Ciro Santilli (@cirosantilli) - undefined articles

Position and momentum space  Updated 2024-12-15  +Created 1970-01-01

One of the main reasons why physicists are obsessed by this topic is that position and momentum are mapped to the phase space coordinates of Hamiltonian mechanics, which appear in the matrix mechanics formulation of quantum mechanics, which offers insight into the theory, particularly when generalizing to relativistic quantum mechanics.

One way to think is: what is the definition of space space? It is a way to write the wave function

ψ_{x} (x)

such that:

the position operator is the multiplication by $x$
the momentum operator is the derivative by $x$

And then, what is the definition of momentum space? It is of course a way to write the wave function

ψ_{p} (p)

such that:

the momentum operator is the multiplication by $p$

physics.stackexchange.com/questions/39442/intuitive-explanation-of-why-momentum-is-the-fourier-transform-variable-of-posit/39508#39508 gives the best idea intuitive idea: the Fourier transform writes a function as a (continuous) sum of plane waves, and each plane wave has a fixed momentum.

Bibliography:

en.wikipedia.org/wiki/Position_and_momentum_space

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Quantum computers as experiments that are hard to predict outcomes  Updated 2024-12-15  +Created 1970-01-01

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One possibly interesting and possibly obvious point of view, is that a quantum computer is an experimental device that executes a quantum probabilistic experiment for which the probabilities cannot be calculated theoretically efficiently by a nuclear weapon.

This is how quantum computing was originally theorized by the likes of Richard Feynman: they noticed that "Hey, here's a well formulated quantum mechanics problem, which I know the algorithm to solve (calculate the probability of outcomes), but it would take exponential time on the problem size".

The converse is then of course that if you were able to encode useful problems in such an experiment, then you have a computer that allows for exponential speedups.

This can be seen very directly by studying one specific quantum computer implementation. E.g. if you take the simplest to understand one, photonic quantum computer, you can make systems for which you need exponential time to calculate the probabilities that photons will exit through certain holes and not others.

The obvious aspect of this idea is by coming from quantum logic gates are needed because you can't compute the matrix explicitly as it grows exponentially: knowing the full explicit matrix is impossible in practice, and knowing the matrix is equivalent to knowing the probabilities of every outcome.

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Quantum electrodynamics  Updated 2024-12-15  +Created 1970-01-01

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Theory that describes electrons and photons really well, and as Feynman puts it "accounts very precisely for all physical phenomena we have ever observed, except for gravity and nuclear physics" ("including the laughter of the crowd" ;-)).

Learning it is one of Ciro Santilli's main intellectual fetishes.

While Ciro acknowledges that QED is intrinsically challenging due to the wide range or requirements (quantum mechanics, special relativity and electromagnetism), Ciro feels that there is a glaring gap in this moneyless market for a learning material that follows the Middle Way as mentioned at: the missing link between basic and advanced. Richard Feynman Quantum Electrodynamics Lecture at University of Auckland (1979) is one of the best attempts so far, but it falls a bit too close to the superficial side of things, if only Feynman hadn't assumed that the audience doesn't know any mathematics...

The funny thing is that when Ciro Santilli's mother retired, learning it (or as she put it: "how photons and electrons interact") was also one of her retirement plans. She is a pharmacist by training, and doesn't know much mathematics, and her English was somewhat limited. Oh, she also wanted to learn how photosynthesis works (possibly not fully understood by science as that time, 2020). Ambitious old lady!!!

Experiments: quantum electrodynamics experiments.

Combines special relativity with more classical quantum mechanics, but further generalizing the Dirac equation, which also does that: Dirac equation vs quantum electrodynamics. The name "relativistic" likely doesn't need to appear on the title of QED because Maxwell's equations require special relativity, so just having "electro-" in the title is enough.

Before QED, the most advanced theory was that of the Dirac equation, which was already relativistic but TODO what was missing there exactly?

As summarized at: youtube.com/watch?v=_AZdvtf6hPU?t=305 Quantum Field Theory lecture at the African Summer Theory Institute 1 of 4 by Anthony Zee (2004):

classical mechanics describes large and slow objects
special relativity describes large and fast objects (they are getting close to the speed of light, so we have to consider relativity)
classical quantum mechanics describes small and slow objects.
QED describes objects that are both small and fast

That video also mentions the interesting idea that:

in special relativity, we have the mass-energy equivalence
in quantum mechanics, thinking along the time-energy uncertainty principle, $Δ E \sim \frac{1}{Δ t}$

Therefore, for small timescales, energy can vary a lot. But mass is equivalent to energy. Therefore, for small time scale, particles can appear and disappear wildly.

