Alright, let's start with something simple since I get nobody wants to waste their time with a thread not even OP contributed to. So how about this: there are basically two ways I have seen people solve the bound delta potential: one is solving the two domains and gluing them together. The other is to use delta distributions. Personally I find the latter more elegant, consider
[eqn]
\hat{H} = -\frac{\hbar^2}{2m} \frac{d^2}{dx^2} -\lambda \delta(x),~(\lambda>0)\\
[/eqn]
Since H = T almost everywhere the solution for E<0 must be exponential exp(k(x)x) where k(x) is piecewise constant.
Since H is symmetric with respect to reflection, the ground state ought to be even, which means k(x) is odd and can be re-written as a constant times sgn(x).
Note that [math] {\rm sgn}(x) = 2\Theta(x) -1[/math].
The wave function also ought to be normalizable which only admits negative k. We can combine all this into one ansatz and apply T:
[eqn]
\hat{T}\Psi =
-\frac{\hbar^2}{2m} \frac{d^2}{dx^2}
\exp(-k|x|) =
-\frac{\hbar^2}{2m} \frac{d}{dx}
\exp(-k|x|)\cdot(-k\cdot {\rm sgn}(x)) \\
=
-\frac{\hbar^2}{2m} \exp(-k|x|)\cdot\left(
(-k\cdot {\rm sgn}(x))^2
+(-k\cdot {\rm sgn}'(x))
\right)\\
=
-\frac{\hbar^2}{2m} \exp(-k|x|)\cdot\left(
k^2-k\cdot 2\delta(x))
\right)
= \left(-\frac{\hbar^2 k^2}{2m} + \frac{\hbar^2 k}{m}\delta(x)\right)\Psi\\
=(E-\lambda\delta(x))\Psi
[/eqn]
Therefor [math]k = -\frac{\lambda m}{\hbar^2}\delta(x)[/math] and [math]E=-\frac{m \lambda^2}{2\hbar^2}[/math]. Compared to the Wikipedia derivation I prefer it.

>>17059960
there's a sign error but you get the idea.

>>17059362
This is much more low IQ than what you are asking for, but I have always found that even in undergraduate Statistics courses and even in high school, what gets taught as “statistics” is way too formal. My first statistics class I took was in the sophomore year of undergrad, and it was highly based on calculus and gaussian integrals. (I had a good background in calculus and differential equations but we never touched on Gaussian integrals)

A good way to understand statistics can be gleaned with absolutely no formalism at all. You just need to understand the Central Limit Theorem, some basic facts about how Gaussian statistics work, and an introduction to Poisson processes and some other non-Gaussian statistics examples. I can explain more if anyone likes but my point is that the formalism presented in standard undergrad curriculum is not a good way to understand statistics and often leads to even high-level scientists ten or 20 years into their careers making ridiculous statistical mistakes

it is literally all over the literature in physics, biology, and every social science. People not understanding how gaussian statistics works, how systematic uncertainties can be correlated or uncorrelared, so on and so forth. These people mostly plug numbers into their computer programs and run a command to generate the numbers they report in the “results” section, but the input numbers are usually only correct in the “observed data” entrt and not in the uncertainties or backgrounds entries.

What I’ve said so far may sound like a criticism of mainstream science, but really what I am saying is that having attained 5+ years as a professional scientist and almost 20 years in high-end academic institutions, I think the entire scientific community would really benefit if I could upload my even very modest intuitive understanding of statistics to the rest of the minds in the field, because they have a warped and bad understanding as is in the world

>>17060696
>You just need to understand the Central Limit Theorem, some basic facts about how Gaussian statistics work, and an introduction to Poisson processes and some other non-Gaussian statistics examples.
Oh, have you ever looked into statistical mechanics literature? This sounds very reminiscent of that, especially when it ties to phenomenological thermodynamics.

>>17060696
>What I’ve said so far may sound like a criticism of mainstream science, but really what I am saying is that having attained 5+ years as a professional scientist and almost 20 years in high-end academic institutions, I think the entire scientific community would really benefit if I could upload my even very modest intuitive understanding of statistics to the rest of the minds in the field, because they have a warped and bad understanding as is in the world
Well upload it. I am curious

>>17059960
On the subject of the delta function, I don't recall if this derivation is in the standard books on electromagnetism but I would very much hope it is, since I found it rather neat. Consider a single point charge in a rest frame with no external fields. In other words J=0 and [math] \rho = Q\delta(\vec{x})[/math].
I am going entirely off memory for this and took some time to verify with pen and paper but it works out.
Fixing the Coulomb gauge what remains of Maxwell's equations is
[eqn]
\vec{\nabla}^2 \phi = -\frac{1}{\varepsilon_0} \rho = -\frac{Q}{\varepsilon_0} \delta(x)\delta(y)\delta(z)
[/eqn]
As so often when dealing with the delta distribution it helps to get an integral form, in this case through the unitary Fourier transform. This simplifies the above by a great deal (I write integrals als multiplicative operators, don't let that confuse you):
[eqn]
-\vec{k}^2 \tilde{\phi} = -\frac{Q}{\sqrt{2\pi}^3\varepsilon_0} \iiint_{\mathbb{R}^3} d^3x \exp(-i\vec{k}\cdot\vec{x}) \delta(x)\delta(y)\delta(z) = -\frac{Q}{\sqrt{2\pi}^3\varepsilon_0}
[/eqn]
let [math] k=|\vec{k}|, r=|\vec{x}|[/math]
Divide by -k^2 and apply the inverse FT:
[eqn]
\phi = \frac{Q}{8\pi^3\varepsilon_0} \iiint_{\mathbb{R}^3} d^3x \exp(i\vec{k}\cdot\vec{x}) \frac{1}{k^2}
[/eqn]
Now comes one insidious trick: since x is arbitrary but fixed from the perspective of the integral, you can align a spherical coordinate system along [math] \hat{x} = \frac{\vec{x}}{r} [/math] as long as x is not 0 (which is pathological anyhow). This gives an integral where we can immediately integrate out one variable (phi).
[eqn]
\phi = \frac{Q}{8\pi^3\varepsilon_0} \int_0^\infty dk \int_0^{2\pi} d\varphi \int_0^\pi d\theta\, k^2 \sin\theta \exp(i kr\cos\theta) \frac{1}{k^2}\\
\phi = \frac{2\pi Q}{8\pi^3\varepsilon_0} \int_0^\infty dk \int_0^\pi d\theta\sin\theta \exp(i kr\cos\theta)
[/eqn]

