Index Notation & Dirac Notation

[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.

 

[TheAustrian]

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?

My problem is with expressions like these:
\left|E L M_{L} S M_{S} \right\rangle

(This expression propped up in atomic physics, and to this day, I have absolutely zero clue as to what it refers to)

 

[Fredrik]

My problem is with expressions like these:
\left|E L M_{L} S M_{S} \right\rangle

(This expression propped up in atomic physics, and to this day, I have absolutely zero clue as to what it refers to)

This isn't an issue with bra-ket notation. If they hadn't been using it, they would have written something like $f_{E L M_{L} S M_{S}}$ instead, which is even worse. The book you were using should have explained the notation.

 

[bhobba]

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.

You need some nifty definitions. First you define a ket acting on a bra as <x|a> = bar <a|x> if its not defined normally ie if both are kets. One obvious way to go further would be given a sequence xn from the test space that converges to the distribution T in the weak toplogy then if <xn|B> converges this defines <T|B>.

Naturally you would need key theorems such are well defined, but I can't recall those from my days investigating the detail of this stuff. The other way is maybe do it on a special case basis and hope they don't overlap. Chapter 7 of Youn's book gives a bit of detail of other ways by, for example, convolution.

Added Later:
Did a bit of reacquainting with this stuff. That definition being well defined is tied up with if a sequence of test functions |xn> goes to zero in weak convergence then <xn|T> needs to go to zero. However that is only guaranteed if |xn> goes to zero in strong convergence - that's so you can use the uniform boundedness principle that guarantees if a sequence converges weakly it converges to an element of the dual. Its basically the old bugbear of reversing limits ie if |ti> goes to |T> weakly then you can reverse <xn|ti>. In physics and applied maths you normally assume that - but if you are being rigorous you cant. My head hurts

Thanks
Bill

 

[TheAustrian]

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

Are you saying that " Ballentine - Quantum Mechanics - A Modern Development" should be the first QM book that I should read? Or should I "warm up" with some other texts first?

 

[micromass]

Are you saying that " Ballentine - Quantum Mechanics - A Modern Development" should be the first QM book that I should read? Or should I "warm up" with some other texts first?

It's definitely not the first QM book that you should read. The book has quite an advanced mathematical formalism and is not easy to understand. I would suggest that you first read an easier, more computational text like Zettili to get the hang of things. Ballentine is an excellent and beautiful book, but I don't think it's for newcomers.

 

[TheAustrian]

It's definitely not the first QM book that you should read. The book has quite an advanced mathematical formalism and is not easy to understand. I would suggest that you first read an easier, more computational text like Zettili to get the hang of things. Ballentine is an excellent and beautiful book, but I don't think it's for newcomers.

Thanks for the clarification. I've only read Griffiths book on QM first (and I think my UG lectures on QM were based on it). I didn't understand anything from QM, I passed exam only with pure maths knowledge.

 

[micromass]

Thanks for the clarification. I've only read Griffiths book on QM first (and I think my UG lectures on QM were based on it). I didn't understand anything from QM, I passed exam only with pure maths knowledge.

Can you explain us why you didn't understand much from Griffiths? What was the problem that you had with the book? Well, aside from the bra-ket notation.

 

[TheAustrian]

Can you explain us why you didn't understand much from Griffiths? What was the problem that you had with the book? Well, aside from the bra-ket notation.

I'm sorry, I have phrased myself badly. I didn't understand anything at University in my QM class, and passed with maths only. After getting that C for that class, I found out about Griffiths book and I have read it. I've managed to grasp some concepts and I understand the basic examples, but for example I do not understand how to treat a multi-electron atom.

 

[TheAustrian]

Micromass is correct.

But as to the texts to build up to it that depends on your background.

If you have the typical first year calculus based physics background then Zettili or Griffiths would be a good text before Ballentine. It would also help to have some linear algebra and mulivariable calculus.

I was a bit different. Because of my math background I had Von Neumann behind me.

