Schrödinger equation for a free one dimensional particle Updated 2025-07-16
Schrödinger equation for a one dimensional particle with . The first step is to calculate the time-independent Schrödinger equation for a free one dimensional particle
Then, for each energy , from the discussion at Section "Solving the Schrodinger equation with the time-independent Schrödinger equation", the solution is:Therefore, we see that the solution is made up of infinitely many plane wave functions.
Schrödinger equation for a one dimensional particle Updated 2025-07-16
We select for the general Equation "Schrodinger equation":giving the full explicit partial differential equation:
- , the linear cartesian coordinate in the x direction
- , which analogous to the sum of kinetic and potential energy in classical mechanics
Equation 1.
Schrödinger equation for a one dimensional particle
. The corresponding time-independent Schrödinger equation for this equation is:
Equation 2.
time-independent Schrödinger equation for a one dimensional particle
. Schrödinger equation solution for the helium atom Updated 2025-07-16
No closed form solution, but good approximation that can be calculated by hand with the Hartree-Fock method, see hartree-Fock method for the helium atom.
Bibliography:
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.
"Making a measurement" for an observable means applying a self-adjoint operator to the state, and after a measurement is done:Those last two rules are also known as the Born rule.
- 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)"
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 .
Science fiction film Updated 2025-07-16
Science is the reverse engineering of nature Updated 2025-07-16
Ciro Santilli had once assigned this as one of Ciro Santilli's best random thoughts, but he later found that Wikipedia actually says exactly that: en.wikipedia.org/wiki/Reverse_engineering ("similar to scientific research, the only difference being that scientific research is about a natural phenomenon") so maybe that is where Ciro picked it up unconsciously in the first place.
Science makes progress funeral by funeral Updated 2025-07-16
Sci-Inspi (YouTube channel) Updated 2025-07-16
Scott Aaronson Updated 2025-07-16
Secondary school Updated 2025-07-16
Second brain Updated 2025-07-16
In the 2020's, this refers to writing down everything you know, usually in some graph structured way.
This is somewhat the centerpiece of Ciro Santilli's documentation superpowers: dumping your brain into text form, which he has been doing through Ciro Santilli's website.
This is also the closest one can get to immortality pre full blown transhumanism.
It is a good question, how much of your knowledge you would be able to give to others with text and images. It is likely almost all of it, except for coordination/signal processing tasks.
His passion for braindumping like this is a big motivation behind Ciro Santilli's OurBigBook.com work.
Bibliography:
Selection rule Updated 2025-07-16
phys.libretexts.org/Courses/University_of_California_Davis/UCD%3A_Physics_9HE_-_Modern_Physics/06%3A_Emission_and_Absorption_of_Photons/6.2%3A_Selection_Rules_and_Transition_Times has some very good mentions:
So it appears that if a hydrogen atom emits a photon, it not only has to transition between two states whose energy difference matches the energy of the photon, but it is restricted in other ways as well, if its mode of radiation is to be dipole. For example, a hydrogen atom in its 3p state must drop to either the n=1 or n=2 energy level, to make the energy available to the photon. The n=2 energy level is 4-fold degenerate, and including the single n=1 state, the atom has five different states to which it can transition. But three of the states in the n=2 energy level have l=1 (the 2p states), so transitioning to these states does not involve a change in the angular momentum quantum number, and the dipole mode is not available.So what's the big deal? Why doesn't the hydrogen atom just use a quadrupole or higher-order mode for this transition? It can, but the characteristic time for the dipole mode is so much shorter than that for the higher-order modes, that by the time the atom gets around to transitioning through a higher-order mode, it has usually already done so via dipole. All of this is statistical, of course, meaning that in a large collection of hydrogen atoms, many different modes of transitions will occur, but the vast majority of these will be dipole.It turns out that examining details of these restrictions introduces a couple more. These come about from the conservation of angular momentum. It turns out that photons have an intrinsic angular momentum (spin) magnitude of , which means whenever a photon (emitted or absorbed) causes a transition in a hydrogen atom, the value of l must change (up or down) by exactly 1. This in turn restricts the changes that can occur to the magnetic quantum number: can change by no more than 1 (it can stay the same). We have dubbed these transition restrictions selection rules, which we summarize as:
Semiconductor device fabrication Updated 2025-07-16
This is the lowest level of abstraction computer, at which the basic gates and power are described.
