The wave equation contains the entire state of a particle.
From mathematical formulation of quantum mechanics remember that the wave equation is a vector in Hilbert space.
And a single vector can be represented in many different ways in different basis, and two of those ways happen to be the position and the momentum representations.
More importantly, position and momentum are first and foremost operators associated with observables: the position operator and the momentum operator. And both of their eigenvalue sets form a basis of the Hilbert space according to the spectral theorem.
When you represent a wave equation as a function, you have to say what the variable of the function means. And depending on weather you say "it means position" or "it means momentum", the position and momentum operators will be written differently.
This is well shown at: Video "Visualization of Quantum Physics (Quantum Mechanics) by udiprod (2017)".
Furthermore, the position and momentum representations are equivalent: one is the Fourier transform of the other: position and momentum space. Remember that notably we can always take the Fourier transform of a function in due to Carleson's theorem.
Then the uncertainty principle follows immediately from a general property of the Fourier transform: en.wikipedia.org/w/index.php?title=Fourier_transform&oldid=961707157#Uncertainty_principle
In precise terms, the uncertainty principle talks about the standard deviation of two measures.
We can visualize the uncertainty principle more intuitively by thinking of a wave function that is a real flat top bump function with a flat top in 1D. We can then change the width of the support, but when we do that, the top goes higher to keep probability equal to 1. The momentum is 0 everywhere, except in the edges of the support. Then:
- to localize the wave in space at position 0 to reduce the space uncertainty, we have to reduce the support. However, doing so makes the momentum variation on the edges more and more important, as the slope will go up and down faster (higher top, and less x space for descent), leading to a larger variance (note that average momentum is still 0, due to to symmetry of the bump function)
- to localize the momentum as much as possible at 0, we can make the support wider and wider. This makes the bumps at the edges smaller and smaller. However, this also obviously delocalises the wave function more and more, increasing the variance of x
Bibliography:
- www.youtube.com/watch?v=bIIjIZBKgtI&list=PL54DF0652B30D99A4&index=59 "K2. Heisenberg Uncertainty Relation" by doctorphys (2011)
- physics.stackexchange.com/questions/132111/uncertainty-principle-intuition Uncertainty Principle Intuition on Physics Stack Exchange
TODO is there any good intuitive argument or proof of conservation of energy, momentum, angular momentum?
Contains the full state of the quantum system.
This is in contrast to classical mechanics where e.g. the state of mechanical system is given by two real functions: position and speed.
The wave equation in position representation on the other hand encodes speed in "how fast does the complex phase spin around", and direction in "does it spin clockwise or counterclockwise", as described well at: Video "Visualization of Quantum Physics (Quantum Mechanics) by udiprod (2017)". Then once you understand that, it is more compact to just view those graphs with the phase color coded as in Video "Simulation of the time-dependent Schrodinger equation (JavaScript Animation) by Coding Physics (2019)".
Analogous problem to the secondary structure of proteins. Likely a bit simpler due to the strong tendency for complementary pairs to bind.
Converts RNA to DNA, i.e. the inverse of transcription. Found in viruses such as Retrovirus, which includes e.g. HIV.
By GitHub around Black Lives Matter, due to a possible ludicrous relationship with slavery of black people:
For the love of God, the word "master" is much more general than black slavery. If you are going to ban it, you might as well ban the word "evil".
TODO year. This was a reply to Microsoft anti-Linux propaganda it seems: www.ubuntubuzz.com/2012/03/truth-happens-redhats-legendary-reply.html
Trascript from: www.dailymotion.com/video/xw3ws
The world is flat. Earth is the centre of the universe. Fact - until proven otherwise.
Despite ignorance. Despite ridicule. Despite opposition. Truth happens.Despite ignorance.
The telephone has too many shortcomings to be seriously considered as a means of communication. /Western Union 1876/
In 1899 the US Patent Commissioner stated, everything that can be invented has been invented.Despite ridicule.
The phonograph has no commercial value at all. /Thomas Edison 1880/
The radio craze will die out in time. /Thomas Edison 1922/
The automobile has practically reached the limit of its development. /Scientific American 1909/Despite it all truth happens.
Man will not fly for fifty years. /Orville Wright 1901/
The rocket will never leave the Earth's atomosphere. /New York Times 1936/
There is a world market for maybe five computers. /IBM's Thomas Watson 1943/
640K Ought to be enough for anybody. /Bill Gates 1981/Then they laugh at you...
We think of linux as competitor in the student and hobbyist market. But I really don't think in the commercial market we'll see it in any significant way. /Bill Gates 2001/Then they fight you...
Linux isn't going away. Linux is a serious competitor. We will rise to this challenge. /Steve Ballmer 2003/Then you win... /Mohandas Gandhi/
There are unlisted articles, also show them or only show them.