# Index Notation & Dirac Notation

1. May 27, 2014

### TheAustrian

Quantum Mechanics using Index notation. Is it possible to do it?

I really don't get the Dirac Notation, and every-time I encounter it, I either avoid the subject, or consult someone who can read it. There doesn't seem to be any worthy explanation about it, and whenever I ask what is the Hilbert Space, the most common answer I get is "only Hilbert would have known".

So Is it possible to do QM using index notation?

For example can something like this exist:

$\int\phi^{*}_{i}\phi_{j} d\tau = \delta_{ij}$?

Last edited by a moderator: May 27, 2014
2. May 27, 2014

### WannabeNewton

You could definitely use index notation in QM and in fact some (not all) geometric approaches to QM sometimes do make heavy use of index notation (although geometric approaches are quite rare at the pedagogical level, see e.g. http://www.phy.syr.edu/~salgado/geroch.notes/geroch-gqm.pdf [Broken]) but, unlike in GR where index notation is the bread and butter of calculations, in QM index notation is unequivocally inferior to Dirac notation so even though you can use it, it is practically useless in this subject apart from trivialities.

Index notation is really only useful in field theoretic contexts (e.g. QFT) and continuum mechanics where one has to constantly deal with vector and tensor fields.

But if you want to learn QM you can't avoid Dirac notation forever, sorry to say. You have to learn it, you simply have no choice. It has disseminated throughout the literature and is the basic calculational language of the theory. Index notation simply won't do for QM. It's also very easy to learn Dirac notation, you're probably just using the wrong resources.

Last edited by a moderator: May 6, 2017
3. May 27, 2014

### TheAustrian

I've been trying to get my head round Dirac Notation since a long time, but possibly, it may be the worst notation of any kind that I've yet seen. It simply does not seem very informative and I find myself turned off by it about an otherwise fascinating subject. I'm already pretty familiar with "normal" Tensor calculus, and I wouldn't mind to continue using it during QM.

4. May 27, 2014

### micromass

I don't disagree...

How familiar are you with linear algebra? Inner-product spaces? Hilbert spaces? Riesz representation theorem? Etc. Knowing this kind of math helped me a lot to grasp Dirac Notation. I still don't find it pretty, but it's something you can get used to.

5. May 27, 2014

### TheAustrian

I'm confident with linear algebra, but I don't understand Hilbert Spaces very much and do not understand any theorems derived from it or connected to it.

6. May 27, 2014

### atyy

QM is just linear algebra in a vector space with an inner product. A translation between some aspects of the various notations, including index notation, is found in http://alexandria.tue.nl/extra1/afstversl/wsk-i/eersel2010.pdf. It is essential to master Dirac notation, but to implement the algorithms numerically the matrix/index notation is also needed.

7. May 27, 2014

### WannabeNewton

I dislike for the notation myself and like you I'm far more comfortable with index notation but Dirac notation is the unequivocal standard for QM so how will you be able to go through any textbook or paper without knowing how to read it? The only textbook I can even think of, apart from Griffiths, that avoids Dirac notation is Weinberg.

Sakurai has a decent enough treatment of Dirac notation so you might check that out. Shankar has an even better treatment of it. If you're still set on using index notation then take a look at Geroch's notes on geometric QM that I linked above. Note however that they are from the 1970s; modern treatments will stick to index-free notation so you're really picking a marginalized language here as far as QM is concerned.

8. May 27, 2014

### TheAustrian

Could you provide the exact title on Weinberg's book? I'm not very familiar with English-language Physics books.

9. May 27, 2014

### WannabeNewton

Last edited by a moderator: May 6, 2017
10. May 27, 2014

### stevendaryl

Staff Emeritus
In my opinion, Dirac's bra and ket notation is extremely ugly in representing operations on product states (such as the state associated with two particles). It seemed to me that something like GR's abstract index notation, with upper indices representing kets and lower indices representing bras, might be a viable alternative. I have not seen anyone try to develop QM using indices, though.

Last edited by a moderator: May 6, 2017
11. May 27, 2014

### TheAustrian

I just genuinely hope that someone will develop it someday... Dirac's notation really makes me want to kill myself at times.

12. May 27, 2014

### kith

As atty remarked, a Hilbert space is basically just a vector space with an inner product (for subtleties see wikipedia). If you know linear algebra, you understand expressions like <v1, Av2> (where v1 and v2 are vectors and A is a matrix corresponding to an endomorphism). In QM, physical observables are represented by self-adjoint endomorphisms. So since <v1, Av2> = <v1A+, v2> = <v1A, v2> it doesn't matter whether you write the A on the right or the left for them.

This inspired Dirac to write <v1, Av2> in the more symmetrical form <v1|A|v2>. Since this expression can be read as (element from dual space) * (matrix) * (element from original vector space), he decided to keep the symbols "|" and ">" resp. "<" to make clear whether a vector v is from the original space (called the "ket" vector, |v>) or from it's dual space (called the "bra" vector, <v|).

13. May 27, 2014

### dauto

I actually find the bra|ket notation very simple and intuitive. It can become confusing if multiple particles are involved in a problem at which point nothing prevents you from combining the bra|ket notation with the abstract index notation. Together they make quite a powerful pair.

14. May 27, 2014

### George Jones

Staff Emeritus
I find Dirac notation very convenient to use for projectors. I find the standard Dirac notation of Hermitian adjoint,

$$\left< \psi_2 | A^\dagger | \psi_1 \right> = \left< \psi_1 | A | \psi_2 \right>*,$$

to be very confusing, and much prefer the standard functional analysis definition.

15. May 27, 2014

### dauto

How is that confusing? The complex conjugate of a product is given by the product of the adjoint factors in backwards order. Simple and intuitive.

16. May 27, 2014

### George Jones

Staff Emeritus
Well, we all have our peccadilloes . The version of the definition that I learned in functional analysis class stuck more than the version that I learned in quantum mechanics class.

$$\left< A^\dagger \psi_1 | \psi_2 \right> = \left< \psi_1 | A \psi_2 \right>$$

17. May 27, 2014

### dauto

They are both true. Same notation, different statements, both true.

18. May 27, 2014

### George Jones

Staff Emeritus
I guess you are saying that Dirac notation is used for both statements. I am not sure I agree.

Yes, they are equivalent true statements. This is why I wrote

It is just that one is much more commonly used in qm books that use Dirac notation, and when I read this one, I have to stop and take a few seconds to translate. When I read the other one, I don't have to pause.

19. May 27, 2014

### micromass

Haha, exactly! Whenever I read something in Dirac notation, I always need to translate it in my usual mathematics language. I make a lot of errors the other way. For me personally, Dirac notation encourages me to make errors and to ignore certain subtleties.

20. May 27, 2014

### TheAustrian

Does anyone have a Dirac Notation to "Normal" notation guidebook/dictionary?