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Schrödinger equation for a one dimensional particle by 
Ciro Santilli 35 Updated 2025-01-22 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-01-22 +Created 1970-01-01We 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
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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: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 .
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:
composite particle made up of an odd number of elementary particles.
Hypothesized as the explanation for continuous electron energy spectrum in beta decay in 1930 by .
First observed directly by the Cowan-Reines neutrino experiment.
Why do the electron and the proton have the same charge except for the opposite signs? by Ciro Santilli 35 Updated 2025-01-22 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-01-22 +Created 1970-01-01Given the view of the Standard Model where the electron and quarks are just completely separate matter fields, there is at first sight no clear theoretical requirement for that.
As mentioned e.g. at QED and the men who made it: Dyson, Feynman, Schwinger, and Tomonaga by Silvan Schweber (1994) chapter 1.6 "Hole theory", Dirac initially wanted to think of the holes in his hole theory as the protons, as a way to not have to postulate a new particle, the positron, and as a way to "explain" the proton in similar terms. Others however soon proposed arguments why the positron would need to have the same mass, and this idea had to be discarded.
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Home 2023:
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2020-12: large-ish chicken, www.youtube.com/watch?v=bJeUb8ToRIw worked very well. Just that after 1 hour it was slightly uncooked in the middle, and 10 minutes later, the top skin burnt a little bit. So next time, use some aluminium foil.
If you pass parallel light.
The basic experiment for a photonic quantum computer.
Can be achieved in two ways it seems:
- macroscopic beam splitter and optical table
- photolithography
Animation of Hong-Ou-Mandel Effect on a silicon like structure by Quantum Light University of Sheffield (2014): www.youtube.com/watch?v=ld2r2IMt4vg No maths, but gives the result clear: the photons are always on the same side.
Quantum Computing with Light by Quantum Light University of Sheffield (2015)
Source. Animation of in-silicon single photon device with brief description of emitting and receiving elements. Mentions:- quantum dot source. TODO how do you produce identical photons from two separate quantum dots? See also: quantum dot single photon source.
- superconducting nanowire detector. So the device has to be cooled then? Video "Jeremy O'Brien: "Quantum Technologies" by GoogleTechTalks (2014)" youtube.com/watch?v=7wCBkAQYBZA&t=2497 however says that semiconducting devices can also be used
Quantum Optics - Beam splitter in quantum optics by Alain Aspect (2017)
Source. More theoretical approach.Building a Quantum Computer Out of Light by whentheappledrops (2014)
Source. Yada yada yada, then at youtu.be/ofg335d3BJ8?t=341 shows optical table and it starts being worth it. Jacques Carolan from the University of Bristol goes through their setup which injects 5 photons into a 21-way experiment. Spontaneous parametric down-conversion by Ciro Santilli 35 Updated 2025-01-22 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-01-22 +Created 1970-01-01Phenomena that produces photons in pairs as it passes through a certain type of crystal.
You can then detect one of the photons, and when you do you know that the other one is there as well and ready to be used. two photon interference experiment comes to mind, which is the basis of photonic quantum computer, where you need two photons to be produced at the exact same time to produce quantum entanglement.
One Photon In, TWO Photons Out by JQInews (2010)
Source. Features Alan Migdall of the National Institute of Standards and Technology. Produced by the Joint Quantum Institute (JQI).
Mentions that this phenomena is useful to determine the efficiency of a single photon detector, as you have the second photon of the pair as a control.
Also briefly describes how the input energy and momentum must balance out the output energy and momentum of the two photons coming out (determined by the output frequency and angle).
Shows the crystal close up of the crystal branded "Cleveland Crystals Inc.". Mentions that only one in a billion photon gets scattered.
Also shows a photomultiplier tube.
Then shows their actual optical table setup, with two tunnels of adjustable angle to get photons with different properties.
How do you produce a single photon? by Physics World (2015)
Source. Very short whiteboard video by Peter Mosley from the University of Bath, but it's worth it for newbs. Basically describes spontaneous parametric down-conversion.
One interesting thing he mentions is that you could get single photons by making your sunglasses thicker and thicker to reduce how many photons pass, but one big downside problem is that then you don't know when the photon is going to come through, that becomes essentially random, and then you can't use this technique if you need two photons at the same time, which is often the case, see also: two photon interference experiment.
Don't force international exchange students to come back early by Ciro Santilli 35 Updated 2025-01-22 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-01-22 +Created 1970-01-01Many of the student exchange programs Ciro witnessed in the 2010's in Brazil were inefficient because they were requiring students to come back immediately after university or PhD in fear that those students will never come back.
This is useless, because you don't learn anything unique during university: the truly valuable knowledge is obtained when you work for several years as a postdoc in a world class research laboratory or as an engineer in a world class company.
