Index Notation & Dirac Notation

[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:

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

 

[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 ) 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.

 

[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.

 

[micromass]

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.

I don't disagree...

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.

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.

 

[TheAustrian]

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.

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.

 

[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.

 

[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.

 

[TheAustrian]

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

 

[WannabeNewton]

https://www.amazon.com/dp/1107028728/?tag=pfamazon01-20

 

[stevendaryl]

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 ) 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.

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.

 

[TheAustrian]

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.

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

 

[kith]

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.

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|).

 

[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.

 

[George Jones]

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.

 

[dauto]

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.

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.

 

[George Jones]

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.

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.

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

 

[dauto]

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.

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

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

 

[George Jones]

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

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

version of the definition

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.

 

[micromass]

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.

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.

 

[TheAustrian]

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

 

[George Jones]

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

Section 3.12 "Physics Notation" from Brian Hall's book "Quantum Theory for Mathematicians"

https://www.amazon.com/dp/146147115X/?tag=pfamazon01-20

contains a partial translation.

 

[bhobba]

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

I love the notation myself - for me its unbelievably beautiful allowing the proof of things in a very elegant way ie check out my proof of the Born rule in the following:
https://www.physicsforums.com/showthread.php?p=4757673#post4757673

Try repeating that in the conventional notation and you will see what I mean eg proving O = ∑ <bi|O|bj> |bi><bj| is trivial using bra-ket - you simply put ∑ |bi><bi| = 1 before and after O.

I started out learning Linear Algebra, Hilbert spaces in analysis courses, many years ago and used its notation when I moved onto QM, the bra-ket notation seemed ugly and awkward at first, but after a while the usual notation seems ugly and awkward.

At a more technical level the real mathematical foundation for QM is not Hilbert spaces, but Rigged Hilbert spaces and most definitely in that area the bra-ket notation is the most natural. Mathematicians who work with Rigged Hilbert spaces even in totally unrelated areas like White Noise theory use it for that very reason eg:
https://www.math.lsu.edu/gradfiles/ngobi.pdf

Although they still tend to use a comma rather than a bar between the bra and ket eg instead of <a|b> they use <a,b> but not (a,b) - although I suppose some would argue its the same as the usual notation, but certainly in my undergrad math days it was (a,b) and not <a,b> for the inner product - maybe its they find the full bra-ket notation just a little too much. Still in Rigged Hilbert spaces the object on the right is an element of the dual and the object on the left an element of the vector space which is more bra-ket like.

Thanks
Bill

 

[TheAustrian]

I love the notation myself - for me its unbelievably beautiful allowing the proof of things in a very elegant way ie check out my proof of the Born rule in the following:
https://www.physicsforums.com/showthread.php?p=4757673#post4757673

I started out learning Linear Algebra, Hilbert spaces in analysis courses, many years ago and used its notation when I moved onto QM, the bra-ket notation seemed ugly and awkward at first, but after a while the usual notation seems ugly and awkward.

Although they still tend to use a comma rather than a bar between the bra and ket eg instead of <a|b> they use <a,b> but not (a,b) - although I suppose some would argue its the same as the usual notation, but certainly in my undergrad math days it was (a,b) and not <a,b> for the inner product - maybe its they find the full bra-ket notation just a little too much. Still in Rigged Hilbert spaces the object on the right is an element of the dual and the object on the left an element of the vector space which is more bra-ket like.

Thanks
Bill

I think you might have pointed me (and people like me) in the right direction. <a,b> for the inner product is a fine notation, but as you said, the full bra-ket thing seems too much and unhelpful. Also, this Rigged Hilbert Space thing seems something much more advanced than what an Undergraduate student would need to know. Personally, I'm just more interested in particles in periodically repeated environments (e.g. things derived from Alexander Lyapunov's Theories, related book: https://openlibrary.org/books/OL1728527M/The_general_problem_of_the_stability_of_motion)

 

[ChrisVer]

Well of course opinions vary. For example, for me the representation of the state as a wavefunction was much more difficult to deal with- when we got into Dirac notation things became nicer.
Dirac notation is even more powerful and insightful over Quantum Mechanics, at least for the undergrads. If you try to solve the Harmonic Oscillator with normal wavefunctions, at least for the 1st try you'll need 3 pages...
In Dirac's notation, this happens in some lines.