Thanks
Bill

I have an OK math background. I'm ok with things like linear algebra, multi-variable calculus, calculus of variations, some kinds of ordinary and partial differential equations,legendre transforms, group theory and various other things.

 

[bhobba]

I have an OK math background. I'm ok with things like linear algebra, multi-variable calculus, calculus of variations, some kinds of ordinary and partial differential equations,legendre transforms, group theory and various other things.

With that background I would give Ballentine a shot.

Thanks
Bill

 

[atyy]

I'm sorry, I have phrased myself badly. I didn't understand anything at University in my QM class, and passed with maths only. After getting that C for that class, I found out about Griffiths book and I have read it. I've managed to grasp some concepts and I understand the basic examples, but for example I do not understand how to treat a multi-electron atom.

It could be helpful to read several treatments of the helium atom, eg.
http://quantummechanics.ucsd.edu/ph130a/130_notes/130_notes.html
http://farside.ph.utexas.edu/teaching/qm/Quantumhtml/index.html

Both those notes have sections on the helium atom. In addition to the usual postulates of quantum mechanics (classical/quantum divide, state is a ray in a vector space, measurement collapses the wave function etc.), an additional postulate for dealing with multiparticle systems is that the basis vectors for the multiparticle state space can be made from the single particle state space.

Two sources that give the fundamental postulates of quantum mechanics are
http://arxiv.org/abs/1110.6815
http://www.theory.caltech.edu/people/preskill/ph229/#lecture

The classical/quantum divide is a quite important assumption. I don't know many books that state it explicitly. Landau and Lifshitz is one, Weinberg's (linked in post WannabeNewtons's #9) is another.

 

[Nugatory]

It's definitely not the first QM book that you should read... Ballentine is an excellent and beautiful book, but I don't think it's for newcomers.

There are arguments both ways, and a lot depends on the degree of mathematical sophistication the newcomer starts with. If you're learning the math as you go, or if your goal is to get to where you can solve problems as quickly as possible, then Ballentine is most certainly not the place to start. But if you're looking for a modern perspective on QM because it's a fascinating and powerful piece of humanity's intellectual property, and you have the necessary math skills... I'd say go for it. The worst that happens is that you have to set the book aside while you work through a less demanding intro, then come back to it.

 

[Jilang]

My problem is with expressions like these:
\left|E L M_{L} S M_{S} \right\rangle

(This expression propped up in atomic physics, and to this day, I have absolutely zero clue as to what it refers to)

I would take that to mean a state with energy E, angular momentum L, z component of angular momentum ML, spin S, z component of spin MS.

 

[TheAustrian]

There are arguments both ways, and a lot depends on the degree of mathematical sophistication the newcomer starts with. If you're learning the math as you go, or if your goal is to get to where you can solve problems as quickly as possible, then Ballentine is most certainly not the place to start. But if you're looking for a modern perspective on QM because it's a fascinating and powerful piece of humanity's intellectual property, and you have the necessary math skills... I'd say go for it. The worst that happens is that you have to set the book aside while you work through a less demanding intro, then come back to it.

I'm fine with that. So far QM seemed more or less just applied mathematics. I don't really find any Physical interpretation of anything in it so far.

 

[TheAustrian]

I would take that to mean a state with energy E, angular momentum L, z component of angular momentum ML, spin S, z component of spin MS.

Thanks for this! This is amazing. How do you know this stuff? More-over, is it possible to write this in "normal" functional form?

 

[Jilang]

They are only labels for the state. I am only guessing, but what else could they mean? What would be a normal functional form - I'm not sure what you mean by this as wouldn't any notation require some sort of labelling?

 

[TheAustrian]

They are only labels for the state. I am only guessing, but what else could they mean? What would be a normal functional form - I'm not sure what you mean by this as wouldn't any notation require some sort of labelling?

I think you have the meanings right, but like I said, I don't know it for sure because I don't understand this notation.
Hmm... I mean if you wanted to write it with not bra-ket notation, but just how you otherwise mathematics?

