia area.

No IP unfortunately.

CC-MAIN-20130516092621-00000-ip-10-60-113-184.ec2.internal.warc.wat.gz (329M, 1.4G unzipped)

A lot of JSON metadata and no contents as desired. Contains IP! Some entries however are humongous with a ton of useless data, that's what bloats these so much:

WARC/1.0
WARC-Type: metadata
WARC-Target-URI: CC-MAIN-20130516092621-00000-ip-10-60-113-184.ec2.internal.warc.gz
WARC-Date: 2013-11-22T14:51:12Z
WARC-Record-ID: <urn:uuid:ec54e493-8965-41be-b344-07596cc30b3a>
WARC-Refers-To: <urn:uuid:cfeff436-7c4c-4119-aaa4-ec2ce27ad3e1>
Content-Type: application/json
Content-Length: 1180

{"Envelope":{"Format":"WARC","WARC-Header-Length":"274","Block-Digest":"sha1:JCZOI4V3UOTXGIRLFMPLW4J2WPLAKGVR","Actual-Content-Length":"372","WARC-Header-Metadata":{"WARC-Type":"warcinfo","WARC-Filename":"CC-MAIN-20130516092621-00000-ip-10-60-113-184.ec2.internal.warc.gz","WARC-Date":"2013-11-22T14:51:12Z","Content-Length":"372","WARC-Record-ID":"<urn:uuid:cfeff436-7c4c-4119-aaa4-ec2ce27ad3e1>","Content-Type":"application/warc-fields"},"Payload-Metadata":{"Trailing-Slop-Length":"0","Actual-Content-Type":"application/warc-fields","Actual-Content-Length":"372","Headers-Corrupt":true,"WARC-Info-Metadata":{"robots":"classic","software":"Nutch 1.6 (CC)/CC WarcExport 1.0","description":"Wide crawl of the web with URLs provided by Blekko for Spring 2013","hostname":"ip-10-60-113-184.ec2.internal","format":"WARC File Format 1.0","isPartOf":"CC-MAIN-2013-20","operator":"CommonCrawl Admin","publisher":"CommonCrawl"}}},"Container":{"Compressed":true,"Gzip-Metadata":{"Footer-Length":"8","Deflate-Length":"453","Header-Length":"10","Inflated-CRC":"866052549","Inflated-Length":"650"},"Offset":"0","Filename":"CC-MAIN-20130516092621-00000-ip-10-60-113-184.ec2.internal.warc.gz"}}

WARC/1.0
WARC-Type: metadata
WARC-Target-URI: http://%20jwashington@ap.org/Content/Press-Release/2012/How-AP-reported-in-all-formats-from-tornado-stricken-regions
WARC-Date: 2013-05-18T05:48:54Z
WARC-Record-ID: <urn:uuid:d519658f-7a63-46c1-849b-4cd92332ddb8>
WARC-Refers-To: <urn:uuid:cefd363b-1fec-4590-8305-4c6fab2e095f>
Content-Type: application/json
Content-Length: 1501

{"Envelope":{"Format":"WARC","WARC-Header-Length":"433","Block-Digest":"sha1:B2B6JDSGWCUQIIUGV54SXEE25RX4SANS","Actual-Content-Length":"302","WARC-Header-Metadata":{"WARC-Type":"request","WARC-Date":"2013-05-18T05:48:54Z","WARC-Warcinfo-ID":"<urn:uuid:cfeff436-7c4c-4119-aaa4-ec2ce27ad3e1>","Content-Length":"302","WARC-Record-ID":"<urn:uuid:cefd363b-1fec-4590-8305-4c6fab2e095f>","WARC-Target-URI":"http://%20jwashington@ap.org/Content/Press-Release/2012/How-AP-reported-in-all-formats-from-tornado-stricken-regions","WARC-IP-Address":"165.1.125.44","Content-Type":"application/http; msgtype=request"},"Payload-Metadata":{"Trailing-Slop-Length":"4","HTTP-Request-Metadata":{"Headers":{"Accept-Language":"en-us,en-gb,en;q=0.7,*;q=0.3","Host":"ap.org","Accept-Encoding":"x-gzip, gzip, deflate","User-Agent":"CCBot/2.0","Accept":"text/html,application/xhtml+xml,application/xml;q=0.9,*/*;q=0.8"},"Headers-Length":"300","Entity-Length":"0","Entity-Trailing-Slop-Bytes":"0","Request-Message":{"Method":"GET","Version":"HTTP/1.0","Path":"/Content/Press-Release/2012/How-AP-reported-in-all-formats-from-tornado-stricken-regions"},"Entity-Digest":"sha1:3I42H3S6NNFQ2MSVX7XZKYAYSCX5QBYJ"},"Actual-Content-Type":"application/http; msgtype=request"}},"Container":{"Compressed":true,"Gzip-Metadata":{"Footer-Length":"8","Deflate-Length":"455","Header-Length":"10","Inflated-CRC":"453539965","Inflated-Length":"739"},"Offset":"453","Filename":"CC-MAIN-20130516092621-00000-ip-10-60-113-184.ec2.internal.warc.gz"}}

