Quantum, Spin, Orbital Angular momentum, operators

[izzy93]

Homework Statement

If a particle has spin 1/2 and is in a state with orbital angular momentum L, there are two basis states with total z-component of angular momentum m*hbar l L,s,Lz,sz > which can be expressed in terms of the individual states ( l L,s,Lz,sz > = l L,Lz > l s,sz > ) as

l a > = l L, 1/2 , m-1/2 , 1/2 > = l L, m-1/2 > l 1/2, 1/2 > = l L, m-1/2 > l alpha >
l b > = l L, 1/2 , m+1/2 , -1/2 > = l L, m+1/2 > l 1/2, -1/2 > = l L, m-1/2 > l beta >

Using these two states (l a > and l b >) as a basis, show that the matrix representation of the operator L.S is: where L.S = 1/2 ( L+S- + L-S+) + LzSz note all operators here with hats

L.S = Hbar ^2 /2 { (m-1/2) [ (L +1/2)^2 - m^2 ]^1/2

[ (L +1/2)^2 - m^2 ]^1/2 (m-1/2) }

Homework Equations

trying to find <a|L.S|a>, <a|L.S|b>, <b|L.S|a>, <b|L.S|b> (those are the matrix elements), by expanding L.S as in the equation, and then using the actions of the given operators on the basis vectors .

The Attempt at a Solution

<alpha|beta> = <beta|alpha> = 0

LzSz |a> = Lz |l,m-> Sz |alpha> = hbar m- |l,m-> hbar/2 |alpha> ...

am confused on how to contract with the other matrix terms,

any help much appreciated

 

[vela]

Homework Statement

If a particle has spin 1/2 and is in a state with orbital angular momentum L, there are two basis states with total z-component of angular momentum m*hbar l L,s,Lz,sz > which can be expressed in terms of the individual states ( l L,s,Lz,sz > = l L,Lz > l s,sz > ) as

l a > = l L, 1/2 , m-1/2 , 1/2 > = l L, m-1/2 > l 1/2, 1/2 > = l L, m-1/2 > l alpha >
l b > = l L, 1/2 , m+1/2 , -1/2 > = l L, m+1/2 > l 1/2, -1/2 > = l L, m-1/2 > l beta >

Using these two states (l a > and l b >) as a basis, show that the matrix representation of the operator L.S is: where L.S = 1/2 ( L₊S_- + L_-S₊) + L_zS_z note all operators here with hats
$$\hat{L}\cdot\hat{S} = \frac{\hbar^2}{2}
\begin{pmatrix}
m-1/2 & \sqrt{(L+1/2)^2 - m^2} \\
\sqrt{(L+1/2)^2 - m^2} & m-1/2
\end{pmatrix}
$$

Homework Equations

trying to find <a|L.S|a>, <a|L.S|b>, <b|L.S|a>, <b|L.S|b> (those are the matrix elements), by expanding L.S as in the equation, and then using the actions of the given operators on the basis vectors.

The Attempt at a Solution

<alpha|beta> = <beta|alpha> = 0

$$\hat{L}_z\hat{S}_z\lvert a \rangle = \hat{L}_z\lvert l,m-\rangle \hat{S}_z\lvert \alpha \rangle = (\hbar m-) \lvert l, m-\rangle \frac{\hbar}{2} |\alpha \rangle$$

am confused on how to contract with the other matrix terms,

any help much appreciated

You have the correct approach. What do the raising and lowering operators do to a state? For instance, what does $\hat{L}_-$ do to $\lvert l, m \rangle$? You should be able to find this in your textbook if you don't already know.