ket in quantum physics

low voltage hobby physica · Anmeldungsdatum: 26.03.2017 Beiträge: 1

Meine Frage:
Hi, I have a problem to understand what really is a ket, i know what is a hilbertspace, here we have skalarproduct and covergence of every caucy secuence. For a (normal) function with for example dependency of x,y,z, I understand this. But what is a ket, here we do not have any dependency from any stationary vector (as x,y,z). What should this be ? A matrix with entries ? What does it say ? and how can any operator be defined that can be applied on a ket ? for normal function f(x,y,z) this is clear.
Every wave function is assigened to a ket how does this work, which information of wave function contains the ket vektor ?
Hope someone can explain me

Meine Ideen:
I have suggestion, that ket is a matrix with entries of different values of the wave function on different places for stationary vector, because there are endless entries possible (stationary verctor is continious), the ket is a matrix (or row vector) with an endless numer of entries. But if this is case, we have a dependency from stationary vector in the ket matrix

\left| n \right> =\begin{pmatrix}till -\infty \\f(x1) \\ f(x2) \\ f(x3) \\till +\infty \end{pmatrix}

jh8979 · Moderator Anmeldungsdatum: 10.07.2012 Beiträge: 8581

Have a look here:
https://en.wikipedia.org/wiki/Bra–ket_notation
especially here:
https://en.wikipedia.org/wiki/Bra–ket_notation#Ket_notation_for_vectors
and -for the connection to wave functions- here:
https://en.wikipedia.org/wiki/Bra–ket_notation#Spinless_position.E2.80.93space_wave_function

Physicshobbyreader · Anmeldungsdatum: 25.03.2017 Beiträge: 9

When I understood it right, there is a hilbert space, in this space we have different Basis possilble, for example

or

where the first related to space, the second to impules.
The ket can be expressed in both basises, it is the same object, but with different representatations in different basises. Because there are endless points in space, the dimension of hilbert space is infinite.
The ket can be expressed in Basis

via integral,

so that the wave function on a defined place (fixed value for x) is only the weight a special basis vector in

is multiplied with. For comparison if we have a normal 3 dim space, the vektor x is expressed as a1*e1+a2*e2*a3*e3, the component a1, a2, a3 are the weight of the basis vektors are mulitpied with. So, now we have the ket, to come back from a ket to the wave function is done via buliding of scalar product

.
Simliar to a normal 3 dim space where for example scalarproduct with basis verktor e1 returns the component a1 the vector is multiplied in the basis, the scalarproduct retuns the components how every "point" of the vector

is "multiplied" (we have integral so multiplied is not fully right) with. Think this should be right.

But I have still a problem to perform the integraton (realy calculte it) in this equation

, how to do it ? I know nothing about

, only that it is a basis in a hilbert space, wich function is

, when doing intergation.

The same Problem I have with equation

, here also I must perform a scalar product with

. Don't know how to introduce

in this equation

Also I like to know more about the definition of this special hilbert space,
and the definition of basis in it. Normally you define a special hilbert space saying for example a special set of functions fulfills the definition to be a hilbert space. But here I read nothing about any such criterias, do we here say,
the space ist spanned by definition of basis ? It might have something to do with delta-function and / or complex exp-function. I don't know. Regarding the basis, this is not any typ of function, so what is

. In endles dimentional spaces, I read it is not alwas prossible to find a basis, but here we have one. How is it enshured, that

and

really form a basis of a Hilbert space.
These two basis kets are also assigened to waveform, ok. you can argue this waveforms span a basis, but how can you argue for the basis kets ?
Thanks for any answers,
Regards

jh8979 · Moderator Anmeldungsdatum: 10.07.2012 Beiträge: 8581

Physicshobbyreader · Anmeldungsdatum: 25.03.2017 Beiträge: 9

Thank you for your answer now I neraly understand.
There is one open piont I still have, the base vectors

and

are still for me only mathematical objects which form a basis of a vector space, in physics they have a real meaning, they are relatet to spce and impulse. So why can we say

has something to do with space, is because of the cotext to the wave function, because this function has the dependency from space. The same for impulse, after your explanations, I
understand what I saw in a book, the same formalism as done for space can also be introduced for impulse, here they call it wave-fucntion in impulse spache. So here it is the same question, is the base vector

related to impulse, because the wavefunction is dependent from impulse.

And here is another point where I'm not fully clear, in this book it is wiritten we assign to every wave funcion a ket, for

they assign the ket to delta function delta(\vec{r}-\vec{r0}) and for impulse the basis ket is assigened to exponential function exp(i/h*\vec{p0} \vec{r}).
Can it be that this special wavefunction for basis ket can be introdruced with the same formalism as for any other ket.

Thank you for reply,
Regards

jh8979 · Moderator Anmeldungsdatum: 10.07.2012 Beiträge: 8581

Physicshobbyreader · Anmeldungsdatum: 25.03.2017 Beiträge: 9

thank you I'm clear with it, if someone explains it is not so confusing,
but I think this (and other) topic(s) about quatum mechanic is (are) explained in literature and scripts not in precise detail, so that I often do not understand.
When you read mathematical stuff everything is explained with exact definitions this is for me better to understand.

One topic I like to ask an additional question do you know any paper or book, where the exact mathematical definition of hilbertspace the kets live in is expained (not wavefunctions but ket), maybe as a topic I read later, when I'm more familiar with this stuff. I read the space of wavefunctions and hilbert space of kets are isomorph, I think it is interesting to know why.
Thanks, best regards

jh8979 · Moderator Anmeldungsdatum: 10.07.2012 Beiträge: 8581

Sakurai: Modern Quantum Mechanics
https://www.amazon.de/Modern-Quantum-Mechanics-San-Tuan/dp/0201539292

Physicshobbyreader · Anmeldungsdatum: 25.03.2017 Beiträge: 9

fine, thanks