Bra Ket Notation

[Xittenn]

I've looked over this one a few times and I'm still not quite sure what to make of it. Reading over the various blurbs on the topic I can say that I am familiar with most of the components in their individual form but I don't understand what Bra Ket is trying to do to put it all together. I was just wondering if someone could give me an immediately obvious example of application and somehow relate it to the more pertinent matters of QM?

 

Looking over the Wiki again, am I simply missing the point that it is an inner product? I mean an inner product is nothing for me to run away from, but Bra Ket has been something I've always felt kind of queezy looking at. And I keep seeing the term linear functional, I mean if this is simply a mapping of vector space onto its scalar components I can understand that in English, but I don't see why or how that it is relevant. And what is the concern with orthogonality, and the importance of the Hilbert Space? I never really found any of this in Messiahs book, it may have been in parts that I couldn't access at the time and I should probably re-review when I have the chance. As far as I had gotten was with the basic building blocks of the wave function, wave-particle duality, and HUP. For anyone who has read my personal rantings I don't disregard HUP while I observe the inner workings of QM, I embrace it and I feel that I actually have a pretty thorough understanding of the concept! I am of coarse, now a little of topic though.

Edited March 6, 2012 by Xittenn

[ajb]

Looking over the Wiki again, am I simply missing the point that it is an inner product?

 

The fact we have an inner product is vital.

 

You can think of the bra ket notation as a way of not having to pick a specific basis for the states. For example you don't need to represent of the sates as Schroedinger wave functions. The best advice here is to pick a good book and follow the examples filling in any gaps.

 

There is of course the usual mathematical problem infinite dimensional spaces and what we mean by their duals, but this is usually glossed over in physics books. So, don't worry about that at the moment.

[mississippichem]

One, IMO, very convenient thing about the bra-ket notation is the way operators can be expressed:

 

[math] \langle \phi | \hat{A} | \psi \rangle = \sum_{i,j} \langle \phi | i \rangle \langle i | \hat{A} | j \rangle \langle j | \psi \rangle [/math]

 

As I'm sure you know this gets quite ugly if you try to write this out with full wavefunctions.

 

Also things like showing whether or not an operator is hermitian is much cleaner rather than writing out huge integrals. Think

 

[math] \langle \phi | \hat{A} | \psi \rangle = \langle \psi | \hat{A}^{\dagger} | \phi \rangle ^{*} [/math]

 

versus

 

[math] \int_{-\infty}^{\infty} \phi \hat{A} \psi \ \mathrm{d} \tau = \int_{-\infty}^{\infty} \psi \hat{A}^{\dagger} \phi \ \mathrm{d} \tau [/math]

 

You get to use all those convenient bra-ket manipulation rules (remembering a few identities) instead of getting bogged down with all the integration.

 

There is much more to it than that [everything I've shown here is ultimately trivial] and I'm sure someone like ajb can put me to shame here, but these are just some of the wonderful time saving things that you can enjoy with bra-ket notation.

 

The notation looks scary at first indeed. The best way I've found to get a feel for it is to just dive in and try to do some things that you already know how to do in the new notation. The Feynman Lectures volume III have a great introduction on this that is very intuitive and requires minimal vector/tensor formalism.

[Xittenn]

DrRocket had suggested the Feynman Lectures Vol. III before also for something else that I had asked, I'll try to move this one to the top of my list. I'll have to go over the Bra Ket manipulation rules. I haven't looked at Bra Ket too closely as of yet because I haven't yet found a formal treatment and I'm not one to study from online sources. I have Linear Operators for Quantum Mechanics by Thomas F. Jordan in my bookmarks as the book to get so I guess I'll just go ahead and do that also. Thanks!

 

[DrRocket]

DrRocket had suggested the Feynman Lectures Vol. III before also for something else that I had asked, I'll try to move this one to the top of my list. I'll have to go over the Bra Ket manipulation rules. I haven't looked at Bra Ket too closely as of yet because I haven't yet found a formal treatment and I'm not one to study from online sources. I have Linear Operators for Quantum Mechanics by Thomas F. Jordan in my bookmarks as the book to get so I guess I'll just go ahead and do that also. Thanks!

 

 

That is a good place to start.

 

A very good introduction to Hilbert spaces, not particularly oriented towards quantum mechanics, is Introduction to Hilbert Space and the Theory of Spectral Multiplicity by Paul Halmos. It is a very short and readable book. It can be found on the used book market (Alibris.com, Amazon.com. Abebooks.com) for about $10 if you search a bit.

Edited March 6, 2012 by DrRocket

[Xittenn]

That is a good place to start.

 

A very good introduction to Hilbert spaces, not particularly oriented towards quantum mechanics, is Introduction to Hilbert Space and the Theory of Spectral Multiplicity by Paul Halmos. It is a very short and readable book. It can be found on the used book market (Alibris.com, Amazon.com. Abebooks.com) for about $10 if you search a bit.

 

Funny that you mention that, it was also on my book list and I ordered it this morning. I hate waiting for my books to show up. :/

 

** as well as Quantum Field Theory by L.H. Ryder, and General Theory of Relativity by P.A.M. Dirac