Lecture Notes#

Angular Momentum Representation Theory#

Because \([\hat J^2,\hat J_z]=0\), we can choose simultaneous eigenstates and build irreducible multiplets algebraically.

Common Eigenstates of \(\hat J^2\) and \(\hat J_z\).

Simultaneous Eigenstates

Let \(\vert j,m\rangle\) satisfy

(41)#\[\begin{split} \begin{split} \hat J^2\vert j,m\rangle &= \hbar^2 j(j+1)\vert j,m\rangle,\\ \hat J_z\vert j,m\rangle &= \hbar m\vert j,m\rangle. \end{split} \end{split}\]

At this stage, \(j,m\in\mathbb{R}\) are labels; quantization is derived below.

Ladder Operator Action.

Ladder Action on \(\vert j,m\rangle\)

(42)#\[\begin{split} \begin{split} \hat J_+\vert j,m\rangle &= \hbar\sqrt{j(j+1)-m(m+1)}\,\vert j,m+1\rangle,\\ \hat J_-\vert j,m\rangle &= \hbar\sqrt{j(j+1)-m(m-1)}\,\vert j,m-1\rangle. \end{split} \end{split}\]

So \(\hat J_+\) raises \(m\) by one and \(\hat J_-\) lowers \(m\) by one.

Derivation: Ladder-Action Coefficients

From \([\hat J_z,\hat J_\pm]=\pm\hbar\hat J_\pm\) and \([\hat J^2,\hat J_\pm]=0\), the states \(\hat J_\pm\vert j,m\rangle\) lie in the same \(j\) multiplet with \(m\to m\pm1\).

For normalization, use (40):

\[ \|\hat J_+\vert j,m\rangle\|^2 =\langle j,m\vert\hat J_-\hat J_+\vert j,m\rangle =\hbar^2\bigl[j(j+1)-m(m+1)\bigr], \]

\[ \|\hat J_-\vert j,m\rangle\|^2 =\langle j,m\vert\hat J_+\hat J_-\vert j,m\rangle =\hbar^2\bigl[j(j+1)-m(m-1)\bigr]. \]

Taking square roots yields (42).

Angular Momentum Quantization.

Quantization Rules

(43)#\[ j\in\left\{0,\tfrac12,1,\tfrac32,2,\ldots\right\}, \qquad m\in\{-j,-j+1,\ldots,j\}. \]

For each \(j\), the multiplet dimension is \(2j+1\).

Proof: Quantum bootstrap argument

The quantum bootstrap is an algebraic method: constrain quantum numbers by positivity of norms. For any operator \(\hat A\) and state \(\vert\psi\rangle\),

\[ \|\hat A\vert\psi\rangle\|^2\ge 0. \]

Applied to angular momentum, this principle fixes the allowed values of \(j\) and \(m\).

From (41), take simultaneous eigenstates \(\vert j,m\rangle\) of \(\hat J^2\) and \(\hat J_z\).

(1) Positivity constraints. Apply positivity to \(\hat A=\hat J_+\) and \(\hat A=\hat J_-\) on \(\vert j,m\rangle\):

\[ \|\hat J_+\vert j,m\rangle\|^2\ge0, \qquad \|\hat J_-\vert j,m\rangle\|^2\ge0. \]

Using (42), these become

(44)#\[\begin{split} \begin{split} \text{(raising)}\;& j(j+1)-m(m+1)\ge 0,\\ \text{(lowering)}\;& j(j+1)-m(m-1)\ge 0. \end{split} \end{split}\]

So \(m\) is bounded: \(-j\le m\le j\).

(2) Ladder unit-step structure. From \([\hat J_z,\hat J_\pm]=\pm\hbar\hat J_\pm\), each ladder action changes \(m\) by exactly one unit. Therefore all \(m\) values in one multiplet differ by integers.

