Bracket VS wavefunction notation in QM

In summary, the conversation revolves around the use of bracket notation and wavefunction notation in explaining quantum mechanics. The bracket notation involves solving the Schrödinger equation and applying the Fourier transform to get the wavefunction, while the wavefunction notation involves expanding a ket in eigenvalues of position. The use of complex variables and Fourier transforms is a standard practice, and understanding linear algebra and distribution theory is crucial in understanding these concepts. The conversation also touches on the topic of probability in quantum mechanics and the importance of avoiding negative probabilities.
  • #71
I don't understand it as a whole, because I don't know what you're trying to do. Why are you using bra vector instead of ket?

olgerm said:
##<base|[i_1](a_{arguments\ of\ wavefunction})=\delta(v(i_1)-a_{arguments\ of\ wavefunction})##

That is not conventional, or (imo) even correct.
 
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  • #72
weirdoguy said:
I don't understand it as a whole
i_1'th base vector, that returns value ##\delta(v(i_1)-a_{arguments\ of\ wavefunction})## if its arguments are ##a_{arguments\ of\ wavefunction}##.
 
  • #73
So why are you using bra there? You seems to be hella confused. You should study a few (not one) textbooks first.
 
  • #74
olgerm said:
##<base|[i_1](a_{arguments\ of\ wavefunction})=\delta(v(i_1)-a_{arguments\ of\ wavefunction})\\

v(i_1)[i_2]=\sum_{i_3=-\infty}^{\infty}(2^{i_3}*((\lfloor i_1*2^{-2*i_3*A+i_2} \rfloor\mod_2)+\sqrt{-1}*(\lfloor i_1*2^{-2*i_3*A+i_2+1} \rfloor\ mod_2)))####\delta## is a function that returns 1 if its argument is 0-vector and 0 otherwise.

##\lfloor \rfloor## is floor function.

##<base|[i_1]## is i'th basevector.since ##\sigma## is not 0 only if ##v=a_{arguments\ of\ wavefunction}##, value of basefunction is 1 only in case of 1 choice of arguments and otherwise it's value is 0. How every ##i_2##'th component of v for ##i_1##'th basevectors is calculated is shown in 2. equation.
I tried to write it as easily as I could. I think I used now only conventional symbols.
How can these be "conventional symbols". I've no clue, where one should find them in any textbook or paper. I cannot make any sense of them :-(.
 
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  • #75
olgerm said:
i_1'th base vector, that returns value ##\delta(v(i_1)-a_{arguments\ of\ wavefunction})## if its arguments are ##a_{arguments\ of\ wavefunction}##.
A wave function for a spinless particle, as treated in the QM 1 lecture, in conventional notation is
$$\psi(\vec{x})=\langle \vec{x}|\psi \rangle.$$
Here ##|\psi \rangle## is a Hilbert-space vector (usually normalized to 1, i.e., ##\langle \psi|\psi \rangle=1##) and ##\langle \vec{x}|## is the generalized eigen-bra of the position operator. In this sense it's indeed conventional that the bras in the definition of the wave function contains the argument of the wave function.

BTW: From which book/manuscript are you studying? Where did you get your notation, which is far from conventional and for me completely incomprehensible.
 
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  • #76
Since my intention is to understand bracket notation(and its relation to normal notation), I did not formulate my equations by using bracket notation, but I used normal convetional mathematical notations. I used as simple notation as I could and explained meaning of some symbols.
 
  • #77
olgerm said:
I used as simple notation as I could and explained meaning of some symbols.

And yet it's convoluted and unreadable.
 
  • #78
Main point is that for every possible arguments ##a_{arguments\ of\ wavefunction}## there is one basefunction that is 1 with these arguments and 0 with other arguments.
 
  • #79
The terminology and notation you're using is not standard, nobody really understands what you're saying. Look at a typical QM or linear algebra textbook and use the notation and terminology that they use in there.
 
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  • #80
olgerm said:
Since my intention is to understand bracket notation(and its relation to normal notation), I did not formulate my equations by using bracket notation, but I used normal convetional mathematical notations. I used as simple notation as I could and explained meaning of some symbols.
Again to put it very clearly: You don't use anything, I've ever seen in the literature. It's unreadable. I don't even know, what you want to write!

You can write QT completely without bras and kets. E.g., Weinberg doesn't use the bra-ket notation at all. However his books are highly readable (and among the best newer textbooks I'm aware of).
 
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  • #81
Which symbols you do not undersatand? I did not invent any original notation, but used mathematical normal notation. I really do not know how to express these relations using more ordinary notation.
If you really do not understand - can you describe me a example of basevectors. by writing
## \vec{e_{base}} [i ] = ... ##, where ## \vec{e_{base}} ##, is i'th basevector. Maybe I could reformulate my equations to be similar to yours.
 
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  • #82
olgerm said:
can you describe me a example of basevectors

Base vectors of which space?
 
  • #83
weirdoguy said:
Base vectors of which space?
The ones that can be used here
DarMM said:
We have some basis set of functions ##\phi_n## and the components of ##\psi## are the terms ##c_n## in the sum:
$$\psi = \sum_{n}c_{n}\phi_{n}$$
 

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