The h Bar

How about this, if I pick a range of frequency $\omega \in [\omega_1, \omega_2]$, the mean square value of $\phi$ is $$\langle \phi^2 \rangle = \int_{\omega_1}^{\omega_2} S(\omega) \, \frac{d\omega}{2\pi} \, .$$

Mithrandir24601

@DanielSank I was working on this all of 3 hours ago O.o

DanielSank

That's the simple meaning of spectral density.

(up to a factor of 2)

Mithrandir24601

5:48 PM

(avoided level crossings in Hamiltonians)

DanielSank

@Mithrandir24601 Good on you.

Important stuff. Note that it's not quantum.

btw someone emailed me about that video you recorded of our seminar.

I guess the audio wasn't that bad.

Mithrandir24601

@DanielSank Yeah, I know - I nearly did a PhD on stuff like that and spent an hour (with the professor) getting told that it's classical as well :)

Bernardo Meurer

@DanielSank Let's build a motorcycle

Mithrandir24601

@DanielSank Oh, nice :) I'm impressed they listened to it - might have been one of the new cohort

Bernardo Meurer

pleasepleasepleasepleaseplease

0celo7

5:52 PM

@BernardoMeurer I may or may not be going to VA tomorrow. Never trust the government.

DanielSank

@BernardoMeurer dude

Bernardo Meurer

@0celo7 What does the government have to do with you going home?

0celo7

@BernardoMeurer not home. I have to prove to the VA government that I'm alive

Bernardo Meurer

@0celo7 Wait, what?

0celo7

@BernardoMeurer my driver's license expired and that's as good as dying

So I have to prove I didn't

Bernardo Meurer

5:54 PM

goodness

Good luck at the DMV

0celo7

They play screeching noises during the hold music to get you to hang up

It's evil

Bernardo Meurer

Hahahahaha

The Portuguese embassy in brazil played this horrible depressive fado for hold music

You'd consider cutting your wrists while waiting for someone to pick up

That's why I can't hear fado anymore

Balarka Sen

MLG airhorn music should be put on as hold music

or some vaporwave

ACuriousMind

@0celo7 What about people who don't get a driver's licence?

Are they legally dead by US standards?

Semiclassical

depends on the state

ACuriousMind

5:59 PM

@Semiclassical That's...not a reassuring answer :D

Semiclassical

and while you'd hardly be considered legally dead, I can well believe it has some real pain-in-the-ass consequences

ACuriousMind

@Slereah I think that unit is called the 'Fadeev'

lılostafa

@DanielSank What is the PSD of the Fourier transform of a white Gaussian noise?

Anonymous

I actually need some book recommendations for Fourier transform of noise and pressure signals (and the procedure to implement it using programming- C/Matlab would be suitable). Any suggestions anyone?

Emilio Pisanty

@DanielSank why exactly are they not quantum?

@DanielSank I have a question but it's not about noise

0celo7

6:08 PM

@ACuriousMind that was a bit of an exaggeration. It's more like I don't legally exist

I have to prove I legally exist

lılostafa

@DanielSank This problem occurs when I measure the spectrum (Fourier transform) of a signal, and want to know the time domain signal. The measured data necessarily have AWGN. I'm not sure of the answer to this (need a proof).

0celo7

I existed legally yesterday. But today I don't. Quite sad.

Semiclassical

compute faster mathematica

Balarka Sen

gotta go faster

Semiclassical

faster faster faster

(alternatively: harder better faster stronger)

Balarka Sen

6:16 PM

sanic wolfram

Semiclassical

alpha sanic

I am having to face the fact that I don't know how to (numerically) obtain the time evolution of an initial wave function in a 1D potential.

I can sorta do it using Mathematica's NDSolve but it's slow and i don't really trust it

0celo7

@Semiclassical just compute e^-iHt

Semiclassical

hmm

0celo7

That was a joke

Although there are methods for computing integral kernels. You can do a greens function method

Fourier transform

Semiclassical

right.

lılostafa

6:21 PM

@0celo7 Thiw is the best way

Semiclassical

I mean, if V(x)=0, then the time evolution in momentum space is just $\phi(k,t)=\phi(k,0)e^{-i k^2 t/2}$ (up to units)

and then one just has to compute the inverse transform

which amounts to getting an integral representation of $\psi(x,t)$ at all times.

(and I mostly am interested in the V(x)=0 case at the moment)

lılostafa

@Semiclassical are the problems you want to solve 1D?

Emilio Pisanty

@Semiclassical so, just do split-step propagation then

29

Q: Are there simple ways to numerically solve the time-dependent Schödinger equation?

