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 $L^2$ 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:

Bibliography:

There is also a time-energy uncertainty principle, because those two operators are also complementary.

See: angular momentum operator.

This is my first working setup, done in a rental friendly way without drilling. I am sure that it is possible to do it cheaper, better and with less work, but it the first one that worked for me, so I will document it.

Cost: 150 Dollars for a relatively large 130 x 158 cm window. They do not sell for windows much larger than that.

Bought from www.orderblinds.co.uk on June 2019.

My window faces East, and in summer the sun rises at around 5AM here, and I am a bit light sensitive and was getting destroyed.

This setup, together with:is good enough to allow me to sleep properly, which is priceless.

Demonstration if everything goes exceptionally well:

However, not all will necessarily be fun and games as in that YouTube video, especially if you are doing it for the first time, and the main point of this article is to make you aware of that.

The first problem is that you may have to remove existing useless "privacy blinds" from the way, which can mean putting effort into learning how they work and, has a risk of damaging the property. So be smart and get a chair and a second person to help you out!

Next, fitting the side rails is not going to be that easy. The thing has to be tight to block the light, right? Careful not to scratch the bottom sill!

Then you will notice that, like in the video at youtu.be/cTVVe7codw4?t=106, you are expected to screw the side rails to the bottom wood / plastic sill of the window, which is not rental acceptable! So I didn't to that, but together with the black tape that I will mention next, it held well enough.

The top part requires hammering nails as shown on the video: youtu.be/cTVVe7codw4?t=24 but this is generally rental acceptable, and you can fill the holes afterwards.

Once you have the setup in place, it is likely that there will be some light gaps still, because it is basically impossible to make such large objects fit perfectly. This was especially true for the top of the window, but also for the sides, so I just used some black tape

So I added some wide (50mm) Figure 4. "Diall PVC repairing black tape" between the blind and the wall to completely seal off the light.

There is still some leakage at the bottom however, which cannot be taped. I mitigated that a bit by putting some black T-shirts on the bottom window sill, and together with the pre-existing rental curtain, it was enough.

Another minor annoyance is that parts of the blind cloth sometimes slip out of the conduit holes in the aluminium side bars. For this reason, I have to always open and close slowly and carefully. But if it does happen, fully opening and closing carefully has solved the problem.

Proof that the probability 1 is conserved by the time evolution:

It can be derived directly from the Schrödinger equation.

Bibliography:

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)".

See form.

Analogous to a linear form, a multilinear form is a Multilinear map where the image is the underlying field of the vector space, e.g. ${\mathbb{R}}^{n_1}\times {\mathbb{R}}^{n_2}\times {\mathbb{R}}^{n_2}\to \mathbb{R}$.

This Heisenberg's breakthrough paper on matrix mechanics which later led to the Schrödinger equation, see also: history of quantum mechanics.

Published on the Zeitschrift für Physik volume 33 page pages 879-893, link.springer.com/article/10.1007%2FBF01328377

Modern overview: www.mat.unimi.it/users/galgani/arch/heisenberg25amer_j_phys.pdf