(no subject)

lucretius was a fuckboy and a flat earther

(no subject)

i dont agree with the usage of “order of magnitude” to mean specifically factor of 10. Just say things are on different orders of magnitude, dont try and count orders of magnitude and say theyre three orders of magnitude apart

(no subject)

thesis: understanding the world through logical and mathematical reasoning (for example plato)

antithesis: understanding the world through observation (for example epicurus)

synthesis: modern physics (for example newton)

(no subject)

The Fourier transform of a standard Gaussian is itself. That is, the standard Gaussian is a fixed point of the Fourier transform.

That's how I thought about it, until I considered units, and I realized they're not really the same thing. Like, a Gaussian wavefunction could map positions to amplitidues in position space. Then, its Fourier transform maps wavenumbers to amplitudes in wavenumber space. This actually made more sense when, instead of considering a standard Guassian, I considered a Gaussian with standard deviation σ, since σ has a unit.

Let φ be a standard Gaussian wavefunction, with unitless input and output. Then the Gaussian wavefunction is

ψ(x) = (1/√σ) φ(x/σ)

The constant takes care of the units. Like, if x is in meters, σ is also in meters, and the input x/σ is unitless. To see that the output is in the right units, consider calculating a probability:

∫ ψ(x)² dx = ∫ (1/σ) φ(x/σ)² dx

The units in 1/σ and dx cancel, and the output is unitless.

Then, the Fourier transform of ψ(x) is

(F ψ)(k) = √σ φ(σ k)

(I use k to mean angular wavenumber.) So again the units in the input cancel. For example if k is in inverse meters, then σ k = meters * inverse meters is unitless.

So ψ and F ψ are not really the same function for units reasons, but they're both expressible in terms of this one unitless function φ.

(no subject)

there's a lot more to say about the wavefunctinos of molecules than that they solve the schrodinger equation. Any quantum system solves the schrodinger equation; molecules are a subset of potentials and you can ask what's distinctive about them.

Here's something. For a molecule with n atoms, the electron density has O(n) critical points.

This is in the first chapter or two of Bader's book. There's generally a critical point at every nucleus, a critical point at every bond, a critical point in the middle of every ring. And then for molecules with cages like cubane, there's a critical point in the middle of the cage. Generally. This is not always the case, but it's pretty much always the case.

That's a limited number of critical points. I mean, I'm sure that the space of possible wavefunctions includes electron densities with arbitrarily high densities of critical points. So this is a very special smoothness property and so epxpansions in the density of critical points or something should allow you to speak very generally about molecules, but say something that's new, something that doesn't just follow from the schrodinger equation.

If this is a standard thing link me to the wikipedia article plz.

(no subject)

so

aRb means "a and b are siblings"

you can represent this relation R as a set of all pairs for which it's true

{
{ Lana Wachowski, Linda Wachowski },
{ Linda Wachowski, Lana Wachowski },
{ Joel Cohen, Ethan Cohen },
{ Ethan Cohen, Joel Cohen }
}

and this looks like a spreadsheet with two columns (a set of two rows)

so a "relational database" stores the data as a collection of spreadsheets.