I would like to run some simple simulations of scattering of wavepackets off of simple potentials in one dimension. Are there simple ways to numerically solve the one-dimensional TDSE for a single particle? I know that, in general, trying to use naïve approaches to integrate partial differential...

Potentially relevant

DanielSank

@lılostafa That'a a really weird question. You want the PSD of a Fourier transform?

@Blue Yes I know a lot about this. What exactly are you trying to do?

Semiclassical

huh, daang

DanielSank

6:32 PM

@EmilioPisanty Because they show up if you study two coupled classical oscillators...

@lılostafa Ah, ok yes I can help with this.

bolbteppa

Gah

Pictures here on Lagrange multipliers are great, why did I always think in the plane medium.com/@andrew.chamberlain/…

DanielSank

But @lılostafa let us be very clear. You measure a time domain signal, yes? Then you compute the Fourier transform of it, yes? and you wish to know something about the statistics of the Fourier coefficients that you have computed?

@bolbteppa Wikipedia's page on Lagrange multipliers has a good diagram too.

@EmilioPisanty Golden rule: ask the damned question.

Semiclassical

@EmilioPisanty thanks. this one is also nice, because of the gifs: mathematica.stackexchange.com/questions/80996/…

bolbteppa

One of the wiki pics is the projection of that links pics on the plane, more confusing to me for some reason

lılostafa

@DanielSank It's basically the same, but I measure the Fourier transform :)

Anonymous

6:38 PM

@DanielSank I'm not completely sure about what I intend to achieve. I saw some recent papers on application of application of machine learning and Fourier transform in signal processing (electrocardiographs, arterial pressure, etc). I'm trying to replicate those results at uni lab and thinking of potential research topics related to that). Basically my experimental physics adviser told me to learn the basics and visit him within 1-2 weeks.

lılostafa

The question ultimately boils down to what is the Fourier transform of an additive white Gaussian noise?

DanielSank

@lılostafa Yes that I can answer very surely.

Note that the Fourier transform is statistical. Think about it: each time you measure a real time trace of a white noise, it is different, so the Fourier transform is not a deterministic quantity.

If you measure white noise, then each Fourier coefficient is a random complex number which has a Gaussian distributed real and imaginary part.

(Not that this is true even if the white noise is not Gaussian)

The widths of the distributions of the real and imaginary parts is related to the variance of the time domain signal.

I have written a very detailed document on this issue. I will send you a link momentarily...

Download and build this TeX file.

Uh... hm... I guess you have to clone the whole repo for that to work. If you want a pdf just give me an email address and I'll send it to you.

I have to go.

lılostafa

6:55 PM

@DanielSank Nice, thanks. I built it.

@Daniel Does the same reasoning apply for continuous Fourier transform of the noise too?

Physics Meta

0

Q: Is it possible to expand the list of migration sites for off-topic posts?

Recently, it seems that we've been getting several questions that would be more appropriate for Cross Validated. This question is only the most recent example. Right now, the only choices for off-topic migrations are to Physics Meta or Mathematics. I think we need more choices - like Cross Valid...

feature-request flagging migration

lılostafa

(although practically we just need the DFT)

Emilio Pisanty

@DanielSank =P

Phase

Could someone please explain what I'm doing wrong here, because it's annoying me that I can't see my basic mistake.

It's for a uniformly charged rod of length a from 0 to a, and it's electric field at a point m from the origin.

I took $\eta = Q/a$ and then $\eta = dQ/da$, rearranged for $\eta da = dQ$, multiplied both sides by k and divided by r^2 on both sides to get $\int \eta k / r^2 da = \int k / r^2 dQ = E$ , with my reasoning for all of this being looking at the right hand side and doing operations on both sides to make the right hand side eventually just be a summation of point cha…

Emilio Pisanty

7:10 PM

What are the equivalents to the Bessel-function sidebands of a particle in a harmonic oscillator that's emitting light along the direction of motion, if you quantize that harmonic oscillator and get a Lamb-dicke hamiltonian?

Phase

oh and I substitued in (m-a) into r.

Emilio Pisanty

I.e. If I'm working in the quantized version, how can I recover the classical Bessel sidebands?

Probably too complicated for chat, though =P, I'll ask it on main soon

Phase

7:27 PM

oh

$\int k/r^2 dQ$ doesn't represent the sum of point charges along a rod does it

lılostafa

7:54 PM

@DanielSank In the paper you have derived the distribution of the transformed noise, which is half the answer; it doesn't say anything about the power spectral density. I need that too if I want to generate the noise numerically.

DanielSank

8:16 PM

@lılostafa I'm pretty sure, yes. You have to learn a little about the Weiner process to understand why. I'm not quite as good at that yet.

@lılostafa Are you essentially asking for the correlation between the Fourier coefficients?