Show that representations of the angular momentum

[latentcorpse]

Show that representations of the angular momentum algebra $[J_i, J_j ] = \epsilon_{ijk}J_k$ act on finite-dimensional vector spaces, $V_j$ , of dimension $2j + 1$, where $j = 0, 1/2, 1, \dots$

This sounds incredibly easy but what is the question actually asking me to do?

 

[fzero]

Do you know what the definition of a representation is? Even if you're not clear on the abstract idea, you've no doubt seen some representations of that algebra. Do you remember how the vector space is constructed from the generators of the representation? Do you have any idea what $$j$$ is?

 

[latentcorpse]

Do you know what the definition of a representation is? Even if you're not clear on the abstract idea, you've no doubt seen some representations of that algebra. Do you remember how the vector space is constructed from the generators of the representation? Do you have any idea what $$j$$ is?

hi. thanks for the reply.

i am not sure on the exact deifnition of a representation. wikipedia offers a very general description. could you provide a more formal definition please?

I think that's a Lie algebra, no? And they are formed by exponentiating the infinitesimal generators aren't they?

j is the angular momentum quantum number, isn't it?

 

[fzero]

hi. thanks for the reply.

i am not sure on the exact deifnition of a representation. wikipedia offers a very general description. could you provide a more formal definition please?

Yes. If you have a linear operator, $$L$$, that acts on a vector space $$V$$, $$L:V\rightarrow V$$, we can define a representation in the following manner. Choose an orthonormal basis $$\{ \mathbf{v}_a \}$$ of $$V$$. For convenience, we'll use the index $$a$$ to label the basis elements, so we're implicitly treating this as if the vector space is finite. Now the action of $$L$$ on the basis vectors can be decomposed in terms of the basis vectors as

$$L( \mathbf{v}_a ) =\sum_b L_{ba} \mathbf{v}_b.$$

The matrix $$L_{ab}$$ is called the representation of $$L$$ on the vector space $$V$$. I believe that I've chosen the order of indices correctly so that it has the proper action on a vector $$w = \sum_c \mathbf{v}_c$$ when $$w$$ is expressed as a column vector.

When we have a collection of operators $$L_i$$, you can show that the matrices that furnish their representation satisfy the same algebra. For your angular momentum case, you can understand the Pauli matrices as a representation.

I think that's a Lie algebra, no? And they are formed by exponentiating the infinitesimal generators aren't they?

Not quite. The operators $$J_i$$ already satisfy a Lie algebra. Exponentiating them would give us the Lie group, but we don't have to do that.

j is the angular momentum quantum number, isn't it?

So can you use any physical or other intuition to say something about states that have different $$j$$ quantum numbers in the vector space $$V$$? You will probably want to recall everything that you know about the angular momentum states, like eigenvalues/vectors and ladder operators.

 

[latentcorpse]

Yes. If you have a linear operator, $$L$$, that acts on a vector space $$V$$, $$L:V\rightarrow V$$, we can define a representation in the following manner. Choose an orthonormal basis $$\{ \mathbf{v}_a \}$$ of $$V$$. For convenience, we'll use the index $$a$$ to label the basis elements, so we're implicitly treating this as if the vector space is finite. Now the action of $$L$$ on the basis vectors can be decomposed in terms of the basis vectors as

$$L( \mathbf{v}_a ) =\sum_b L_{ba} \mathbf{v}_b.$$

The matrix $$L_{ab}$$ is called the representation of $$L$$ on the vector space $$V$$. I believe that I've chosen the order of indices correctly so that it has the proper action on a vector $$w = \sum_c \mathbf{v}_c$$ when $$w$$ is expressed as a column vector.

When we have a collection of operators $$L_i$$, you can show that the matrices that furnish their representation satisfy the same algebra. For your angular momentum case, you can understand the Pauli matrices as a representation.



Not quite. The operators $$J_i$$ already satisfy a Lie algebra. Exponentiating them would give us the Lie group, but we don't have to do that.



So can you use any physical or other intuition to say something about states that have different $$j$$ quantum numbers in the vector space $$V$$? You will probably want to recall everything that you know about the angular momentum states, like eigenvalues/vectors and ladder operators.

ok thanks for that stuff on representations. I'm sure i can waffle all the stuff i know aobut angular momentum states for the last bit about states with different j numbers.

Have i actually answered the original question though? Have i shown that the matrices act on V_j of dim 2j+1? Wouldn't that require the Pauli matrices to be (2j+1)x(2j+1) - this doesn't make sense though because we know the Pauli matrices are all 2x2. hmmm.

 

[fzero]

The Pauli matrices are a representation for a specific value of $$j$$. You will need to explain why $$V$$ can be written as a sum of the spaces $$V_j$$ and why the $$V_j$$ have dim $$2j+1$$.

 

[latentcorpse]

The Pauli matrices are a representation for a specific value of $$j$$. You will need to explain why $$V$$ can be written as a sum of the spaces $$V_j$$ and why the $$V_j$$ have dim $$2j+1$$.

ok well given a specific j value, there are 2j+1 corresponding allowed values of m and therefore 2j+1 quantum states associated ot that particular j. This means that V_j is the space of all quantum states associated to an angular momentum quantum number of j.
This means that V is all the possible quantum states of the system (direct sum since there will be no intersection of states with different j values).

But I'm still confused - the Pauli matrices are 2x2 so surely they can only act on 2d vector spaces. for example it doesn't make sense to try and multiply a pauli matrix with a vector in R^4, does it?

 

[fzero]

j does not have to be an integer.

 

[latentcorpse]

j does not have to be an integer.

i understand that. but to me it seems that this theory only works for j=1/2.

what if, say, j=3/2. then what happens?
do the pauli matrices no longer apply?
surely we would need a set of 4x4 matrices as our representation?

 

[fzero]

Yes, the Pauli matrices are a representation of spin 1/2. Any other j will correspond to a representation by some other matrices of rank 2j+1.