Quantum mechanics

le cat is le dead and le alive at le same time
>bazoingo

quantum mechanics is a fucking meme, and a stale one at that

>>8750691
you do know there are other interpretations than Copenhagen, right?

>>8750613
Could you be a bit more specific?
What are you learning it for? What exactly in quantum mechanics do you want to learn about (it's a HUGE field)? What is your background? To what depth do you want to study?

>>8750613
Sure thing senpai.

The basic object of study in QM is the "state", normally denoted by a ket [math] | \psi \rangle [/math], this is a vector (often called an "abstract vector" in some introductory text, but I find this name retarded and insulting so it'll be avoided). the ket can define a matrix in a discrete case and an integral in the continuous case.

You might be wondering how we define an inner product with kets, well to do that we need the ket's dual, the bra [math] \langle \phi | [/math] we stick the two together as [eqn] \langle \phi | \psi \rangle = \left ( \langle \psi | \phi \rangle \right )^* \in \mathbb { C } [/eqn] Furthermore [math] | | \psi \rangle |^2 = \langle \psi | \psi \rangle [/math] but normalisations are physically meaningless, so we may as well choose one that makes our life easier, in that case we pick [math] \langle \psi | \psi \rangle = 1 [/math].

Now if we have state vector [math] | \psi \rangle [/math] we can define a basis for this in the following way [eqn] | \psi \rangle = \sum _{n} \psi _{n} | n \rangle [/eqn] and we want the basis to be orthonormal, so we enforce [math] \langle m | n \rangle = \delta _{m,n} [/math] so that we have [eqn] \langle m | \psi \rangle = \sum _n \psi _n \delta _{m,n} = \psi _m [/eqn] we call [math] \langle m | \psi \rangle [/math] the "probability amplitude", moreover we have the following relation, called the "completeness relation" [eqn] \hat { 1 } = \sum _{n} | n \rangle \langle n | [/eqn]

When it comes tp extracting values from QM we use operators called "observables" )or perhaps that would be more accurate to say that observable are operators with certain properties), so if we have an observable [math] \xi _n [/math] in the basis [math] | n \rangle [/math] the we can construct the "spectral decomposition as:

>>8750714
[eqn] \hat { \xi } = \sum _m \xi _m \langle m | n \rangle [/eqn] Which satisfies [eqn] 1.) ~ | n \rangle ~ \text { are the eigenvectors (often called eigenstates) and } ~ \xi _n ~ \text { are eigenvalues } \\ 2.) ~ \hat { \xi } ~ \text {are linear } \\ 3.) ~ \hat { \xi } ~ \text { is Hermitian } [/eqn]Hence we can think of the measurement process as projecting the state [math] | \psi \rangle [/math] onto a specific eigenstate [math] | n \rangle [/math], to this end we define the projection operator [eqn] \hat { P } _n = | n \rangle \langle n | [/eqn] And so after measurement the state is no longer [math] | \psi \rangle [/math] but [math] \hat { P } _n | \psi \rangle [/math].

Now lets define the average value of many measurements of the same state [eqn] \bar { \xi } = \sum _{n} | \langle n | \psi \rangle |^2 = \sum _n \langle \psi | n \rangle \langle n | \psi \rangle = \sum _n \langle \psi | ( \hat { \alpha } _n | n \rangle \langle n |) \psi \rangle = \langle \psi | \hat { \xi } | \psi \rangle [/eqn]

>>8750700
le alternate uniberses
le infinite superpositions

i don`t think there`s a more fedora tipping branch of science than quantum physics

>>8750718
In the continuous case we pretty much just swap the summation with an integral, The definition of ket then becomes [eqn] | \psi \rangle = \int _{a} ^{b} \phi (x) | x \rangle ~ dx [/eqn] and we can enforce by insisting that [math] \langle x | x' \rangle = \delta ( x - x') [/math] we can see that this produces the expected result [eqn] \langle x' | \phi \rangle = \int _{a} ^{b} \phi (x) \langle x' | x \rangle ~ dx = \int _{a} ^{b} \phi (x) \delta ( x -x') dx = \phi (x') [/eqn]Likewise we define a completeness relation by [eqn] \int _{a} ^{b} | x \rangle \langle x | dx = \hat { 1 } [/eqn]Lets use our machinery to define the position operator [eqn] \hat { x } = \int _{a} ^{b} dx x | x \rangle \langle x | \\ \implies \hat { x } | x \rangle = \int _{a} ^{b} dx' x' | x \rangle \langle x' | x \rangle = \int _{a} ^{b} dx' x' | x' \rangle \delta ( x' - x )=x | x \rangle [/eqn]Finally we can define the continuous analogue of the average value of many measurements as [eqn] \langle \phi | x | \phi \rangle = \int _{a} ^{b} dx \langle \phi | x \rangle x \langle x| \phi \rangle = \int _{a} ^{b} dx \phi ^{*} x \phi[/eqn]

And that (along with the commutator) is pretty much the heart of QM, would you like to know more?

>>8750713
Sure, i want to learn quantum computing and I want to learn the basics principles

>>8750714
>>8750718
Physicists are disgusting. No discussion on how/why these series actually converge.

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>>8750873
whats the matter anon? cant stand a lack of mathematical rigor?

>>8750873
It's simple anon, it's physics so they all must converge. If it helps, just take convergence as an axiom :^)

>>8750880
Is for a work

>>8750880
Omitting a discussion of convergence leads to students having inherent misconceptions on basic properties of these series.

People end up thinking they always behave the same way as finite sums and fuck things up.

>>8750745
ty

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>>8750873

>finite sum
>discussion of convergence

>>8750873
I'm pretty sure a finite sum always converge

>>8750905
can you give an example of a mistake one could make by doing that?

>>8750725
quantum physics doesn't care about any of that. it cares about the math.
what it actually means is a question for philosophy

>>8750700
HAHAHAHAHAHAHAHAHAHAHA

>>8752420
>>8752425
If you are assuming finite sums then you are assuming a finite dimensional hilbert space and therefore omitting a lot of possible QM systems.

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>>8750613

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>>8755557

>>8753771
>what are integrals and L2 functions

>>8755557
This looks so cool.

>>8755654
>Every infinite dimensional Hilbert space is an L2 space

>>8755998
>implying I said that

What are you even objecting to? By definition a Hilbert space has a finite inner product...

>>8750613
Read any freshman physics textbook.

Einberg and Resnick is p good. Currently finishing up Ch 5, the examples are very easy to follow which is always good