QED is the first quantum field theory fully developed. That framework was later extended to also include the weak interaction and strong interaction. As a result, it is perhaps easier to just Google for "Quantum Field Theory" if you want to learn QED, since QFT is more general and has more resources available generally.

Like in more general quantum field theory, there is on field for each particle type. In quantum field theory, there are only two fields to worry about:

photon field
electromagnetism field

Video 1.

Lecture 01 | Overview of Quantum Field Theory by Markus Luty (2013)

Source. This takes quite a direct approach, one cool thing he says is how we have to be careful with adding special relativity to the Schrödinger equation to avoid faster-than-light information.

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Quantum key distribution  Updated 2024-12-15  +Created 1970-01-01

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Man-in-the-middle attack

quantumcomputing.stackexchange.com/questions/142/advantage-of-quantum-key-distribution-over-post-quantum-cryptography/25727#25727 Advantage of quantum key distribution over post-quantum cryptography has Ciro Santilli's comparison to classical encryption.

BB84 is a good first algorithm to look into.

Long story short:

QKD allows you to generate shared keys without public-key cryptography. You can then use thses shared keys
QKD requires authentication on a classical channel, exactly like a classical public-key cryptography forward secrecy would. The simplest way to do this is a with a pre-shared key, just like in classical public key cryptography. If that key is compromised at any point, your future messages can get man-in-the-middle'd, exactly like in classical cryptography.

QKD uses quantum mechanics stuff to allow sharing unsnoopable keys: you can detect any snooping and abort communication. Unsnoopability is guaranteed by the known laws of physics, up only to engineering imperfections.

Furthermore, it allows this key distribution without having to physically take a box by car somewhere: once the channel is established, e.g. optical fiber, you can just keep generating perfect keys from it. Otherwise it would be pointless, as you could just drive your one-time pad key every time.

However, the keys likely have a limited rate of generation, so you can't just one-time pad the entire message, except for small text messages. What you would then do is to use the shared key with symmetric encryption.

Therefore, this setup usually ultimately relies on the idea that we believe that symmetric encryption is safer than , even though there aren't mathematical safety proofs of either as of 2020.

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Real world applications of the Lebesgue integral  Updated 2024-12-15  +Created 1970-01-01

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In "practice" it is likely "useless", because the functions that it can integrate that Riemann can't are just too funky to appear in practice :-)

Its value is much more indirect and subtle, as in "it serves as a solid basis of quantum mechanics" due to the definition of Hilbert spaces.

Bibliography:

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Relativistic quantum mechanics  Updated 2024-12-15  +Created 1970-01-01

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The first really good quantum mechanics theory made compatible with special relativity was the Dirac equation.

And then came quantum electrodynamics to improve it: Dirac equation vs quantum electrodynamics.

TODO: does it use full blown QED, or just something intermediate?

www.youtube.com/watch?v=NtnsHtYYKf0 "Mercury and Relativity - Periodic Table of Videos" by Periodic Videos (2013). Doesn't give the key juicy details/intuition. Also mentioned on Wikipedia: en.wikipedia.org/wiki/Relativistic_quantum_chemistry#Mercury

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Richard Feynman  Updated 2024-12-15  +Created 1970-01-01

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Some of Feynman's key characteristics are:

obsession with understanding the experiments well, see also Section "How to teach and learn physics"
when doing more mathematical stuff, analogous obsession about starting with a concrete example and then generalizing that into the theory
liked to teach others. At Surely You're Joking, Mr. Feynman for example he mentions that one key problem of the Institute for Advanced Study is that they didn't have to teach, and besides that making you feel useless when were not having new ideas, it is also the case that student's questions often inspire you to look again in some direction which sometimes happens to be profitable
He hated however mentoring others one to one, because almost everyone was too stupid for him
interest in other natural sciences, and also random art and culture (and especially if it involves pretty women)

Some non-Physics related ones, mostly highlighted at Genius: Richard Feynman and Modern Physics by James Gleick (1994):

Feynman was a huge womanizer during a certain period of his life
he hated pomp, going as far as seeming uneducated to some people in the way he spoke, or going out of his way to look like that. This is in stark contrast to "rivals" Murray Gell-Mann and Julian Schwinger, who were posh/snobby.

Even Apple thinks so according to their Think different campaign: www.feynman.com/fun/think-different/

quantum electrodynamics lectures:

Richard Feynman Quantum Electrodynamics Lecture at University of Auckland (1979)

Feynman was apparently seriously interested/amused by computer:

Video "Los Alamos From Below by Richard Feynman (1975)" see description for the human emulator
quantum computers as experiments that are hard to predict outcomes was first attributed to Feynman
www.youtube.com/watch?v=EKWGGDXe5MA Richard Feynman Computer Heuristics Lecture (1986)

Video 1.