Therefore, Brazil should learn from the Chinese exchange system, which lets students go do whatever they want, and once they are Gods of the domain, entices them back with great positions and pay as heads of laboratory back in China. Just don't do fraudulent stuff like this like China did, or else you will get a bad rep.
To help this university collaboration happen, we should create communication channels between exchange students and professors of the origin country who work on the same domain so that they can discuss the subject. For example, once Ciro Santilli wanted to contact some of his former teachers at the University of São Paulo about "advanced" topics he had been exposed to as part of his job. However, they didn't even reply to his email, and Ciro didn't know who else to contact. This must never happen. We need a way to informally contact several professors of a given domain informally, to increase the chances that at least one might be interested. It is pointless to just let students loose abroad and hope that they will bring things back to their home country: a more cohesive infrastructure is needed to nurture that.
There is basically one sane way to achieve these goals: the exchange programs must be organized at a national level, not in an ad-hoc per-university manner.
Another good idea is to have taxes that depend on your nationality alone and which only start collecting when you reach a very high amount of net worth. So e.g. if someone leaves the country and makes it big, then and only then does the Government starts clawing back the benefits of its investments in the person. Furthermore, such taxes could be reduced if the person brings some of the business back to the country. And mandatory taxes should be charged if the person decides to drop their nationality at some point.
The above points would also be greatly eased by having a national-level exchange program. E.g. in Brazil in the 2010's which Ciro experience, every university had different terms and conditions, which made everything a mess. Exchange programs must be treated as a unified federal policy.
Ciro actually had to return for just six months from the École Polytechnique to the University of São Paulo, to finish a course he had only done the generic Maths/Physics introduction to. Students from other Brazilian universities were forced to return for up to 3 years even to get their Brazilian diplomas! Ciro was lucky that his teachers understood the situation, and allowed him to develop online learning projects instead of his supposed control engineering projects, which hopefully will have led to changing the world with motivation one day. And for this, Ciro is eternally thankful.
This shows the complete and total lack of any Brazilian strategy to send its students abroad to really learn valuable things and then come back. There is no strategy at all. Things have just reached an equilibrium point of bureaucracies, Brazilian universities trying to bring students back to validate useless diploma pieces of paper, and foreign universities no caring about that, and just wanting the students to say abroad forever.
Ciro was once talking about why so few Brazilians go study abroad compared to the Chinese. Besides the likely true "there are a lot of Chinese" argument, his wife made another: good point Brazil is not so bad to live in, because you have good food and freedom, while China only has good food.
But Ciro still fells bad that so few of his University of São Paulo colleagues, who learnt automation and control engineering, are doing deep tech. Nor physical engineering. They have all basically become computer people like Ciro.
This is not their fault. They basically don't have a choice: all physical science and technology is done in rich countries.
Yes, someone has to implement the newest tech to improve local country efficiency in projects that will never spread abroad.
But who will be left then for the next big thing problems that would really make Brazil richer? 6 out of 30 person class ended up working on a gaming company at one point, even though they were not crazy passionate about the field! What could possibly be a worst investment for society?
This lack of technological innovation can also be clearly seen when you research investment options available in Brazil. Huge emphasis is put on fixed return financial products (often inflation adjusted) linked to base non-tech business such as housing market and agriculture. And when you look to the returns of the stock market on s&P 500-analogue backed exchange-traded funds, they do not seem obviously better, especially considering inflation and taxation benefits that exist for some of the other investment possibilities.
When the companies of a country are not clearly the best investment, you know that something is wrong. They are highly specialized money making machines, remember! And housing and agriculture are not such innovative markets where people can hugely influence efficiency.
When it is best to send students is a good question. Undergrad studies could be easily reproduced in poor countries if they had any intelligence at all, since even in rich countries laboratory usage is always limited. Masters and PhD are generally more valuable moments to send people out. The question is if the students will actually have a fighting chance without having been out, in particular in terms of language skills. Ciro feels that Masters are a good focus point for entry, as that is where PhD links are more actively done.
Faster-than-light implies time travel by Ciro Santilli 35 Updated 2025-01-22 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-01-22 +Created 1970-01-01
Spacetime diagram illustrating how faster-than-light travel implies time travel
. Legend an explanation are shown in this answer.Bibliography:
- physics.stackexchange.com/questions/13001/does-superluminal-travel-imply-travelling-back-in-time/615079#615079
- physics.stackexchange.com/questions/574395/why-would-ftl-imply-time-travel
- physics.stackexchange.com/questions/516767/how-does-a-tachyonic-antitelephone-work
- www.physicsmatt.com/blog/2016/8/25/why-ftl-implies-time-travel shows the causality violation on a Spacetime diagram
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