Exploring the Equivalence of Different Representations in Quantum Mechanics

[SemM]

Hi, I found this article very interesting, given the loads of question I have posted in this regard in the last months. I cannot recall where I got the link from, and if it came from Bill Hobba in some discussion, thanks Bill! If not, thanks anyway for your answers and contributions.

Here is the article.
http://www.phy.ohiou.edu/~elster/lectures/qm1_1p2.pdf

Cheers

 

[dextercioby]

If the paper is using bra/ket notation, then it is not about Hilbert spaces.

 

[SemM]

If the paper is using bra/ket notation, then it is not about Hilbert spaces.

It's called "Quantum Mechanics in Hilbert spaces", maybe that is a blend of the two. What should rather be used?

 

[dextercioby]

No, you did not get my point. I would keep the text and title and simply drop the bra/ket notation, that is all. This is because, if a unique mathematical justification of the bra/ket notation exists (I doubt it), then it necessarily goes beyond the mathematics of Hilbert spaces.

 

[SemM]

No, you did not get my point. I would keep the text and title and simply drop the bra/ket notation, that is all. This is because, if a unique mathematical justification of the bra/ket notation exists (I doubt it), then it necessarily goes beyond the mathematics of Hilbert spaces.

I am not confident I know enough on the subject to say that "I see what you mean", but I appreciate your explanation. I have always thought that the bra-ket notation was a QM formality introduced by Dirac, which designates integrals of hermitian pairs, observables etc, and had no mathematical meaning in itself.

 

[vanhees71]

This is nonsense. The Dirac bra-ket notation is just a notation, which you might like or not, but it's describing the mathematics of (rigged) Hilbert space, admittedly in physics books with not too much mathematical rigor, but this is not due to the bra-ket notation but due to the habits of physicists to be more interested in physics than in mathematical subtleties (sometimes with not so favorable consequences ;-)).

 

[SemM]

This is nonsense. The Dirac bra-ket notation is just a notation, which you might like or not, but it's describing the mathematics of (rigged) Hilbert space, admittedly in physics books with not too much mathematical rigor, but this is not due to the bra-ket notation but due to the habits of physicists to be more interested in physics than in mathematical subtleties (sometimes with not so favorable consequences ;-)).

But if I wrote all the integrals such as:

\begin{equation}
\int \psi p \psi^{*}dx
\end{equation}

instead of

\begin{equation}
\langle \psi | p | \psi^{*} \rangle
\end{equation}

Wouldn't we be able to do exactly the same anyway?

 

[vanhees71]

Well (1) is the position representation of (2), i.e., in (1) you have realized the abstract separable rigged Hilbert space of the Dirac bra-ket formalism as the Hilbert space of square integrable functions. Since all separable Hilbert spaces are the same, up to isomorphy, of course, you can do any calculation in the one or the other formalism. For the same reason also the Heisenberg-Born-Jordan version of QT ("matrix mechanics"), using a harmonic-oscillator basis to represent the separable Hilbert space in terms of square summable sequences, is equivalent to Schrödinger's wave mechanics. The Dirac formalism is simply the representation-free formulation and thus the most flexible one. You can often shortcut a calculation in, say, wave mechanics, by first analyze a problem in the Dirac formalism and only finally to write the model in terms of wave mechanics.

Some people don't like the bra-ket notation as, e.g., Weinberg, who seems to be a bit reserved against Dirac in general, given his remarks about him in both his QFT book vol. 1 and in the QM book. He presents the representation-free formalism in another notation. Of course, everything is independent on the notation, and it's just a matter of preference, how you write down your equation.