Murray Gell-Mann talks about Richard Feynman's intentional anecdote creation

. Source. TODO original interviewer, date and source. Very amusing, he tells how Feynman wouldn't brush his teeth, or purposefully forget to wear jacket and tie when going to the faculty canteen where it was required and so he would use ugly emergency jacket the canteen offered to anyone who had forgotten theirs.

Video 2.

Murray Gell-Mann talks about Feynman's partons by Web of Stories (1997)

Source. Listener is likely this Geoffrey West. Key quote:

Feynman of course, as usual, put it in a form so that the common people could use it, and experimentalists all over the world now thought they understood things because Feynman had put it in such simple language for them.

Two official websites?

www.richardfeynman.com/ this one has clearly superior scientific information.
www.feynman.com/

High level timeline of his life:

In 1948 he published his reworking of classical quantum mechanics in terms of the path integral formulation: journals.aps.org/rmp/abstract/10.1103/RevModPhys.20.367 Space Time Approach to nonrelativistic quantum mechanics (paywalled 2021)

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Schrödinger equation  Updated 2024-12-15  +Created 1970-01-01

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The partial differential equation of non-relativistic quantum mechanics.

Experiments explained:

via the Schrödinger equation solution for the hydrogen atom it predicts:
- spectral line basic lines, plus Zeeman effect
Schrödinger equation solution for the helium atom: perturbative solutions give good approximations to the energy levels
double-slit experiment: I think we have a closed solution for the max and min probabilities on the measurement wall, and they match experiments

Experiments not explained: those that the Dirac equation explains like:

fine structure
spontaneous emission coefficients

To get some intuition on the equation on the consequences of the equation, have a look at:

The easiest to understand case of the equation which you must have in mind initially that of the Schrödinger equation for a free one dimensional particle.

Then, with that in mind, the general form of the Schrödinger equation is:

i ℏ \frac{\partial ψ ( x , t )}{\partial t} = \hat{H} [ψ (x, t)]

(1)

Equation 1.

Schrodinger equation

.

where:

$ℏ$ is the reduced Planck constant
$ψ$ is the wave function
$t$ is the time
$\hat{H}$ is a linear operator called the Hamiltonian. It takes as input a function $ψ$ , and returns another function. This plays a role analogous to the Hamiltonian in classical mechanics: determining it determines what the physical system looks like, and how the system evolves in time, because we can just plug it into the equation and solve it. It basically encodes the total energy and forces of the system.

The

x

argument of

ψ

could be anything, e.g.:

we could have preferred polar coordinates instead of linear ones if the potential were symmetric around a point
we could have more than one particle, e.g. solutions of the Schrodinger equation for two electrons, which would have e.g. $x_{1}$ and $x_{2}$ for different particles. No matter how many particles there are, we have just a single $ψ$ , we just add more arguments to it.
we could have even more generalized coordinates. This is much in the spirit of Hamiltonian mechanics or generalized coordinates

Note however that there is always a single magical

t

time variable. This is needed in particular because there is a time partial derivative in the equation, so there must be a corresponding time variable in the function. This makes the equation explicitly non-relativistic.

The general Schrödinger equation can be broken up into a trivial time-dependent and a time-independent Schrödinger equation by separation of variables. So in practice, all we need to solve is the slightly simpler time-independent Schrödinger equation, and the full equation comes out as a result.

Existence and uniqueness: mathoverflow.net/questions/212913/existence-and-uniqueness-for-two-dimensional-time-dependent-schr%C3%B6dinger-equation

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Spectral line  Updated 2024-12-15  +Created 1970-01-01

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A single line in the emission spectrum.

So precise, so discrete, which makes no sense in classical mechanics!

Has been the leading motivation of the development of quantum mechanics, all the way from the:

Schrödinger equation: major lines predicted, including Zeeman effect, but not finer line splits like fine structure
Dirac equation: explains fine structure 2p spin split due to electron spin/orbit interactions, but not Lamb shift
quantum electrodynamics: explains Lamb shift
hyperfine structure: due to electron/nucleus spin interactions, offers a window into nuclear spin

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Why you should give money to Ciro Santilli  Updated 2024-12-15  +Created 1970-01-01

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The Fourier transform is a bijection in

L^{2}

 Updated 2024-12-15  +Created 1970-01-01

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As mentioned at Section "Plancherel theorem", some people call this part of Plancherel theorem, while others say it is just a corollary.

This is an important fact in quantum mechanics, since it is because of this that it makes sense to talk about position and momentum space as two dual representations of the wave function that contain the exact same amount of information.

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