 

[micromass]

I think you have the meanings right, but like I said, I don't know it for sure because I don't understand this notation.
Hmm... I mean if you wanted to write it with not bra-ket notation, but just how you otherwise mathematics?

Bra-ket notation is certainly not necessary here. The mathematics behind QM is essentially functional analysis. And most functional analysis texts do not use the bra-ket notation. They use a notation that I think is much simpler and isn't prone to errors. You can try out the excellent book by Kreyszig: https://www.amazon.com/dp/0471504599/?tag=pfamazon01-20

However, most physics texts do use bra-ket notation. I think this is a sad thing, but it's the way it is. It has become standard in physics. So you need to learn it anyway. It will really harm you later down the road not to learn and get used to bra-ket notation. Only a very few books don't use the notation and they are usually written for mathematicians, like https://www.amazon.com/dp/146147115X/?tag=pfamazon01-20 and https://www.amazon.com/dp/0486453278/?tag=pfamazon01-20

 

[Jilang]

I think you have the meanings right, but like I said, I don't know it for sure because I don't understand this notation.
Hmm... I mean if you wanted to write it with not bra-ket notation, but just how you otherwise mathematics?

You would write a ψ with the same subscripts I would think. They may be replaced with i,j,k,l,m etc, but standing for the same physical labels.

 

[TheAustrian]

You would write a ψ with the same subscripts I would think. They may be replaced with i,j,k,l,m etc, but standing for the same physical labels.

Thanks. So basically, that expression is an insanely complicated wavefunction, that which we do not know how it looks like?

 

[Jilang]

Lol, that involves solving the Schroedinger equation. This can be easy or incredibly hard! A good start is Griffiths.

 

[TheAustrian]

I see. I know, it's usually not possible to find analytical solution, I'm aware, I just had problems in understanding what the notation meant, thanks for the clarification :D

 

[Matterwave]

Thanks. So basically, that expression is an insanely complicated wavefunction, that which we do not know how it looks like?

There just happens to be 5 different operators which are diagonalized by the same set of basis states in this problem. The energy, the square of the total orbital angular momentum, the z-component of the orbital angular momentum, the square of the total spin, and the z-component of the spin.

Really, the first 3 variables have to do with the wave function in real space, while the last 2 have to do with the internal degrees of freedom of the particle (spin).

Perhaps a more common notation one might see would be:

$$\left|n,l,m_l\right>\otimes\left|s,m_s\right>$$

This would explicitly separate the internal degrees of freedom from the external ones. Without using bra-ket notation, one might instead construct this state from a wave function $\psi_{n,l,m}$ and a spinor $\chi_{s,m_s}$:

$$\psi_{n,l,m}\otimes\chi_{s,m_s}$$

You might be able to simplify this a little by noting the super selection rule that superpositions do not exist between states of different total spin (e.g. an electron will always have spin 1/2, this number is intrinsic to this particle), so that if we specify the species of particle, we implicitly know the eigenvalue of s, so we may simply write, in either case:

$$\left|n,l,m\right>\otimes\left|m_s\right>$$
or
$$\psi_{n,l,m}\otimes\chi$$

 

[bhobba]

I'm fine with that. So far QM seemed more or less just applied mathematics. I don't really find any Physical interpretation of anything in it so far.

If that's how you found it then Ballentine is most certainly for you.

Here its developed in a fairly rigorous way from just two axioms.

And once you feel comfortable with the stuff in Ballentine do a post and I can explain how it can be reduced to just one axiom. QM from just one axiom? Well of course not - its just each step seems very reasonable and natural - but that is a revelation for the future.

Thanks
Bill

 

[WannabeNewton]

I'm fine with that. So far QM seemed more or less just applied mathematics. I don't really find any Physical interpretation of anything in it so far.

I thought I was the only one who felt this way. Then I asked almost all the people in my QM class and they all felt the same way. Compared to subjects like classical mechanics, EM, statistical mechanics, and GR I would have to agree with you wholeheartedly that QM just seems like applied math and apparently so do the people in my class.

However QM comes to life brilliantly and beautifully in statistical mechanics :)