Let's beautify one of them to see it better:


{
  "Envelope": {
    "Format": "WARC",
    "WARC-Header-Length": "274",
    "Block-Digest": "sha1:JCZOI4V3UOTXGIRLFMPLW4J2WPLAKGVR",
    "Actual-Content-Length": "372",
    "WARC-Header-Metadata": {
      "WARC-Type": "warcinfo",
      "WARC-Filename": "CC-MAIN-20130516092621-00000-ip-10-60-113-184.ec2.internal.warc.gz",
      "WARC-Date": "2013-11-22T14:51:12Z",
      "Content-Length": "372",
      "WARC-Record-ID": "<urn:uuid:cfeff436-7c4c-4119-aaa4-ec2ce27ad3e1>",
      "Content-Type": "application/warc-fields"
    },
    "Payload-Metadata": {
      "Trailing-Slop-Length": "0",
      "Actual-Content-Type": "application/warc-fields",
      "Actual-Content-Length": "372",
      "Headers-Corrupt": true,
      "WARC-Info-Metadata": {
        "robots": "classic",
        "software": "Nutch 1.6 (CC)/CC WarcExport 1.0",
        "description": "Wide crawl of the web with URLs provided by Blekko for Spring 2013",
        "hostname": "ip-10-60-113-184.ec2.internal",
        "format": "WARC File Format 1.0",
        "isPartOf": "CC-MAIN-2013-20",
        "operator": "CommonCrawl Admin",
        "publisher": "CommonCrawl"
      }
    }
  },
  "Container": {
    "Compressed": true,
    "Gzip-Metadata": {
      "Footer-Length": "8",
      "Deflate-Length": "453",
      "Header-Length": "10",
      "Inflated-CRC": "866052549",
      "Inflated-Length": "650"
    },
    "Offset": "0",
    "Filename": "CC-MAIN-20130516092621-00000-ip-10-60-113-184.ec2.internal.warc.gz"
  }
}

Fuck no IP addresses either. But other entries do have it, why not this one?

The reason these can be huge is the HTML-Metadata section which contain all outlinks! gist.github.com/Smerity/e750f0ef0ab9aa366558#file-bbc-pretty-wat-L34

CC-MAIN-20130516092621-00000-ip-10-60-113-184.ec2.internal.warc.gz ()

Obtain:

aws s3 cp s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368696381249/warc/CC-MAIN-20130516092621-00000-ip-10-60-113-184.ec2.internal.warc.gz .

 Read the full article

Computational problem  Updated 2025-07-11  +Created 1970-01-01

 View more

The list: complexityzoo.uwaterloo.ca/Complexity_Zoo

 Read the full article

Fandom (website)  Updated 2025-07-11  +Created 1970-01-01

 Read the full article

Puzzle  Updated 2025-07-11  +Created 1970-01-01

 Read the full article

Yitang Zhang's theorem  Updated 2025-07-11  +Created 1970-01-01

 View more

There are infinitely many primes with a neighbor not further apart than 70 million. This was the first such finite bound to be proven, and therefore a major breakthrough.

This implies that for at least one value (or more) below 70 million there are infinitely many repetitions, but we don't know which e.g. we could have infinitely many: