Angular momentum Operators and Commutation

[cooev769]

So I understand the commutation laws etc, but one thing I can't get my head around is the fact that L^2 commutes with Lx,y,z but L does not.

I mean if you found L^2 couldn't you just take the square root of it and hence know the total angular momentum. It seems completely ridiculous that you could know the square of the total angular momentum is 100, but not know that the angular momentum is therefore 10. Either that, or the textbook explains it horrifically and you can know the total angular momentum squared which is a scalar and hence you can know L scalar but L generally has direction, so they use L^2 to be explicit that it's a scalar.

Please elaborate. Cheers.

 

[MisterX]

You've pretty much got it. The scalar $L = \sqrt{\mathbf{L}^2}$ commutes with everything $\mathbf{L}^2$ does. The vector $\mathbf{L}$ does not.

 

[cooev769]

Oh okay, then why the hell didn't they just put that in the book, he didn't explain it at all, just said L^2 does but L doesn't. Extremely ambiguous. Stupid Griffiths.

 

[WannabeNewton]

You do realize that $\hat{L}$ is an operator right...? Do you understand how "square roots" of operators actually work? No one talks about the "square root" of $\hat{L}^2$ they talk about $\hat{L}^2$ and $\hat{L}$ alone, including Griffiths. Here $\hat{L} = (\hat{L}_x,\hat{L}_y,\hat{L}_z)$ so clearly the different components do not commute with one another.

As an aside, you calling him "stupid" is extremely unwarranted given that you do not yet understand basic operator theory.

 

[cooev769]

Good point, but yes I do understand that, but he clearly said, "meaning we can know the square of the angular momentum but not the angular momentum itself". So I'm just thinking well if you can know both the angular momentum squared and a component at the same time, you could square root the square and get angular momentum. Then that must mean given they are both able to be observed simultaneously, that the L operator for total angular momentum and Lz a component must commute.

 

[Ishida52134]

I don't know what you're talking about wbn, but clearly you can multiply operators together and get $\hat{L}^n$ and that commutes with all $\hat{L}^k$.

 

[cooev769]

Btw, Jimmy page Madison Square Garden nice.

 

[cooev769]

Yeah the Linear algebra is really not covered enough in this textbook. I mean I've done linear algebra and was decent at it, but it would be nice to have the explanations there to refresh on. I feel like I need to re-read my real analysis and linear algebra textbooks just so I can piece together everything.

 

[cooev769]

Btw WannabeNation, me calling him stupid was a joke, clearly the man is not stupid he is a well regarded physics author and his EM book is hailed as potentially the best ever. I was just making a joke so you would know which textbook I am using, you do not need to rub my ignorance to operator theory in my face, I'm an undergraduate student and I'm still learning.

 

[micromass]

I don't know what you're talking about wbn, but clearly you can multiply operators together and get $\hat{L}^n$ and that commutes with all $\hat{L}^k$.

You do realize that $\mathbf{L}^2$ is not the square of $\mathbf{L}$, right?

 

[Matterwave]

One should realize that:

$$\sqrt{L^2}=\sqrt{L_x^2+L_y^2+L_z^2} \neq \bf{L}=L_x+L_y+L_z$$

Taken as an operator, it is not even immediately clear what $\sqrt{L^2}$ even means. One can take the Taylor expansion of the square root function, to try to define this operator, but as the series is not convergent everywhere, one runs into problems immediately.

 

[micromass]

One should realize that:

$$\sqrt{L^2}=\sqrt{L_x^2+L_y^2+L_z^2} \neq \bf{L}$$

Taken as an operator, it is not even immediately clear what $\sqrt{L^2}$ even means. One can take the Taylor expansion of the square root function, to try to define this operator, but as the series is not convergent everywhere, one runs into problems immediately.

The spectral theorem for unbounded operators allows you to define this square root.

