Ciro Santilli @cirosantilli 37

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Schrödinger picture Updated 2025-07-16

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 $ϕ$ .

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TransferWise Updated 2025-07-16

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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.

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Computer security researcher Updated 2025-07-16

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Ciro Santilli found out that he likes computer security researchers and vice versa.

It's a bit the same reason why he likes physicists: you can't bullshit with security.

You can't just talk nice and hope for people to belive you.

You can't not try to break things and just keep everyone happy in their false illusion of safety.

You can't do a half job.

If you do any of that, you will get your ass handed to you in a little gift bag.

All of this is closely linked to Ciro Santilli's self perceived creative personality and being naughty and creative are correlated.

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python/sympy_cheat/logarithm_integral.py Updated 2025-07-16

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Handscroll Updated 2025-07-16

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Ciro Santilli's favorite. A mixture of painting and book.

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History of the Internet Updated 2025-07-16

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SQL histogram Updated 2025-07-16

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OK, there's a billion questions:

SQL Server
- stackoverflow.com/questions/485409/generating-a-histogram-from-column-values-in-a-database OP did not know the difference between count and histogram :-) But it's the number one Google result.
- stackoverflow.com/questions/19103991/create-range-bins-from-sql-server-table-for-histograms has a minor extra group by twist, but otherwise fine
- stackoverflow.com/questions/16268441/generate-histogram-in-sql-server
SQLite
- stackoverflow.com/questions/67514208/how-to-optimise-creating-histogram-bins-in-sqlite perf only, benchmarking would be needed. SQLite.
- stackoverflow.com/questions/32155449/create-a-histogram-with-a-dynamic-number-of-partitions-in-sqlite variable bin size, same number of entries per bin
- stackoverflow.com/questions/60348109/histogram-for-time-periods-using-sqlite-regular-buckets-1h-wide time
MySQL: stackoverflow.com/questions/1764881/getting-data-for-histogram-plot MySQL appears to extend ROUND to also round by integers: ROUND(numeric_value, -2), but this is not widely portable which is a shame
stackoverflow.com/questions/72367652/populating-empty-bins-in-a-histogram-generated-using-sql specifically asks about empty bins, which is amazing. Amazon Redshift dialect unfortunately, but answer provided works widely, and Redshift was forked from PostgreSQL, so there's hope. Those newb open source server focused projects that don't use AGPL!

Let's try it on SQLite 3.40.1, Ubuntu 23.04. Data setup:

sqlite3 tmp.sqlite 'create table t(x integer)'
sqlite3 tmp.sqlite <<EOF
insert into t values (
  0,
  2,
  2,
  3,

  5,
  6,
  6,
  8,
  9,

  17,
)
EOF
sqlite3 tmp.sqlite 'create index tx on t(x)'

For a bin size of 5 ignoring empty ranges we can:

sqlite3 tmp.sqlite <<EOF
select floor(x/5)*5 as x,
       count(*) as cnt
from t
group by 1
order by 1
EOF

which produces the desired:

0|4
5|5
15|1

And to consider empty ranges we can use SQL genenerate_series + as per stackoverflow.com/questions/72367652/populating-empty-bins-in-a-histogram-generated-using-sql:

sqlite3 tmp.sqlite <<EOF
select x, sum(cnt) from (
  select floor(x/5)*5 as x,
         count(*) as cnt
    from t
    group by 1
  union
  select *, 0 as cnt from generate_series(0, 15, 5)
)
group by x
EOF

which outputs the desired: