Spin - any book recommendations?

[dreamspy]

Hi

I am currently taking my first course in Quantum Mechanics, and I'm having difficulty getting a grip on the spin concept. The book I am reading is "Introduction to Quantum Mechanics" by Griffiths. This is a great book, but the coverage on spin isn't as clear as the rest of the book. Can anyone recommend me another source of information, some introductory text on the concept of spin.

Regards
Frímann Kjerúlf

 

[mccoy1]

Hi

I am currently taking my first course in Quantum Mechanics, and I'm having difficulty getting a grip on the spin concept. The book I am reading is "Introduction to Quantum Mechanics" by Griffiths. This is a great book, but the coverage on spin isn't as clear as the rest of the book. Can anyone recommend me another source of information, some introductory text on the concept of spin.

Regards
Frímann Kjerúlf

Electron has spin up and spin down; it works, so go and use it.

Okay ...well...In my opinion, there's no book that can do a very good job in helping you understand what actually spin is. You just have to accept it like a religion as an intrinsic property of an electron(just like its mass).

Good luck in applying the concept of spin up/down.

 

[dreamspy]

I'm mostly looking for a book that goes well into the notation of spin, and maybe gives some basic examples/problems so I can "learn by doing".

 

[dextercioby]

My advice would be to read the 3rd volume of the Feynman lectures on Physics. And if you want rigorous notation, then Sakurai's book should be the choice.

 

[ansgar]

Sakurai chapter 3 is the best reference I have tried/read regarding spin and angular momentum in QM. It is simply a representation of an underlying symmetry.

 

[ansgar]

Electron has spin up and spin down; it works, so go and use it.

Okay ...well...In my opinion, there's no book that can do a very good job in helping you understand what actually spin is. You just have to accept it like a religion as an intrinsic property of an electron(just like its mass).

Good luck in applying the concept of spin up/down.

why should one "just accept it"? It is very simple to derive and to experimentrally observe. In my previous answer, I told OP that spin is a representation/manifestation of an underlying symmetry.

When you come to relativistic QM, you will find that spin (spionors) are just the irreducible representation of the Lorentz Group, i.e. related to symmetry.

 

[Ben Niehoff]

I'll second (third?) Sakurai. The very first chapter has some thought experiments that help clarify what spin really is.

 

[Spinnor]

Hi

I am currently taking my first course in Quantum Mechanics, and I'm having difficulty getting a grip on the spin concept. The book I am reading is "Introduction to Quantum Mechanics" by Griffiths. This is a great book, but the coverage on spin isn't as clear as the rest of the book. Can anyone recommend me another source of information, some introductory text on the concept of spin.

Regards
Frímann Kjerúlf

As a second opinion try:

Quantum Mechanics (Student Physics Series) by Paul Davies (Paperback - Sep 1988) ISBN-10: 0710099622
11 Used & new from $1.31 (Amazon used books)

Make sure you can answer any test questions, some math involved. Get a grip latter %^)

 

[mccoy1]

why should one "just accept it"? It is very simple to derive and to experimentrally observe. In my previous answer, I told OP that spin is a representation/manifestation of an underlying symmetry.

You don't derive spin man..One can only derive the mathematical representation of spin.
What the poster wants is not to "derive spin" because griffiths has that calculation. However, he wants explanation of it is.Okay, now that you have experimentally observed it, can you explain what you have observed to others who want to learn it? And as I read your post, you haven't done a good job in explaining it to him. It's easy for some people to pretend that they intuitively understand "spin" as in QM, but the reality is that they don't.

I think those who recommended some books have done a good job. Let him read by himself to have a feel of what spin is actually is.

 

[ansgar]

You don't derive spin man..One can only derive the mathematical representation of spin.
What the poster wants is not to "derive spin" because griffiths has that calculation. However, he wants explanation of it is.Okay, now that you have experimentally observed it, can you explain what you have observed to others who want to learn it? And as I read your post, you haven't done a good job in explaining it to him. It's easy for some people to pretend that they intuitively understand "spin" as in QM, but the reality is that they don't.