 

[cooev769]

Ahaha thanks for all the replies. Slightly two sided answer. Yes, I know you cannot just say L operator is the square of L^2 operator. But what I'm saying is do L operator and Lz operator commute, if that makes it easier. Can I know the magnitude of the total angular momentum and one component at the same time in other words, given I can know the square of the total angular momentum and a component at the same time, given that this is the definition of commuting operators, they share eigenfunctions and you can know both.

 

[Matterwave]

The spectral theorem for unbounded operators allows you to define this square root.

Do you know what it's actually defined as then? Would this operator have the same eigenfunctions as L^2, with eigenvalue equal to sqrt(l(l+1))?

 

[cooev769]

Yes that's exactly what I'm asking, would the operator L commute with L^2, and would it have the eigenvalue sqrt(h^2l(l+1)

 

[strangerep]

Do you know what it's actually defined as then?

(Sigh.) Of course Micromass knows.

Did you and cooev769 actually make an effort to look up the "spectral theorem", and see how it answers the question in the 2nd part of your post?

Hint: how does one represent a self-adjoint operator on Hilbert space in terms of its eigenvectors?

 

[cooev769]

I don't have the maths knowledge to even understand what you're saying let alone figure it out for myself. Hence why I've asked here. I just want a yes or no answer, not to go and learn some linear algebra which we don't do in our physics degree's here and aren't expected to understand. I don't have the maths ability to tie the two together.

 

[Matterwave]

(Sigh.) Of course Micromass knows.

Did you and cooev769 actually make an effort to look up the "spectral theorem", and see how it answers the question in the 2nd part of your post?

Hint: how does one represent a self-adjoint operator on Hilbert space in terms of its eigenvectors?

I asked as a point of curiosity since I have not seen, in my QM studies, an operator L=sqrt(L^2) defined. Unfortunately, I don't have time right now to study the spectral theorem and come up with a definition. Well, I can try. On a finite dimensional space, I know that a self-adjoint operator has a set of orthonormal eigenfunctions which span the space, thus given:

$$\hat{A}\left|\psi_n\right>=A_n\left|\psi_n\right>$$

For n=1,...,d where d is the dimensionality of the finite dimensional space. We can construct the operator as:

$$\hat{A}=\sum_{n=1}^d A_n\left|\psi_n\right>\left<\psi_n\right|$$

For a positive integer p, we have that A commutes with A^p (since A commutes with any function of itself), and therefore:

$$\hat{A}^p=A_n^p\left|\psi_n\right>$$

So that:

$$\hat{A}^p=\sum_n^d A_n^p \left|\psi_n\right>\left<\psi_n\right|$$

I suppose we can define this formula to work for real p as well? If we make such a definition, then:

$$\hat{A}^{1/2}=\sum_n^d A_n^{1/2}\left|\psi_n\right>\left<\psi_n\right|$$

It seems that in order for this operator to be self-adjoint as well, we must demand:

$$A_n^{1/2}=(A_n^{1/2})^*$$

Which would be true if all $A_n\geq 0$. Is this generally how a square root of an operator is defined? Does this extend to the infinite dimensional case without problem? In that case, a direct substitution $\hat{A}\rightarrow\hat{L^2}$ would lead to:

$$\hat{L}\equiv\sqrt{\hat{L}^2}=\sum_{l,m}^\infty \sqrt{\hbar^2 l(l+1)}\left|l,m\right>\left<l,m\right| $$

Which would have indeed the same eigenfunctions, with eigenvalue sqrt(l(l+1)) by construction. Now, my question would be, are there no issues with convergences and the like when moving from the finite dimensional case to the infinite dimensional case?

 

[strangerep]

Does this extend to the infinite dimensional case without problem?

It depends on the details of the operator. This opens a large can of worms.

are there no issues with convergences and the like when moving from the finite dimensional case to the infinite dimensional case?

Yes, convergence issues are critical -- one must specify the topology in which one is working. But I won't give you a course in General Topology here.

For compact s.a. operators on (inf-dim) Hilbert space (in which one usually assumes the standard norm topology when discussing convergence issues), one finds a set of eigenvectors that constitute an orthormal basis. Further, the sequence of eigenvalues tends to 0.