Except for that, as an undergrad, it's only by using the bra-ket notation that you can get the spin automatically on Hydrogen atom. If you write wavefunctions in space using spinors, your calculations can be nasty while writing the spinor part in normal bracket notation, you can find probabilities about spins much easier...

And because symmetries are contained in group theory, and from group theory we know how certain representations act on vector spaces, someone would prefer the bra-ket notation (vectors) to speak and clarify symmetries in a more fundamental way.

So it's a must.
It's also elegant.
It's also easier (well that depends on the person, some people prefer functions, others prefer linear algebra).
Insightful...

But I don't understand what is so difficult to grasp about it in the undergrad level? Instead of using the wavefunction to represent states, you use vectors. That way it's much easier to get results by taking the inner product... eg if you write a state at an orthonormal basis:
$|\psi > = a |0> + b|1>$
it's much easier to see that the possibility for it to be at 0 is $|<0|\psi>|^{2}=|a|^{2}$ which is just a projection. If instead of 0 and 1 states you'd have to put eigenstates as functions, then the result wouldn't be obvious from the beginning (you'd have to multiply with the eigenstate function's conjugate, take an integral to kill the 1 term and leave the other and so on).

Also you should sit and question yourself, if you really understood QM so far by using the functions...

 

[stevendaryl]

And because symmetries are contained in group theory, and from group theory we know how certain representations act on vector spaces, someone would prefer the bra-ket notation (vectors) to speak and clarify symmetries in a more fundamental way.

Well, this claim is mixing up two different things: (1) Viewing states as vectors and co-vectors, and (2) the Dirac notation for dealing with vectors and co-vectors. The vector approach doesn't require using the Dirac notation--the alternative might be some variant of Einstein's abstract index notation.

Bill Hobba points out how the Dirac notation allows very powerful reasoning (such as the insertion of the identity operator written in terms of a basis). I agree that whatever alternative is developed should allow the same sort of reasoning. Since I haven't seen QM developed using abstract indices, I can't know whether it's possible to do the same thing with abstract indices.

 

[bhobba]

I think you might have pointed me (and people like me) in the right direction. <a,b> for the inner product is a fine notation, but as you said, the full bra-ket thing seems too much and unhelpful. Also, this Rigged Hilbert Space thing seems something much more advanced than what an Undergraduate student would need to know. Personally, I'm just more interested in particles in periodically repeated environments (e.g. things derived from Alexander Lyapunov's Theories, related book: https://openlibrary.org/books/OL1728527M/The_general_problem_of_the_stability_of_motion)

Yea - Rigged Hilbert spaces are a bit 'out there'.

If I hadn't studied analysis and reading that link I would have gone - ahhhhh - please someone put me out of my misery.

Mate use whatever you are comfortable with. Initially I stuck to the standard math notation and came around to the bra-ket's charms.

Thanks
Bill

 

[TheAustrian]

Yea - Rigged Hilbert spaces are a bit 'out there'.

If I hadn't studied analysis and reading that link I would have gone - ahhhhh - please someone put me out of my misery.

Mate use whatever you are comfortable with. Initially I stuck to the standard math notation and came around to the bra-ket's charms.

Thanks
Bill

Do you work with QM in Industry/Research? You seem really well knowledged in the subject, especially after reading the materials you've presented. (Although I do not necessarily understand much of them)

 

[bhobba]

Do you work with QM in Industry/Research? You seem really well knowledged in the subject, especially after reading the materials you've presented. (Although I do not necessarily understand much of them)

No.

I am simply a guy with a degree in math and an interest in QM.

I taught myself QM many moons ago from Dirac and Von Neumann.

Having a math background I was really annoyed by this Dirac Delta function thing. Von Neumann was of course utterly rigorous and more than acceptable to someone with a background in Hilbert spaces, analysis etc like I had in my undergrad training. No issues there. However Von Neumann was very critical, correctly, of Dirac's approach. Extremely elegant, but mathematically a crock of the proverbial. Yet it worked.