I think those who recommended some books have done a good job. Let him read by himself to have a feel of what spin is actually is.

Just because you don't know what spin is does not mean that everyone else doesn't know.

Here lies the heart in all physics understanding, if one understands the mathematical representation of a physical phenomena one understands the physics. We can not have any other understanding of physics (except for things like Newtonian gravity and basic electricity, fluid dynamics etc hehe) but the mathematical "mapping".

As pointed out, first chapter in sakurai does a good job in explaining spin from an experimental point of view.

In a very similar way, electromagnetism is due to a symmetry principle, gauge symmetry - which relates E and B. Now would you claim that this is just a mathematical representation, that we have no understanding of what EM is?

 

[sokrates]

I vehemently disagree.

Just because you get good at some math tricks, some taylor expansions, infinitesmall rotations, or translations does not at all mean you understand what spin is. Why don't you "derive" spin in some spare time so everyone benefits?

While Math is the only way we know of representing reality (i.e predicting/explaining experiments), it hardly means it is the only way.

There are at least 10 ways of "defining", "adjusting", "tuning" the math to get to the SAME experimental fact in many cases, (Sakurai himself talks about his "way" of defining angular momentum as the generator of rotations) so this by itself is an indication of
the fact that a single theory or representation is probably not uniquely related to what we call the "underlying reality".

To the OP: Sakurai - Feynman is probably a very good combination. Read Sakurai as much as you can and resort to Feynman when it becomes very hard.

 

[ansgar]

I vehemently disagree.

Just because you get good at some math tricks, some taylor expansions, infinitesmall rotations, or translations does not at all mean you understand it. Why don't you "derive" spin in some spare time so everyone benefits?

While it is the only way we know of representing reality, it hardly means it is the only way

There are at least 10 ways of "defining", "adjusting", "tuning" the math to get to the SAME experimental fact (Sakurai himself talks about his "way" of defining spin etc...)

As I said, spin is a consequense of Lorentz symmetry.

Of course on can argue forever what criterion for "understanding" one should employ.

 

[sokrates]

I don't think Lorentz symmetry is the only way one can use to decide on the consequences.

One can argue forever about even that.

 

[ansgar]

I don't think Lorentz symmetry is the only way one can use to decide on the consequences.

One can argue forever about even that.

The issue is that in non-relativistic QM, spin can be a somewhat ad-hoc / fuzzy thing - on that I agree - but in relativistic QM/QFT it is transparent.

And even if Lorentz Symmetry is just ONE way to deduce spin - a priori there might be in principle several way to obtain spin-1/2 representations of particles. But that does not per se mean that we don't know what spin is. Every question concerning understanding should be asked within a paradigm. If not, then discussions are pointless. Now the current paradigm of physics at scales larger than String Theory regimes - is that the Poincare Symmetry should hold. I.e. particles and dynamics should be related to representations of the Poincare group.

One can of course ask if a human beeing can understand anything - i.e. are there any objective criteria one should fulfil in order to gain understanding which is applicable to all regimes of human activity?

I would say that one understands a physical phenomena if one can derive and manipulate the mathematical representation of it and appreciate the experimental predictions and outcome of measurements if that phenomena.

And there are of course different levels regarding understanding, the teacher (very often) has a deeper and wider understanding of what he teach than the students who gains the highest grade in his class.

 

[dreamspy]

Gentlemen, I want to thank you for your answers.

 

[ansgar]

Gentlemen, I want to thank you for your answers.

I hope you find a good book, I am embarrassed due to the first reply of mccoy1 - certainly he is probable just a troublemaker who has not gone far in QM business. So he have to impose this on everyone else, he can't just accept that he is the one having an issue.

Now besides Sakurai, you can try the book by Shankar, or Ballentine.

 

[dreamspy]

Well there are always black sheeps. But must admit that I was surprized to this kind of discussion on this forum.