For noncompact bounded s.a. operators on (inf-dim) Hilbert space, things are trickier. Basically, one constructs an integral measure over the space of eigenvalues, which generalizes the formulas you wrote.

For unbounded operators, things are trickier still, since one must specify domains carefully. (Such operators are not well-defined everywhere on the Hilbert space.) That's why some authors generalize to use a so-called "Rigged Hilbert Space" (kinda like a generalization of the Schwarz theory of distributions). Within that framework, there's another theorem, the so-called "nuclear spectral theorem" that establishes the existence of an integral measure that allows one to do QM.

Ballentine gives a brief, but useful, introduction to the notions (and motivation for) Rigged Hilbert Space. (Every serious student of QM should have a copy of Ballentine...)

 

[strangerep]

I just want a yes or no answer

Tbh, I'm not really interested in teaching a parrot.

But... more constructively...

not to go and learn some linear algebra which we don't do in our physics degree's here and aren't expected to understand.

You aren't expected to understand any linear algebra for your physics degree? Sorry, but I think that's rubbish. Of course a physics student needs to understand some linear algebra, just as they need to understand some calculus. There is no hope of understanding QM properly if you don't remedy your lack of knowledge of linear algebra, and soon.

Axler's "Linear Algebra Done Right" might be a good starting, or maybe "Linear Algebra" in the Schaum Outline series. I see Schaum now even has a "Linear Algebra -- Crash Course" volume.

I don't have the maths ability to tie the two together.

Anyway, Matterwave has given a sketch of how one represents operators in terms of their eigenvectors.

 

[cooev769]

I'm not a fan of just blindly accepting something either, I am a very conceptual learner. But right now, it is just a small question I would like a yes or no answer to so that if it is a no I can understand why.

Yes we do linear algebra Math202, but it is a beginners course and it is not even compulsory for a Bsci, majoring in physics in New Zealand. I will have a look at the linear algebra textbooks. But can somebody for the love of god answer my question with a yes or no, once I have the answer I can then go and search for the why.

Given that there is an operator L^2, with eigenvalue l(l+1), can you define an operator L with eigenvalue square root of l(l+1) when applied to an eigenfunction, thus telling you the total angular momentum of the system?

 

[cooev769]

I have done the math 202 paper and did extremely well in it, but as I said it is a beginners course and was last year. I do recall the spectral theorem having something to do with the diagonalisation of a matrix, but that's about it.

 

[cooev769]

What I mean is this is an under graduate introduction to linear algebra, I probably do not have the mathematical ability nor the time to obtain the ability to answer this question for myself and being the curious person I am this is frustrating me that I don't have a yes or no answer.

 

[Matterwave]

It depends on the details of the operator. This opens a large can of worms.

Yes, convergence issues are critical -- one must specify the topology in which one is working. But I won't give you a course in General Topology here.

For compact s.a. operators on (inf-dim) Hilbert space (in which one usually assumes the standard norm topology when discussing convergence issues), one finds a set of eigenvectors that constitute an orthormal basis. Further, the sequence of eigenvalues tends to 0.

For noncompact bounded s.a. operators on (inf-dim) Hilbert space, things are trickier. Basically, one constructs an integral measure over the space of eigenvalues, which generalizes the formulas you wrote.

For unbounded operators, things are trickier still, since one must specify domains carefully. (Such operators are not well-defined everywhere on the Hilbert space.) That's why some authors generalize to use a so-called "Rigged Hilbert Space" (kinda like a generalization of the Schwarz theory of distributions). Within that framework, there's another theorem, the so-called "nuclear spectral theorem" that establishes the existence of an integral measure that allows one to do QM.

Ballentine gives a brief, but useful, introduction to the notions (and motivation for) Rigged Hilbert Space. (Every serious student of QM should have a copy of Ballentine...)