Semiconductor equipment maker Updated 2025-07-16
As mentioned at youtu.be/16BzIG0lrEs?t=397 from Video "Applied Materials by Asianometry (2021)", originally the companies fabs would make their own equipment. But eventually things got so complicated that it became worth it for separate companies to focus on equipment, which then then sell to the fabs.
Send free emails from Heroku Updated 2025-07-16
Arghh, why so hard... tested 2021:
- SendGrid: this one is the first one I got working on free tier!
- Mailgun: the Heroku add-on creates a free plan. This is smaller than the flex plan and does not allow custom domains, and is not available when signing up on mailgun.com directly: help.mailgun.com/hc/en-us/articles/203068914-What-Are-the-Differences-Between-the-Free-and-Flex-Plans- And without custom domains you cannot send emails to anyone, only to people in the 5 manually whitelisted list, thus making this worthless. Also, gmail is not able to verify the DNS of the sandbox emails, and they go to spam.Mailgun does feel good otherwise if you are willing to pay. Their Heroku integration feels great, exposes everything you need on environment variables straight away.
- CloudMailin: does not feel as well developed as Mailgun. More focus on receiving. Tried adding TXT xxx._domainkey.ourbigbook.com and CNAME mta.ourbigbook.com entires with custom domain to see if it works, took forever to find that page... www.cloudmailin.com/outbound/domains/xxx Domain verification requires a bit of human contact via email.
Separation of variables Updated 2025-07-16
Technique to solve partial differential equations
Naturally leads to the Fourier series, see: solving partial differential equations with the Fourier series, and to other analogous expansions:
One notable application is the solution of the Schrödinger equation via the time-independent Schrödinger equation.
Sequelize transaction retry Updated 2025-07-16
Transaction retries are inevitable, as some sQL isolation levels
Doesn't seem to have any simple built-in mechanism?
Shel Kaphan Updated 2025-07-16
He looks like an older and more experienced dude compared to Bezos at the time.
Bibliography:
. www.geekwire.com/2011/meet-shel-kaphan-amazoncom-employee-1/2/ also mentions that unlike California, there's no sales tax in the state of Washington, which is important for selling books.
. www.geekwire.com/2011/meet-shel-kaphan-amazoncom-employee-1/2/ also mentions that unlike California, there's no sales tax in the state of Washington, which is important for selling books.
- a few mentions at: Video "Jeff Bezos presentation at MIT (2002)"
Amazon.com Continues to Grow by NBC 15 (2014)
Source. Features short excerpt of filmed interview with Shel.:format(webp)/cdn.vox-cdn.com/uploads/chorus_image/image/66308477/175972523.jpg.0.jpg)
Single particle double slit experiment Updated 2025-07-16
This experiment seems to be really hard to do, and so there aren't many super clear demonstration videos with full experimental setup description out there unfortunately.
For single-photon non-double-slit experiments see: single photon production and detection experiments. Those are basically a pre-requisite to this.
photon experiments:
- aapt.scitation.org/doi/full/10.1119/1.4955173 "Video recording true single-photon double-slit interference" by Aspden and Padgetta (2016). Abstract says using spontaneous parametric down-conversion detection of the second photon to know when to turn the camera on
Non-elementary particle:
- 2019-10-08: 25,000 Daltons
- interactive.quantumnano.at/letsgo/ awesome interactive demo that allows you to control many parameters on a lab. Written in Flash unfortunately, in 2015... what a lack of future proofing!
Skelet machine #1 Updated 2025-07-16
Unlisted articles are being shown, click here to show only listed articles.