I'we located all the books recommended in her, except Ballentine, so now it's only a matter of "time" :)

Thanks again for your answers.

 

[ansgar]

Well there are always black sheeps. But must admit that I was surprized to this kind of discussion on this forum.

I'we located all the books recommended in her, except Ballentine, so now it's only a matter of "time" :)

Thanks again for your answers.

There is also a science book discussion forum here:


Academic Guidance (44 Viewing)
Which college and degree? Grad school and PhD help
Educators & Teaching - Science Book Discussion

Please have a look there also, recommendations of QM books is a frequently asked question.

 

[mccoy1]

Just because you don't know what spin is does not mean that everyone else doesn't know.

Here lies the heart in all physics understanding, if one understands the mathematical representation of a physical phenomena one understands the physics. We can not have any other understanding of physics (except for things like Newtonian gravity and basic electricity, fluid dynamics etc hehe) but the mathematical "mapping".

As pointed out, first chapter in sakurai does a good job in explaining spin from an experimental point of view.

Doing the maths and understanding are two completely different things. You can do/calculate QM but you still don’t understand it. Mostly those who assume they understand QM because they can follow calculations tend to be the very people who just don’t understand QM. In fact, it’s safe to say no one understand QM, not even the fine physicist, Einstein.
Okay spin was deduced from the stern – Gerlach experiment...so, can you explain it to us or ‘derive’ its explanation as you just said?
bye

 

[ansgar]

Doing the maths and understanding are two completely different things. You can do/calculate QM but you still don’t understand it. Mostly those who assume they understand QM because they can follow calculations tend to be the very people who just don’t understand QM. In fact, it’s safe to say no one understand QM, not even the fine physicist, Einstein.
Okay spin was deduced from the stern – Gerlach experiment...so, can you explain it to us or ‘derive’ its explanation as you just said?
bye

The question was related to a book recommendation, if you can't answer then don't write anything in the first place.

It depends on, as I said earlier, what definition and criteria one has for "understanding". OF course we can not understand QM on the same level as we understand that hitting someone with a bat in the head is going to hurt really really bad. But one can not see things as black and white as you do, that if something can not be related to intuitive feelings, then it is not something that can be understandable.

Understanding in physics is, as from my viewpoint, if one can relate theory and experiments.

 

[mccoy1]

I am embarrassed due to the first reply of mccoy1 - certainly he is probable just a troublemaker who has not gone far in QM business. So he have to impose this on everyone else, he can't just accept that he is the one having an issue.

Yes sir, can you please do me a favour and explain what spin is?

 

[ansgar]

Yes sir, can you please do me a favour and explain what spin is?

Have you done any relativistic QM?

 

[mccoy1]

Have you done any relativistic QM?

Don't worry about that...
I just asked you a favour to explain to me that property of electron called spin.

 

[Matterwave]

spin is a property of all particles, not just electrons ;)

 

[ansgar]

spin is a property of all particles, not just electrons ;)

exactly ! All particles have intrinsic angularmomentum, i.e. is a symmetry "eigenstate" to the Lorentz group, which consists of boosts and rotations.

 

[mccoy1]

spin is a property of all particles, not just electrons ;)

I know...electron is one of those particles, so there's no a big deal here.
So explanation?

 

[ansgar]

Don't worry about that...
I just asked you a favour to explain to me that property of electron called spin.

why should I waste my valuable time to preach from the R-QM textbooks? I have already said that intrinsic angular momentum (called spin) is a manifestation due to Lorentz Symmetry: Which we have pretty good proofs that this symmetry is realized in nature.

All particles have intrinsic angular momentum; fermions have 1/2, i.e. transform under the irreducible representation of the Lorentz group. Scalars have 0, i.e. transforms as singlets. Vectors are particles that transforms as four-vectors, tensor-2 particles are particles that transforms as rank-2 tensors etc etc.

So compact: Intrinsic Angular momentum / spin is a symmetry manifestation.

Analogy: electric charge is a symmetry manifestation of the gauge transformations in electromagnetism.