I was thinking about this problem on my daily commute, and here's my analysis that I came up with, tell me if I'm going in the wrong direction:

So, we have two different (but hopefully equivalent) ways of expressing L^2, either through its complete and orthonormal set of eigenfunctions (spherical harmonics) or through the spatial derivatives (using spherical coordinates):

$$L^2=\sum_{l=0}^\infty\sum_{m=-l}^l \hbar^2l(l+1)\left|l,m\right>\left<l,m\right|$$

Or:

$$L^2=-\hbar^2\left(\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right)+\frac{1}{\sin^2\theta} \frac{\partial^2}{\partial\phi^2}\right)$$

Let us use the first representation, and look at the expectation value of this operator given some valid wavefunction:

$$\left<L^2\right>=\left<\Psi\right|L^2\left|\Psi\right>=\sum_{l=0}^∞ \sum_{m=-l}^l \hbar^2l(l+1)\left<\Psi|l,m\right>\left<l,m|\Psi\right>$$

Define the (complex) numbers:

$$a_{l,m}\equiv\left<l,m|\Psi\right>$$

And we have then:

$$\left<L^2\right>=\sum_{l=0}^\infty\sum_{m=-l}^l \hbar^2l(l+1)|a_{l,m}|^2$$

In order for this sum to converge, we must require that $|a_{l,m}|^2\rightarrow 0$ as $l\rightarrow\infty$. Moreover, we must require that $|a_{l,m}|^2$ goes to zero faster than $l^2$ diverges by at least more than 1 power of l (e.g. it goes to zero as a power of 1/l^4, 1/l^3 wouldn't work I think since the harmonic series does not converge).

However, our usual definition of a Hilbert space, the space from which $\left|\Psi\right>$ can be picked from only requires that $\left|\Psi\right>$ is normalizable. In other words, our definition of a Hilbert space only requires:

$$\left<\Psi|\Psi\right><\infty$$

This constraint is equivalent to only the constraint:

$$\sum_{l=0}^\infty\sum_{m=-l}^l |a_{l,m}|^2<\infty$$

This is not a sufficient constraint to guarantee that our angular momentum operator has a well defined expectation value over all of the Hilbert space, for consider if:

$$|a_{l,m}|^2=\frac{1}{1+l^2},\quad m=0$$
$$\qquad else\quad a_{l,m}=0$$

This series converges, meaning our wave function would be normalizable, but the series for the expectation value of L^2 does not converge:

$$\sum_{l=0}^\infty\sum_{m=-l}^l \hbar^2l(l+1)|a_{l,m}|^2=\infty$$

This finally leads me to the conclusion that the Hilbert space is "too big" of a space for our definition of L^2 (big might not be the right word to use here, as L^2 might be defined for particular functions not in our Hilbert space, I have to think about this scenario with more detailed analysis), and looking back at the derivative definition of L^2, this is obvious. L^2 will not work for non twice-differentiable functions, but there are certainly non twice-differentiable functions which are normalizable (e.g. a square wave). This leads me to conclude, therefore, that physical states should be at least twice-differentiable (else, they would have infinite L^2). In fact, if we require L^p to exist for all p, we must require physical states to be infinitely differentiable. Therefore, can we say "physical wave functions must be normalizable, as well as smooth"? What about adding the requirement that they be analytic?

This leads me to the curious side-conclusion that the more times our wave function is differentiable, the faster $|a_{l,m}|$ will approach 0. If we require our wave function to be smooth, then $|a_{l,m}|$ must approach 0 faster than any polynomial function diverges. Is this result correct? It appears to be non-trivial (or maybe it is, but I'm just not smart enough).

 

[HomogenousCow]

So I understand the commutation laws etc, but one thing I can't get my head around is the fact that L^2 commutes with Lx,y,z but L does not.

I mean if you found L^2 couldn't you just take the square root of it and hence know the total angular momentum. It seems completely ridiculous that you could know the square of the total angular momentum is 100, but not know that the angular momentum is therefore 10. Either that, or the textbook explains it horrifically and you can know the total angular momentum squared which is a scalar and hence you can know L scalar but L generally has direction, so they use L^2 to be explicit that it's a scalar.

Please elaborate. Cheers.

