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- TL;DR Summary
- Theoretical exercise on the addition of an angular momentum operator and a spin operator, derivation of eigenvalues and basis. Normalization of the eigenstates.
While studying for my exam I came across the addition of two angular momenta, in the simple case of ##J=L+S## where ##L## is the angular momentum operator and ##S## is the spin (in this case a fermion with ##s=1/2##). I have some doubts on the derivation of the basis and the eigenvalues.
From the start we know that ##J=L+S## and by this ##J^2=(L+S)^2##. We should start by the commutator ##[J_m,J_n]##
##
[J_m,J_n]=[L_m+S_m,L_n+S_n]=[L_m,L_n]+[S_m,S_n]=i\hbar\epsilon_{jmn}L_j+i\hbar\epsilon_{jmn}S_j=i\hbar\epsilon_{jmn}(L_j+S_j)=
##
##
=i\hbar\epsilon_{jmn}J_j
##
that shows that $J$ is still an angular momentum. By this we know that all the properties holding for both ##L## and ##S## apply also for ##J##, for example: ##[J,J^2]=0##.
Now we can study the commutators ##[J_m,L_n],\ [J_m,L^2],\ [J_m,S_n],\ [J_m,S^2],\ [J^2, L_n],\ [J^2,S_n],\ [J^2,L^2],####\ [J^2,S^2]##:
##
[J_m,L_n]=[L_m+S_m,L_n]=[L_m,L_n]\neq 0
##
##
[J_m,L^2]=\sum_n [J_m, L_n^2]=\sum_n\left( L_n[J_m,L_n]+[J_m,L_n]L_n\right)=\sum_n (i\hbar\epsilon_{jmn}L_n L_j+i\hbar\epsilon_{jmn}L_j L_n)=...=0
##
##
[J_m,S_n]=...\neq 0
##
##
[J_m,S^2]=...=0
##
##
[J^2,L_n]=[L^2+S^2+2L\cdot S,L_n]=[L^2,L_n]+\sum_m[L_m,L_n]=0+i\hbar\epsilon_{jmn}L_j\neq 0
##
##
[J^2,S_n]=...\neq 0
##
##
[J^2,L^2]=[L^2+S^2+2L\cdot S,L^2]=[L,L^2]\cdot S=0
##
##
[J^2,L^2]=...=0
##
Since ##J^2,\ J_m,\ L^2,\ S^2## all commute, for the third theorem of hermitian operators they feature a common base of eigenstates. Since ##J## has the form of an angular momentum we can assume that ##J_3## is associated to an eigenvalue ##\hbar m_j## and ##J^2## is associated to an eigenvalue ##\hbar j(j+1)##.
Hence the basis associated to operators ##J_m,\ J^2,\ L^2,\ S^2## is in Dirac notation: ##|j, m_j, l, s\rangle##.
About the properties of the eigenvalues:
##
J_3\left|j, m_j, l, s\right\rangle=(L_3+S_3)\left|j, m_j, l, s\right\rangle=(\hbar m+\hbar m_s)\left|j, m_j, l, s\right\rangle=\hbar m_j \left|j, m_j, l, s\right\rangle \rightarrow m_j=m+m_s
##
##
J^2\left|j, m_j, l, s\right\rangle=\hbar^2j(j+1)\left|j, m_j, l, s\right\rangle \ \text{with}\ j>0 \\
##
##
L^2\left|j, m_j, l, s\right\rangle=\hbar^2l(l+1)\left|j, m_j, l, s\right\rangle \\
##
##
S^2\left|j, m_j, l, s\right\rangle=\hbar^2s(s+1)\left|j, m_j, l, s\right\rangle \\
\rightarrow l-s< j<l+s
##
also ##-j\leq m_j\leq j##
We can see that the new basis of eigenvalues features a degeneracy for ##m+1,\ s=1/2## and ##m,\ s=-1/2## this means that the new basis can be written as
$$
\left|j, m_j, l, s\right\rangle=\sum_{m,m_s}^\ast C(m,m_s)\left|l,m\right\rangle \left|s,m_s\right\rangle
$$
The superscript * means that ##m+m_s## is constant.
Using spectrum generating algebra we can define ##J_+=L_++S_+## and ##J_-=L_--S_-##.
##
J_+\left|l,l\right\rangle\left|s,\frac{1}{2}\right\rangle=0 \\
##
##
J_-\left|l,-l\right\rangle\left|s,-\frac{1}{2}\right\rangle=0 \\
##
Taking the two states ##j=l+1/2## and ##j=l-1/2## can give us the coefficients for the double series.
$$
\begin{cases}
\left|l-1/2, m_j, l, s\right\rangle=\alpha_-\left|l,m_j-1/2\right\rangle \left|s,1/2\right\rangle+\beta_-\left|l,m_j+1/2\right\rangle \left|s,-1/2\right\rangle \\
\left|l+1/2, m_j, l, s\right\rangle=\alpha_+\left|l,m_j-1/2\right\rangle \left|s,1/2\right\rangle+\beta_+\left|l,m_j+1/2\right\rangle \left|s,-1/2\right\rangle
\end{cases}
$$
Normalizing gets us that
$$
\alpha_\pm=\pm\sqrt{\frac{l\pm m_j+1/2}{2l+1}}=\pm\beta_\mp
$$
This should conclude the derivation. Is this exercise correct? If not can anyone point out how this is wrong and why?