Physics is about symmetries, once you know the symmetries in nature, we can deduce the dymanics.

 

[mccoy1]

why should I waste my valuable time to preach from the R-QM textbooks? I have already said that intrinsic angular momentum (called spin) is a manifestation due to Lorentz Symmetry: Which we have pretty good proofs that this symmetry is realized in nature.

All particles have intrinsic angular momentum; fermions have 1/2, i.e. transform under the irreducible representation of the Lorentz group. Scalars have 0, i.e. transforms as singlets. Vectors are particles that transforms as four-vectors, tensor-2 particles are particles that transforms as rank-2 tensors etc etc.

So compact: Intrinsic Angular momentum / spin is a symmetry manifestation.

Analogy: electric charge is a symmetry manifestation of the gauge transformations in electromagnetism.

Physics is about symmetries, once you know the symmetries in nature, we can deduce the dymanics.

hahahaha, what a waste of time. I thoght you actually knew it. hint: if one didn't understand the philosophical point of view of something, then one didn't understand that thing.
byee dude.

 

[ansgar]

hahahaha, what a waste of time. I thoght you actually knew it. hint: if one didn't understand the philosophical point of view of something, then one didn't understand that thing.
byee dude.

Do you have a reference to the claim "if one didn't understand the philosophical point of view of something, then one didn't understand that thing." ?

Define "philosophical point of view"

I am also a trained philosopher... what you just wrote makes no sense.

What you are trying to do is a mapping of metaphysics into physics, it is like asking if one can measure God in a laboratory experiment.

If relating physical quantities to symmetries in nature is not a valid preposition to understanding things, then we can't understanding anything.

I beg you to define the philosophical point of view of "torque" which we have in classical mechanics, go ahead.

 

[Matterwave]

I think he means, you must have a "deeper" understanding of the physics than just the mathematical model involved.

It's very subjective, imo. Some people may find Newton's laws "deeper", while some may find the principle of least action "deeper". I don't see how one could judge either way...

 

[ansgar]

I think he means, you must have a "deeper" understanding of the physics than just the mathematical model involved.

It's very subjective, imo. Some people may find Newton's laws "deeper", while some may find the principle of least action "deeper". I don't see how one could judge either way...

Exactly, it is very subjective. Concepts of force, and fields etc, are very (or rather were) very debated entities back in those days.

I would say that deriving things from symmetries are probably the deepest thing one can do in physics, the simplicity and beauty of nature is simply manifested. And it is even more beautiful and deep if we can perform measurements and verify that the imposed symmetry of nature actually seems to be there, i.e. realized in nature.

If one can not understand what spin is, then we can not understand what electric charge is either...

Things like Angular momentum and Linear momentum are also related to symmetries, that is how one formally derive those quantities. So by saing that P = mv^2 / 2 is momentum is not a very deep answer, but if one says derives P from - assuming- that nature is invariant under linear translations, then one has a deeper connection between P and nature - i.e. one can argue that we have understood what P is - a manifestation of a underlying symmetry in nature. Now that is what i call philosophy!

 

[alxm]

I don't think there's a good answer to the OP's question really. If he's reading Griffiths, which is an introductory textbook, then there's not much you can do to 'explain' spin at that level.

It's a relativistic property (as pointed out), and can't really be 'explained' in a rigorous manner without getting into QM that's pretty far beyond the introductory level.

There's a good reason why most introductory textbooks simply assert it as a separate postulate.

 

[ansgar]

I don't think there's a good answer to the OP's question really. If he's reading Griffiths, which is an introductory textbook, then there's not much you can do to 'explain' spin at that level.

It's a relativistic property (as pointed out), and can't really be 'explained' in a rigorous manner without getting into QM that's pretty far beyond the introductory level.

There's a good reason why most introductory textbooks simply assert it as a separate postulate.