So I went on a detour to get to the bottom of it investigating Rigged Hilbert spaces and such. By dint of effort I came out the other end with the issue resolved, but won't put myself through that again. It was HARD. As part of that investigation I found out its also used in another interest on mine at the time - Stochastic modelling - hence my knowledge of white noise functionals and Hida distributions - which also have application to QM in rigorously defining the path integral.

My advice to those interested in QM is not to go down my path. THE book to get, that even gives a brief outline of how my issues are resolved with Rigged Hilbert Spaces, is Ballentine - Quantum Mechanics - A Modern Development:
https://www.amazon.com/dp/9810241054/?tag=pfamazon01-20

I wish I started with that book.

Once you understand it you can branch out into issues of foundations (that's what interests me these days), mathematical foundations (I am over that now), applications, or whatever.

If you would like advise on building up to Ballentine, its graduate level, but explains exactly what's going on the best I have ever seen, do a post with your background and me and others can give you some recommendations.

Thanks
Bill

 

[stevendaryl]

Yea - Rigged Hilbert spaces are a bit 'out there'.

If I hadn't studied analysis and reading that link I would have gone - ahhhhh - please someone put me out of my misery.

Mate use whatever you are comfortable with. Initially I stuck to the standard math notation and came around to the bra-ket's charms.

Thanks
Bill

Okay, I took the 15 minute whirlwind tour of rigged hilbert spaces, and I think I understand the motivation, but I'm a little confused about how it relates to Dirac notation.

First, instead of the full Hilbert space $\mathcal{H}$, we look at a smaller subset $\Phi$ that consists of the exceptionally well-behaved elements of $\mathcal{H}$. These elements all have well-defined expectation values for all combinations of the observables. The example I saw of a function that was in $\mathcal{H}$ but not in $\Phi$ was $\psi(x) = \dfrac{1}{x+i}$. $\psi(x)$ is square-integrable, so it's in $\mathcal{H}$, but it has no expectation value for the position operator, and so is not in $\Phi$.

In terms of the well-behaved functions $\Phi$, we can define the "bras" as linear functionals on $\Phi$, and the kets to be anti-linear functionals on $\Phi$. Every element $\phi$ of $\Phi$ corresponds to a linear functional $F_\phi$ as follows:

$F_\phi(\psi) = \int \phi^*(x) \psi(x) dx$

As a "bra", it would be written as $F_\phi = \langle \phi |$

Similarly, every element of $\Phi$ corresponds to an antilinear functional

$F'_\phi(\psi) = \int \psi^*(x) \phi(x) dx$

As a "ket", this would be written as $F'_\phi = | \phi \rangle$

But there are additional elements that don't correspond to any element of $\Phi$, for example:

$F(\psi) = \psi(x)$

This is the "bra" $|x\rangle$

$F(\psi) =$ Fourier transform of $\psi$ evaluated at $k$

This is the "bra" $|k\rangle$

This all makes perfect sense to me. However, this understanding of bras and kets only justifies expressions of the form

$\langle F|\phi \rangle$

and

$\langle \phi|F \rangle$

where $\phi$ is one of the well-behaved elements of $\Phi$, and $F$ is a general functional. It doesn't seem like it justifies expressions of the form

$\langle F | F' \rangle$

where both the bra and the ket are generalized functionals. For example:

$\langle x | k \rangle$

and

$\langle x | x' \rangle$

The operator $\langle x |$ as a functional only applies to well-behaved functions, in $\Phi$; it doesn't apply to other functionals.

 

[micromass]

It doesn't seem like it justifies expressions of the form

$\langle F | F' \rangle$

where both the bra and the ket are generalized functionals. For example:

$\langle x | k \rangle$

and

$\langle x | x' \rangle$

The operator $\langle x |$ as a functional only applies to well-behaved functions, in $\Phi$; it doesn't apply to other functionals.

Well, the operator $\langle x |$ does apply to something a tad more general than to well-behaved functions in $\Phi$. But you're right. Something like $\langle F |F^\prime\rangle$ doesn't make sense in this approach. But is there ever a situation that we want this to make sense?

 

[stevendaryl]

Well, the operator $\langle x |$ does apply to something a tad more general than to well-behaved functions in $\Phi$. But you're right. Something like $\langle F |F^\prime\rangle$ doesn't make sense in this approach. But is there ever a situation that we want this to make sense?