How bout you read this article http://en.wikipedia.org/wiki/Angular_momentum_operator and calm down a bit.

 

[strangerep]

Given that there is an operator L^2, with eigenvalue l(l+1), can you define an operator L with eigenvalue square root of l(l+1) when applied to an eigenfunction, thus telling you the total angular momentum of the system? [...]

[...] this is frustrating me that I don't have a yes or no answer.

Matterwave already gave you an answer in post #18. That's why I didn't elaborate.

If you still don't understand why post #18 answers your question, then feel free to ask further questions, quoting the context of that post.

Btw, in case you think I'm being too tough on you, let me say that I actually understand your situation, having been through it myself. In one u'grad course, they presumed knowledge of group theory even though the course sequence I'd been allowed to choose did not include group theory up to that point. I was close to a nervous breakdown by the end, but got through eventually.

Take a few deep breaths, and then continue...

 

[strangerep]

[...]

This finally leads me to the conclusion that the Hilbert space is "too big" of a space for our definition of L^2 (big might not be the right word to use here, as L^2 might be defined for particular functions not in our Hilbert space, I have to think about this scenario with more detailed analysis), and looking back at the derivative definition of L^2, this is obvious. L^2 will not work for non twice-differentiable functions, but there are certainly non twice-differentiable functions which are normalizable (e.g. a square wave). This leads me to conclude, therefore, that physical states should be at least twice-differentiable (else, they would have infinite L^2). In fact, if we require L^p to exist for all p, we must require physical states to be infinitely differentiable. Therefore, can we say "physical wave functions must be normalizable, as well as smooth"? What about adding the requirement that they be analytic?

This leads me to the curious side-conclusion that the more times our wave function is differentiable, the faster $|a_{l,m}|$ will approach 0. If we require our wave function to be smooth, then $|a_{l,m}|$ must approach 0 faster than any polynomial function diverges. Is this result correct? It appears to be non-trivial (or maybe it is, but I'm just not smart enough).

Oh, I'm sure you're smart enough -- since you've essentially arrived at the standard motivation for using so-called "Rigged Hilbert Spaces" instead of ordinary Hilbert spaces. The basic idea is that we need a "small" space $\Omega$ on which all the observables are well-defined everywhere. This means that we must be able to operate with any observable arbitrarily many times on any element of $\Omega$, and still get a result that's also in $\Omega$. One may complete $\Omega$ in norm topology to get a Hilbert space $H$. One can also construct the dual space of $\Omega$, denoted $\Omega'$. This triple of spaces, i.e., $\Omega \subset H \subset \Omega'$ is known as a "Gel'fand triple" or "Rigged Hilbert Space".

I think you'd really enjoy ch1 of Ballentine, especially section 1.4. Can you access a copy?

Have you studied distribution theory yet? The Schwarz theory of distributions is essentially a prototypical case of Rigged Hilbert Spaces, applicable to the usual position/momentum observables.

 

[Matterwave]

Sounds interesting. I have not really studied distribution theory (I know roughly what they are since I've encountered them several times, but not much more than that). Unfortunately I only have finite time in which to study, and right now I'm busy reviewing other material. Hopefully when I'm not so busy I'll get back to look into this some more. In the mean time, I'll see if I can locate a copy of Ballentine, thanks for the recommendation. :)

 

[dextercioby]

@Post #24 above: One trivially gets that the L^2 angular momentum operator in L^2(R^3) is unbounded, hence cannot be defined everywhere in the Hilbert space. See sections 10.2 and 11.5 of Blank et al's book "Hilbert Space Operators in Quantum Physics".

 

[Matterwave]

@Post #24 above: One trivially gets that the L^2 angular momentum operator in L^2(R^3) is unbounded, hence cannot be defined everywhere in the Hilbert space. See sections 10.2 and 11.5 of Blank et al's book "Hilbert Space Operators in Quantum Physics".

Ok, thanks for calling my argument trivial, I put a lot of thought and effort into it! It wasn't trivial to me anyways lol.

I'll eventually get around to reading some of this stuff...