Index Notation & Dirac Notation

[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 :)

 

[bhobba]

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 on that and apparently so do the people in my class.

That's because a good treatment hides the difficult issues of interpretation.

Its how it should be BTW - don't get caught up in that quagmire until you understand its formalism.

Thanks
Bill

 

[atyy]

That's nonsensical. GR is applied math too! QM is a wonderful theory, because it is an "effective" theory, ie. one that is not complete. It has the classical/quantum cut, and its most natural interpretation is "instrumental". (It might be possible that many-worlds makes QM a complete theory, and indeed there are versions that seem very satisfactory, but there is still no consensus on whether many-worlds really works. And if it does, that makes QM even more interesting!)

QM is a theory which is foundationally interesting, in that it (1) has a measurement problem (2) violates the Bell inequalities indicating some sort of nonlocality, yet can be made consistent with relativistic locality (3) together with gravity, and thermodynamics forms the "information loss" problem. It shows the importance of experiments, because I don't think anyone would accept this theory as reasonable, except that it is so successful in describing experiments.

 

[Matterwave]

In my view, any science that deals with probabilities will run into interpretational difficulties (unless all you care about are the average values, like in statistical mechanics). Deterministic theories are much easier to work with conceptually. A leads to B is a very easy logical thing to think about. A might lead to B, C, and D, with probabilities B*, C*, and D* is much more difficult conceptually in my opinion. Even the definition of a probability itself (other than the strict mathematical definition) is mired in conflicting views.

 

[Nugatory]

That's nonsensical. GR is applied math too!

There's a strong element of personal taste here, and... de gustibus non est disputandum.

But I have to say that I'm siding with what I'm hearing from TheAustrian and WbN. Classical mechanics, E&M, SR/GR, I look at the math and map it back into a physical interpretation of how the world works and I think I understand, but QM... Not so much. It's only in QM that "shut up and calculate" is sound advice, and only in QM that all interpretations fall short in one way or another and I'm stuck with what I get when I calculate.

 

[bhobba]

In my view, any science that deals with probabilities will run into interpretational difficulties

Hmmmm. Yes and no.

Probabilities have difficult interpretational issues that are philosophically - how to put it - non trivial - eg the frequentest interpretation in circular unless you base it on something else that isn't such as the Kolmogorov axioms..

But these days with the axiomatic foundation via the Kolmogorov axioms they are manageable in a number of ways eg by the so called Cox axioms or a reasonable implementation of those axioms to applied problems.

In a certain sense Ballentines statistical interpretation is a frequentest view of probabilities in QM, Copenhagen more Bayesian.

At the axiomatic formal level of QM I simply stick to the Kolmogorov axioms to avoid any issues with interpretation.

Thanks
Bill

 

[bhobba]

That's nonsensical. GR is applied math too! QM is a wonderful theory, because it is an "effective" theory, ie. one that is not complete.

I don't think its complete either (specifically I want some mechanism for an improper mixed state to become a proper one), but that is a debatable point.

Personally I think discussions on issues like that don't really go anywhere. They are fine for highlighting the issues involved, but beyond that - its just a talkfest - what we want is experiments.

Thanks
Bill

 

[bhobba]

But I have to say that I'm siding with what I'm hearing from TheAustrian and WbN.

Make that four, and I suspect many more ascribe to it as well.

Thanks
Bill

 

[TheAustrian]

Make that four, and I suspect many more ascribe to it as well.

Thanks
Bill

I'm glad to hear I'm not alone. If I would have said this to my lecturer back then, he would have torn my head off for it.

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 :)

I'm okay with interpreting Relativity. It makes sense how it applies to the Physical world. But not QM.

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

I will give it a read in the upcoming month of June.

 

[atyy]

Query about Ballentine: Does the book even mention that when one does QM, one divides the universe into classical and quantum parts?

Query to TheAustrian: Did your lecturer even mention the need to divide the world into classical and quantum parts? Do you know that this is part of the basic procedure of QM?

 

[TheAustrian]

Query about Ballentine: Does the book even mention that when one does QM, one divides the universe into classical and quantum parts?

Query to TheAustrian: Did your lecturer even mention the need to divide the world into classical and quantum parts? Do you know that this is part of the basic procedure of QM?

Our lecturer hated maths. So he just talked a lot about various things. He mentioned that you can learn QM without knowing anything else about Physics, and that when you do QM, you can pretty much forget about everything you've learned so far. We also made some connections to the Lagrangian, and some research paper by Richard Feynman.

To be honest, he was not a good lecturer, and less than 60% of the class passed.