Well, in the common sorts of things we do with Dirac notation, we do form terms like $\langle x|k \rangle$, which is the product of two generalized functionals, $\langle x|$ and $|k\rangle$. Of course, in this case we can understand it as the expression $e^{i k x}$, but the rigged hilbert space approach doesn't actually justify allowing a generalized bra acting on a generalized ket.

 

[bhobba]

Okay, I took the 15 minute whirlwind tour of rigged hilbert spaces, and I think I understand the motivation, but I'm a little confused about how it relates to Dirac notation.

In a Hilbert space the bounded functionals are in 1-1 correspondence with the space - that's the Rietz Representation theorem. This means you can forget about the difference and treat them as the same thing. That's why a Hilbert space sits in the middle of a Gelfland triple.

But if you look at the functionals of sub-spaces then they are not in 1-1 correspondence, so the Dirac notation where you have two different objects - the vector space objects - the kets - and the functional - the bras - is the natural way to look at it.

At the basic level of looking at it; as the functionals of nice behaving functions, so you can define differentiation etc - easy peasy - its fairly straightforward and everyone should be aware of that. A good, and fairly rigorous book at that level is:
https://www.amazon.com/dp/0521558905/?tag=pfamazon01-20

By far the best way to do Fourier analysis - very elegant.

But it gets hairy when you do things like define a completion with respect to a Hilbert-Schmidt operator and get an infinite number of norms you take the intersection of to get a so called Nuclear space. You need a bit of a background in more advanced analysis for that. Its a bit of a specialised area of math (I don't mean Nuclear spaces - that's very specialised - but rather the general functional space analysis stuff to understand it). I was very fortunate to have done it in my math degree - and it wasn't a particularly popular one either. Their were a whole three of us in that class, and the lecturer was shocked even at that many.

But, and here is the thing, its what you need to prove the generalised eigenfunction theorem:
http://mathserver.neu.edu/~king_chris/GenEf.pdf

That's what separates math nerds from physicists. Physicists may simply accept there is this important generalised eigenfunction theorem. Math nerds - well as Hilbert said - 'We must know, we shall know'. If you have that bent like I did, and may still do, then it becomes imperative and you go on this long detour in understanding such things.

Thanks
Bill

 

[stevendaryl]

But if you look at the functionals of sub-spaces then they are not in 1-1 correspondence, so the Dirac notation where you have two different objects - the vector space objects - the kets - and the functional - the bras - is the natural way to look at it.

But still, it seems to me that $|p\rangle$ and $|x\rangle$ are not vector space objects, so how does it make sense to form terms such as $\langle p | x \rangle$ or $\langle x | p \rangle$?

 

[bhobba]

But still, it seems to me that $|p\rangle$ and $|x\rangle$ are not vector space objects, so how does it make sense to form terms such as $\langle p | x \rangle$ or $\langle x | p \rangle$?

The kets are a vector space - the bras act on the kets as linear functionals. Actually they are reflexive - if you take the kets as your base space the linear functionals if I recall correctly are the bra's.

Sometimes one can define a bra acting on a bra - but that is a matter of special definition eg <x| and <p| are bras and one can define <x|p>. But not always eg <x|x> isn't defined - well you can in non standard analysis if you want to down that path.

Thanks
Bill

 

[Fredrik]

The kets are a vector space - the bras act on the kets as linear functionals. Actually they are reflexive - if you take the kets as your base space the linear functionals if I recall correctly are the bra's.

This is certainly true if we just use ket notation for elements of a Hilbert space H, and bra notation for elements of its dual space H*. We write |f> instead of f and denote the map $f\mapsto \langle g,f\rangle$ by <g| instead of something like $\langle g,\cdot\rangle$.

But if we're using a rigged Hilbert space to ensure that the set of kets contains objects like |x> and |p>, things are pretty different. There's still a Hilbert space H involved, but there's also a vector space $\Omega$ that's a vector subspace of H. Now the bras are linear functionals on $\Omega$, not H, and the kets are antilinear functionals on $\Omega$ rather than just elements of $\Omega$ or H. So products like $\langle x||p\rangle$ still need to be explained, but I have to admit, I still haven't studied the details, so I don't know how this is done.