(3) Finite-dimensional irreducibility. In a finite-dimensional irrep, the ladder must terminate at top and bottom states:

\[ \hat J_+\vert j,m_{\max}\rangle=0, \qquad \hat J_-\vert j,m_{\min}\rangle=0. \]

By (42), vanishing coefficients imply

\[ m_{\max}=j, \qquad m_{\min}=-j. \]

Hence the ladder runs

\[ m=-j,-j+1,\ldots,j, \]

with \(N=2j+1\) states.

Since \(N\in\mathbb N\), we must have \(2j\in\mathbb N_0\). Therefore

\[ j\in\left\{0,\tfrac12,1,\tfrac32,\ldots\right\}, \qquad m\in\{-j,-j+1,\ldots,j\}. \]

So quantization follows from algebra + positivity + finite-dimensional irreducibility.

Integer vs. half-integer: Orbital angular momentum (\(\hat{\boldsymbol{L}}=\hat{\boldsymbol{r}}\times\hat{\boldsymbol{p}}\)) gives only integer \(\ell\) from single-valued wavefunctions, while spin admits half-integer \(j\).

Matrix Representations.

For fixed \(j\), the operators act on a \((2j+1)\)-dimensional space.

Spin-1/2: Pauli Matrices

For \(j=1/2\):

\[\begin{split} \hat J_z=\frac{\hbar}{2} \begin{pmatrix}1&0\\0&-1\end{pmatrix}, \quad \hat J_x=\frac{\hbar}{2} \begin{pmatrix}0&1\\1&0\end{pmatrix}, \quad \hat J_y=\frac{\hbar}{2} \begin{pmatrix}0&-\mathrm{i}\\\mathrm{i}&0\end{pmatrix}. \end{split}\]

Spin-1: 3\(\times\)3 Matrices

For \(j=1\):

\[\begin{split} \hat J_z=\hbar \begin{pmatrix}1&0&0\\0&0&0\\0&0&-1\end{pmatrix}, \quad \hat J_x=\frac{\hbar}{\sqrt2} \begin{pmatrix}0&1&0\\1&0&1\\0&1&0\end{pmatrix}, \quad \hat J_y=\frac{\hbar}{\sqrt2} \begin{pmatrix}0&-\mathrm{i}&0\\\mathrm{i}&0&-\mathrm{i}\\0&\mathrm{i}&0\end{pmatrix}. \end{split}\]

Discussion: integer vs half-integer angular momentum

Why do orbital and spin angular momentum components differ in their allowed quantum numbers? Orbital angular momentum must have integer \(\ell\) to ensure single-valued wavefunctions in physical space, while spin can have half-integer \(j\). Yet both satisfy the exact same commutation algebra \([\hat{J}_i, \hat{J}_j] = \mathrm{i}\hbar\epsilon_{ijk}\hat{J}_k\). Does this mean the algebra alone cannot distinguish integer from half-integer representations? What is the fundamental reason—is it topology, boundary conditions, or something deeper about the nature of rotation in quantum mechanics?

Poll: Ladder operator eigenvalue shift

Angular momentum operators satisfy \([\hat{J}_x, \hat{J}_y] = \mathrm{i}\hbar\hat{J}_z\). The raising operator is \(\hat{J}_+ = \hat{J}_x + \mathrm{i}\hat{J}_y\). If \(\hat{J}_z\vert j, m\rangle = \hbar m\vert j, m\rangle\), what is \(\hat{J}_z(\hat{J}_+\vert j, m\rangle)\) proportional to?

(A) \(\hat{J}_+\vert j, m\rangle\) with eigenvalue \(\hbar m\) (unchanged).

(B) \(\hat{J}_+\vert j, m\rangle\) with eigenvalue \(\hbar(m+1)\) (raised by 1).

(C) Zero (raising operators are orthogonal to Z eigenstates).

(D) \((\hat{J}_+)^2\vert j, m\rangle\) (squared).

