Used to explain the black-body radiation experiment.
Published as: On the Theory of the Energy Distribution Law of the Normal Spectrum by Max Planck (1900).
The Quantum Story by Jim Baggott (2011) page 9 mentions that Planck apparently immediately recognized that Planck constant was a new fundamental physical constant, and could have potential applications in the definition of the system of units (TODO where was that published):This was a visionary insight, and was finally realized in the 2019 redefinition of the SI base units.
Planck wrote that the constants offered: 'the possibility of establishing units of length, mass, time and temperature which are independent of specific bodies or materials and which necessarily maintain their meaning for all time and for all civilizations, even those which are extraterrestrial and nonhuman, constants which therefore can be called "fundamental physical units of measurement".'
TODO how can it be derived from theoretical principles alone? There is one derivation at; en.wikipedia.org/wiki/Planck%27s_law#Derivation but it does not seem to mention the Schrödinger equation at all.
Quantum Mechanics 2 - Photons by ViaScience (2012)
Source. Contains a good explanation of how discretization + energy increases with frequency explains the black-body radiation experiment curve: you need more and more energy for small wavelengths, each time higher above the average energy available.The Royal Society's Nobel Prize.
royalsociety.org/grants-schemes-awards/awards/copley-medal/ says it is now open to international citizens, but having a quick look at the 2010 awards still suggests that it is very British centric, or at least anglophone centric, much like the society fellowship itself. That's likely the reason why the Nobel prize won, being much more international from the start.
More like a "lifetime achievement" though, rather than the Nobel Prize, which tends to be for more specific achievements.
The traffic is designed for cars, which makes many red stops for bicycles completely stupid.
In a bicycle you just have too much more control and awareness than in a car, so if the way is completely clear, you should be allowed to stop, look if the way is clear, and then run reds.
Of course, this does increase the chances of hitting pedestrians a little bit. But the risk change feels so little that it would be worth it. Studies quoted by Wikipedia corroborate. It just feels extremely unintuitive to make cyclists stop in certain places when the street is clear.
- www.theguardian.com/cities/2015/oct/27/cyclists-run-red-lights-paris-london-san-francisco Should cyclists be allowed to run red lights? by The Guardian 2015
- www.vox.com/2014/5/9/5691098/why-cyclists-should-be-able-to-roll-through-stop-signs-and-ride
Let's start with the one dimensional case. Let the and a Functional defined by a function of three variables :
Then, the Euler-Lagrange equation gives the maxima and minima of the that type of functional. Note that this type of functional is just one very specific type of functional amongst all possible functionals that one might come up with. However, it turns out to be enough to do most of physics, so we are happy with with it.
Given , the Euler-Lagrange equations are a system of ordinary differential equations constructed from that such that the solutions to that system are the maxima/minima.
In the one dimensional case, the system has a single ordinary differential equation:
By and we simply mean "the partial derivative of with respect to its second and third arguments". The notation is a bit confusing at first, but that's all it means.
Therefore, that expression ends up being at most a second order ordinary differential equation where is the unknown, since:
- the term is a function of
- the term is a function of . And so it's derivative with respect to time will contain only up to
Now let's think about the multi-dimensional case. Instead of having , we now have . Think about the Lagrangian mechanics motivation of a double pendulum where for a given time we have two angles.
Let's do the 2-dimensional case then. In that case, is going to be a function of 5 variables rather than 3 as in the one dimensional case, and the functional looks like:
This time, the Euler-Lagrange equations are going to be a system of two ordinary differential equations on two unknown functions and of order up to 2 in both variables:At this point, notation is getting a bit clunky, so people will often condense the vectoror just omit the arguments of entirely:
This is about functions that take functions as input or output.
This section is about telecommunication systems that are based on top of telephone lines.
Telephone lines were ubiquitous from early on, and many technologies used them to send data, including much after regular phone calls became obsolete with VoIP.
These market forces tended to eventually crush non-telephone-based systems such as telex. Maybe in that case it was just that the name sounded like a thing of the 50's. But still. Dead.
Long Distance by AT&T (1941)
Source. youtu.be/aRvFA1uqzVQ?t=219 is perhaps the best moment, which attempts to correlate the exploration of the United States with the founding of the U.S. states. Does the exact position of vertices matter in Feynman diagrams? Updated 2025-03-28 +Created 1970-01-01
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