Yes it is simply postulated, but one can at least provide a clear presentation :)

 

[dreamspy]

Thats right, it's an introductory class. At the moment I'm mostly interested in beeing able to calculate simple questions to pass my test. Deeper understanding will be needed when I go to other QM classes.

 

[Fredrik]

Spin is not a relativistic property. The existence of the spin operators can be derived from the assumption that space is rotationally invariant. This is how the argument goes (details omitted because of time and because I only know some of them):

A symmetry is a bijection on the set of unit rays, that preserves probabilities, i.e. S is a symmetry if $|\langle\psi'_1|\psi'_2\rangle|^2=|\langle\psi_1|\psi_2\rangle|^2$ for all $\psi_1,\psi_2\in \mathcal R$ and all $\psi'_1,\psi'_2\in S(\mathcal R)$. The idea that space is rotationally invariant is incorporated into QM as the assumption that there's a group homomorphism from SO(3) into the group of symmetries.

Now the question is, does this mean that there also exists a group homorphism from SO(3) into the group of unitary linear operators on our Hilbert space H, i.e. is there a representation of the symmetry group on the Hilbert space of state vectors? Wigner showed that the answer is no, and also that it's "almost" yes. We can define a function from the set of continuous curves that start at the identity of SO(3), into the group of unitary linear operators on H. I'll write such operators as U(C), where C is the curve. Note that the other endpoint of C is a rotation matrix. It turns out that these curves can be divided into two equivalence classes, which I'll call 0 and 1, and that U(C) only depends on the endpoint and the equivalence class. So I'll write U([C],R) instead of U(C). It's also possible to prove that U(0,R)=-U(-1,R).

For every R, there's exactly one curve in 0 and one curve in 1 that goes from the identity to R. We can try to use only the 0 curves, hoping that this will enable us to drop the C's completely and just write U(R), but it turns out that we still won't get the simple relationship U(R')U(R)=U(R'R). The U(C) operators satisfy

$$U(C',R')U(C,R)=U(C'*C,R'R)$$

where C'*C is the curve that goes through C at twice the normal speed, and then through a translated version of C' from R to R'R at twice the speed. The problem is that C'*C might be in 1 even if C and C' are both in 0. This means that we can write

$$U(0,R')U(0,R)=U(s,R'R)=(-1)^s U(0,R'R)$$

where s=[C'*C], and this can't be simplified any further.

This is why we take the symmetry group to be SU(2) instead of SO(3). SO(3) is homeomorphic to a 3-sphere with opposite points identified, and SU(2) is homeomorphic to a 3-sphere. This difference means that there will only be one equivalence class of curves in SU(2) instead of two, and that enables us to write U(r')U(r)=U(r'r), where r and r' are members of SU(2) that correspond to rotation operators, and r and -r correspond to the same rotation operator.

The R's and the r's can be expressed as functions of three parameters (e.g. Euler angles), and therefore both U(R) and U(r) can be expanded in Taylor series:

[tex]U(r)=1+\frac i 2\omega_{ij}M_{ij}+\cdots[/itex]

where second-order terms have been omitted and repeated indices are summed over, and $\omega$ is the first-order term in a series expansion of r. The fact that $\omega$ is symmetric implies that M can be defined as symmetric. The spin operators can now be defined as

$$S_i=\epsilon_{ijk}M_{jk}$$

where the epsilon is the Levi-Civita symbol. The commutation relations

$$[S_i,S_j]=i\epsilon_{ijk}S_k$$

can be derived from the fact operators must transform as

$$X\rightarrow U(r)^\dagger X U(r)$$

in order to make expectation values invariant.

As you guys can see, this stuff is pretty complicated. I think that's the reason why a proper discussion of spin is usually delayed until an advanced class where no one is going to be bothered by the fact that we consider the restricted Poincaré group instead of just SO(3) or SU(2). SO(3) is a subgroup of the restricted Poincaré group, so when we do the relativistic version of this, we get everything I said above and then some. That includes the concepts concepts "momentum", "energy" and "mass", and we end up with a proper definition of a quantum mechanical "particle".