>>17061583
Theta is just integration by substitution.
[eqn]
\phi = -\frac{Q}{4\pi^2\varepsilon_0} \int_0^\infty dk \frac{1}{ikr} \int_0^\pi d\theta \frac{d}{d\theta} \exp(i kr\cos\theta) \\
\phi = \frac{Q}{2\pi^2\varepsilon_0} \int_0^\infty dk \frac{\exp(ikr)-\exp(-ikr)}{2ikr} = \frac{Q}{2\pi^2\varepsilon_0} \int_0^\infty dk \frac{\sin(kr)}{kr}
[/eqn]
The integrand is an even function so we can double the range if we divide the prefactor by two, and do a simple substitution of [math] k = \frac{\pi l}{r}[/math] to reduce the result to the normalized sinc function
[eqn]
\phi = \frac{Q}{4\pi\varepsilon_0 r} \int_{-\infty}^\infty dl \frac{\sin(\pi l)}{\pi l} = \frac{Q}{4\pi\varepsilon_0 r},
[/eqn]
which completes the derivation.

>>17060809
I meant “upload” figuratively, as in “upload your consciousness”.

As an example, consider this statistical puzzle. There are experiments for Dark Matter and, in the last, for proton decay (P+D), where the experiment is set up so carefully that without a genuine DM or P+D signal, you realisticallg expect that there are zero background events, or basically that your detector sees no bright flashes of light (and in some cases, there may be rare spurious ones but they can be identified as a fake and vetoed). But let’s say you see one event that cannot be ruled out (as is the recent case in LUX-Zeppelin).

In normal experiments where you are in a Gaussian statistics regime, with a similar understanding of the background rate of a poisson process, it is easy to evaluate what statistical significance (in “sigma”) or what p-value you assign to observations.

But in this case of a zero-background experiment, how to you assign a significance to one positive DM observation? Or one positive P+D observation (where the signal would be much more clearly creating secondary particles which add up to an invariant mass matching the proton mass)?

Seriously, is the LZ “one event” a five sigma observation? a one sigma observation? What? is the P so low that you reject the H0?

This depends on your *priors* and how you evaluate the systematic uncertainties on your background model. If people could honestly evaluate these things then I am pretty sure this LZ announcement is basically a total stupid publicity stunt and has basically less than one sigma significance

The problem is I know people on LZ and none of them actually understand statistics. They are not statisticians, and even if they were then they would also need to understand the systematic effects in their own experiment, and usually on big collaborations the people who understand those things are not on the same team who write the papers (or press releases) for the bulk of what gets published

Show your working playboy

>>17061709
>Or one positive P+D observation
nta but I would say you can't at all, personally. In such events I would argue that a bayesian approach is more natural. You obviously agree since you speak in terms of priors. But ANY physicist who has two braincells to rub together should know that statistical processes converge like ass and that a single event is never ever ever gonna cut it. If they don't grasp that then how did they not fail stat mech?

>>17064827
You are entirely right but you need to look at what the LUX-Zeplin people publicized in the last couple weeks
https://www6.slac.stanford.edu/news/2026-09-01-lz-sees-surprising-result-search-dark-matter
they are doing a huge publicity stunt on literally ONE EVENT. this implies that their way of interpreting priors, or alternatively in their way of doing background estimations in a frequentist approach, is clearly leaning in some direction where they think **ONE EVENT** is worthy of creating a media hype storm.

Why? probably funding shit due to Trump cuts and the Genesis shit. Maybe they know statistics and they are just doing bad science as a ploy to trick the trump DOE/NSF. maybe. so they are either doing a dumb mistake on purpose to defraud the government, or they are making a dumb mistake by mistake because they are retarded

>>17065116
>Stanford
Maybe I am mistaken but isn't Stanford full of AI bullshit artists as well? I think they drank too much of the high impact kool aid but I digress. I genuinely hope it is marketing only

>>17065539
Stanford definitely is high on their own supply of stupid marketing bullshit. They had a good run ever since what—the 1980s?— when West Coast schools could even be considered in the same mindset as real Ivy League East Coast elite schools like Harvard or Princeton.

Unfortunately the rot that brought down Stanford applies equally to the Ivy League as well. Harvard and Princeton will survive based on brand loyalty but I wouldn’t be surprised if Stanford gets lowered to second-tier status like Brown or Dartmouth (places ostensibly top-tier but they kind of are a joke in the end)