Some source material I found online are linked below.
From the start we know that ##J=L+S## and by this ##J^2=(L+S)^2##. We should start by the commutator ##[J_m,J_n]##
##
[J_m,J_n]=[L_m+S_m,L_n+S_n]=[L_m,L_n]+[S_m,S_n]=i\hbar\epsilon_{jmn}L_j+i\hbar\epsilon_{jmn}S_j=i\hbar\epsilon_{jmn}(L_j+S_j)=
##
##
=i\hbar\epsilon_{jmn}J_j
##
that shows that $J$ is still an angular momentum. By this we know that all the properties holding for both ##L## and ##S## apply also for ##J##, for example: ##[J,J^2]=0##.
Now we can study the commutators ##[J_m,L_n],\ [J_m,L^2],\ [J_m,S_n],\ [J_m,S^2],\ [J^2, L_n],\ [J^2,S_n],\ [J^2,L^2],####\ [J^2,S^2]##:
##
[J_m,L_n]=[L_m+S_m,L_n]=[L_m,L_n]\neq 0
##
##
[J_m,L^2]=\sum_n [J_m, L_n^2]=\sum_n\left( L_n[J_m,L_n]+[J_m,L_n]L_n\right)=\sum_n (i\hbar\epsilon_{jmn}L_n L_j+i\hbar\epsilon_{jmn}L_j L_n)=...=0
##
##
[J_m,S_n]=...\neq 0
##
##
[J_m,S^2]=...=0
##
##
[J^2,L_n]=[L^2+S^2+2L\cdot S,L_n]=[L^2,L_n]+\sum_m[L_m,L_n]=0+i\hbar\epsilon_{jmn}L_j\neq 0
##
##
[J^2,S_n]=...\neq 0
##
##
[J^2,L^2]=[L^2+S^2+2L\cdot S,L^2]=[L,L^2]\cdot S=0
##
##
[J^2,L^2]=...=0
##
Since ##J^2,\ J_m,\ L^2,\ S^2## all commute, for the third theorem of hermitian operators they feature a common base of eigenstates. Since ##J## has the form of an angular momentum we can assume that ##J_3## is associated to an eigenvalue ##\hbar m_j## and ##J^2## is associated to an eigenvalue ##\hbar j(j+1)##.
Hence the basis associated to operators ##J_m,\ J^2,\ L^2,\ S^2## is in Dirac notation: ##|j, m_j, l, s\rangle##.
About the properties of the eigenvalues:
##
J_3\left|j, m_j, l, s\right\rangle=(L_3+S_3)\left|j, m_j, l, s\right\rangle=(\hbar m+\hbar m_s)\left|j, m_j, l, s\right\rangle=\hbar m_j \left|j, m_j, l, s\right\rangle \rightarrow m_j=m+m_s
##
##
J^2\left|j, m_j, l, s\right\rangle=\hbar^2j(j+1)\left|j, m_j, l, s\right\rangle \ \text{with}\ j>0 \\
##
##
L^2\left|j, m_j, l, s\right\rangle=\hbar^2l(l+1)\left|j, m_j, l, s\right\rangle \\
##
##
S^2\left|j, m_j, l, s\right\rangle=\hbar^2s(s+1)\left|j, m_j, l, s\right\rangle \\
\rightarrow l-s< j<l+s
##
also ##-j\leq m_j\leq j##
We can see that the new basis of eigenvalues features a degeneracy for ##m+1,\ s=1/2## and ##m,\ s=-1/2## this means that the new basis can be written as
$$
\left|j, m_j, l, s\right\rangle=\sum_{m,m_s}^\ast C(m,m_s)\left|l,m\right\rangle \left|s,m_s\right\rangle
$$
The superscript * means that ##m+m_s## is constant.
Using spectrum generating algebra we can define ##J_+=L_++S_+## and ##J_-=L_--S_-##.
##
J_+\left|l,l\right\rangle\left|s,\frac{1}{2}\right\rangle=0 \\
##
##
J_-\left|l,-l\right\rangle\left|s,-\frac{1}{2}\right\rangle=0 \\
##
Taking the two states ##j=l+1/2## and ##j=l-1/2## can give us the coefficients for the double series.
$$
\begin{cases}
\left|l-1/2, m_j, l, s\right\rangle=\alpha_-\left|l,m_j-1/2\right\rangle \left|s,1/2\right\rangle+\beta_-\left|l,m_j+1/2\right\rangle \left|s,-1/2\right\rangle \\
\left|l+1/2, m_j, l, s\right\rangle=\alpha_+\left|l,m_j-1/2\right\rangle \left|s,1/2\right\rangle+\beta_+\left|l,m_j+1/2\right\rangle \left|s,-1/2\right\rangle
\end{cases}
$$
Normalizing gets us that
$$
\alpha_\pm=\pm\sqrt{\frac{l\pm m_j+1/2}{2l+1}}=\pm\beta_\mp
$$
This should conclude the derivation. Is this exercise correct? If not can anyone point out how this is wrong and why?
Some source material I found online are linked below.
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