Homework#

1. Angular momentum algebra. Verify the commutation relations \([\hat{J}_x, \hat{J}_y] = \mathrm{i}\hbar \hat{J}_z\) (and cyclic) by explicit computation for the spin-1/2 representation \(\hat{J}_i = \frac{\hbar}{2}\hat{\sigma}^i\).

2. Commutation relations. Show that \([\hat{J}^2, \hat{J}_z] = 0\) by expanding \(\hat{J}^2 = \hat{J}_x^2 + \hat{J}_y^2 + \hat{J}_z^2\) and using the fundamental commutation relations. Why is this result essential for labeling states by \(j\) and \(m\) simultaneously?

3. Eigenvalue problem. Using the definitions \(\hat{J}_\pm = \hat{J}_x \pm \mathrm{i}\hat{J}_y\), derive \([\hat{J}_+, \hat{J}_-] = 2\hbar\hat{J}_z\) and \([\hat{J}_z, \hat{J}_\pm] = \pm\hbar\hat{J}_\pm\).

4. Ladder operator action. Use \(\hat{J}_+\vert j, m\rangle = \hbar\sqrt{(j-m)(j+m+1)}\vert j, m+1\rangle\) to explain why \(\hat{J}_+\vert 1, 1\rangle = 0\). Compute \(\hat{J}_-\vert 1, 0\rangle\) and verify using the spin-1 matrix representation.

5. Angular momentum spectrum. For a state \(\vert j, m\rangle\), compute \(\langle \hat{J}_x\rangle\) and \(\langle \hat{J}_y\rangle\) using \(\hat{J}_x = \frac{1}{2}(\hat{J}_+ + \hat{J}_-)\). Explain physically why both vanish.

6. Quantum rotation. Show that the identity \(\hat{J}_-\hat{J}_+ = \hat{J}^2 - \hat{J}_z^2 - \hbar\hat{J}_z\) holds by expanding the left side. Use it to derive \(\|\hat{J}_+\vert j, m\rangle\|^2 = \hbar^2(j-m)(j+m+1)\).

7. Clebsch-Gordan relation. Explain why orbital angular momentum only has integer \(\ell\) (hint: single-valuedness of \(Y_\ell^m(\theta, \varphi)\) on the sphere) while spin can be half-integer. How does this connect to the spin-statistics theorem?

8. Quantum bootstrap. Two operators \(\hat{\alpha}\) and \(\hat{\beta}\) satisfy the algebraic relations

where \([\hat{A}, \hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}\) is the commutator, \(\{\hat{A}, \hat{B}\} = \hat{A}\hat{B} + \hat{B}\hat{A}\) is the anticommutator, and \(\hat{I}\) is the identity operator.

(a) Show that \(\hat{\beta}\) is Hermitian.

(b) Express \(\hat{\alpha}^\dagger\hat{\alpha}\) and \(\hat{\alpha}\hat{\alpha}^\dagger\) separately as linear combinations of \(\hat{\beta}\) and \(\hat{I}\).

(c) Show that \((\hat{\alpha}^\dagger)^2 \hat{\alpha}^2 = \hat{\beta}^2 - \hat{I}\) by expressing the left-hand side in terms of \(\hat{\beta}\). Using the positivity conditions \(\hat{\alpha}^\dagger\hat{\alpha} \geq 0\), \(\hat{\alpha}\hat{\alpha}^\dagger \geq 0\), and \((\hat{\alpha}^\dagger)^2\hat{\alpha}^2 \geq 0\), determine the feasible eigenvalues of \(\hat{\beta}\).

(d) Identify the angular momentum operators that \(\hat{\alpha}\) and \(\hat{\beta}\) correspond to (with \(\hbar = 1\)). What spin representation does this algebra describe? Write the \(2\times 2\) matrix representations of \(\hat{\alpha}\) and \(\hat{\beta}\) in the eigenbasis of \(\hat{\beta}\).