I Addition of angular momenta

alebruna
Messages
6
Reaction score
0
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.
 

Attachments

Last edited:
Physics news on Phys.org
\textbf{1. Angular Momentum and Spin Operators:}

You correctly started from the fact that you have two angular momentum operators:

\[
\mathbf{L} \quad \text{(orbital angular momentum)}, \quad \mathbf{S} \quad \text{(spin angular momentum)},
\]

and you define the total angular momentum operator as

\[
\mathbf{J} = \mathbf{L} + \mathbf{S}.
\]

Because both \(\mathbf{L}\) and \(\mathbf{S}\) satisfy the angular momentum commutation relations,

\[
[J_i, J_j] = i \hbar \epsilon_{ijk} J_k,
\]

it follows that \(\mathbf{J}\) also behaves like an angular momentum operator.

---

\textbf{2. Commutation and Common Eigenbasis:}

You mentioned studying the commutators involving \(J^2\), \(L^2\), and \(S^2\):

\[
[J^2, L^2] = 0, \quad [J^2, S^2] = 0, \quad [L^2, S^2] = 0.
\]

Since these operators commute pairwise, they can share a common eigenbasis. This is essential in constructing the joint eigenstates of these observables.

---

\textbf{3. Quantum Numbers and Dirac Notation:}

The standard notation for the joint eigenstates of \(J^2\) and \(J_z\) is

\[
|j, m_j\rangle,
\]

where \(j\) is the total angular momentum quantum number and \(m_j\) is its projection along the \(z\)-axis.

Similarly, the eigenstates of \(\mathbf{L}^2\) and \(L_z\) are

\[
|l, m_l\rangle,
\]

and for \(\mathbf{S}^2\) and \(S_z\),

\[
|s, m_s\rangle.
\]

For a spin-\(\frac{1}{2}\) particle (like a fermion), \(s = \frac{1}{2}\) and \(m_s = \pm \frac{1}{2}\).

---

\textbf{4. Addition of Angular Momenta:}

When combining \(\mathbf{L}\) and \(\mathbf{S}\), the possible total angular momentum quantum numbers \(j\) satisfy

\[
j = |l - s|, |l - s| + 1, \ldots, l + s.
\]

For example, if \(l = 1\) and \(s = \frac{1}{2}\), then \(j\) can be \(\frac{1}{2}\) or \(\frac{3}{2}\).

This gives the \emph{degeneracy} you mentioned: multiple \(m_j\) values correspond to each \(j\).

---

\textbf{5. Basis Transformation and Clebsch-Gordan Coefficients:}

The original basis is the \emph{uncoupled} basis:

\[
|l, m_l\rangle |s, m_s\rangle,
\]

where angular momenta are considered separately. The new basis after addition is the \emph{coupled} basis:

\[
|j, m_j; l, s\rangle,
\]

which are eigenstates of \(\mathbf{J}^2\) and \(J_z\).

The transformation between these two bases is given by the Clebsch-Gordan coefficients \(C^{j,m_j}_{l,m_l;s,m_s}\):

\[
|j, m_j; l, s\rangle = \sum_{m_l, m_s} C^{j,m_j}_{l,m_l;s,m_s} |l, m_l\rangle |s, m_s\rangle.
\]

These coefficients can be found in tables or calculated using standard recursion relations.

---

\textbf{6. Spectrum Generating Algebra and Ladder Operators:}

You also correctly introduced the ladder operators:

\[
J_\pm = J_x \pm i J_y,
\]

which raise or lower the \(m_j\) value by one:

\[
J_\pm |j, m_j\rangle = \hbar \sqrt{j(j+1) - m_j(m_j \pm 1)} |j, m_j \pm 1\rangle.
\]

The use of ladder operators is fundamental in constructing the entire multiplet of states starting from the highest weight state \( |j, j\rangle \).

---

\textbf{7. Normalization:}

The normalization condition

\[
\langle j, m_j; l, s | j, m_j; l, s \rangle = 1
\]

ensures that these states form an orthonormal basis. When constructing states via Clebsch-Gordan coefficients, the coefficients themselves satisfy orthogonality and normalization relations ensuring this.

---

\textbf{8. Is the Exercise Correct?}

Your exercise and derivation are essentially correct and follow the standard textbook approach for adding angular momenta in quantum mechanics. The key points are:

\begin{itemize}
\item Verify the commutation relations hold for \(\mathbf{J}\).
\item Identify the allowed \(j\) values.
\item Understand the transformation between uncoupled and coupled bases via Clebsch-Gordan coefficients.
\item Use ladder operators to generate all states.
\item Normalize states properly.
\end{itemize}

Your summary touches on all of these points appropriately.

---

\textbf{Additional Suggestions:}

If you want to deepen your understanding, I recommend:

\begin{itemize}
\item Reviewing a detailed example, e.g., adding \(l=1\) and \(s=1/2\).
\item Studying explicit Clebsch-Gordan tables.
\item Working through problems that require expanding states in both bases.
\item Consulting standard textbooks such as Sakurai's \emph{Modern Quantum Mechanics} or Cohen-Tannoudji's \emph{Quantum Mechanics}.
\end{itemize}
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. Towards the end of the first lecture for the Qiskit Global Summer School 2025, Foundations of Quantum Mechanics, Olivia Lanes (Global Lead, Content and Education IBM) stated... Source: https://www.physicsforums.com/insights/quantum-entanglement-is-a-kinematic-fact-not-a-dynamical-effect/ by @RUTA
If we release an electron around a positively charged sphere, the initial state of electron is a linear combination of Hydrogen-like states. According to quantum mechanics, evolution of time would not change this initial state because the potential is time independent. However, classically we expect the electron to collide with the sphere. So, it seems that the quantum and classics predict different behaviours!

Similar threads

Back
Top