I have seen the proof that for fermions a rotation of $2 \pi$ does not return a spin angular momentum eigenstate to its original form, but instead multiplies the wavefunction by $-1$.
Here is an abbreviated derivation:
\begin{equation}
U(\vec \alpha) = \exp (-i \vec{\alpha} \cdot \vec{J}) = \sum_{n=0}^{\infty} \frac{(-i)^n \left[\vec{\alpha} \cdot \frac{1}{2} \vec{\sigma}^n \right]}{n!} = I \cos{\left(\frac{\alpha}{2} \right)} – i \hat{\alpha} \cdot \vec{\sigma} \sin{\left(\frac{\alpha}{2} \right)}
\end{equation}
where I have used the fact that $\vec{J} = \vec{S} = \frac{1}{2} \vec \sigma$ for a spin-half particle at rest. For $|\alpha| = 2 \pi$, we have:
\begin{equation}
U(\vec \alpha) = -I
\end{equation}
I understand this is a special state for fermionic spin, and not the case for a state with zero spin.
Is there an equivalent proof that a rotation of $2 \pi$ will return a state with just orbital angular momentum to its original state?
Best Answer
The identity you used, $$ \exp(i\theta \, \hat s)=\cos(\theta)+i\sin(\theta)\,\hat s, \tag{$\ast$} $$ is crucially dependent on the operator $\hat s$ being idempotent, and particularly on the fact that $\hat{s}^2=\mathbb1$. This is generally not the case for angular momenta other than spin-1/2. In general, the total angular momentum is a scalar within the representation (i.e. $\hat{\mathbf J}^2=\hbar^2j(j+1)\times\mathbb 1$) but outside of spin-1/2 it is no longer the case that each component is idempotent, so if $\hat{\mathbf n}$ is a unit vector, $\hat{\mathbf n}\cdot\hat{\mathbf J}$ no longer squares to a multiple of $\mathbb 1$. In that case, the rotation operator is still given by $$ \hat U(\vec \alpha)=\exp(-i\vec \alpha\cdot\hat{\mathbf J}/\hbar) $$ but no further simplifications are possible.
That said, if you have a specific $j$ you want to investigate, there is probably an analogous formula to $(\ast)$, because the $(2j+1)$th and higher powers must be linear combinations of the first $2j$ powers and the identity, so you can again fold the exponential series into $2j+1$ terms, $$ \exp(-i\theta \hat{\mathbf n}\cdot\hat{\mathbf J})=\sum_{k=0}^{2j}f_k(\theta) (\hat{\mathbf n}\cdot\hat{\mathbf J})^k, $$ where the $f_k$ are given by appropriate series. Depending on what comes out, this may or may not be useful, but again it is a lot of work for each specific $j$ you're interested in.
To be a bit more explicit, let me show how this works for the simplest nontrivial orbital angular momentum, $j=1$. If you work in the $z$ direction you get, for the different powers of $\hat J_z$, $$ \mathbb 1=\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix} ,\quad \hat J_z=\hbar\begin{pmatrix}1&0&0\\0&0&0\\0&0&-1\end{pmatrix} ,\quad \hat J_z^2=\hbar^2\begin{pmatrix}1&0&0\\0&0&0\\0&0&1\end{pmatrix} ,\quad $$ and it's only then that you get stuck in a loop: $\hat J_z^{2+n}=\hbar^{n+1} \hat J_z$ for odd $n\geq1$ and $\hat J_z^{2+n}=\hbar^{n} \hat J_z^2$ for even $n\geq0$. This means that the exponential folds again, but not as neatly: $$ \exp(-i\theta \hat J_z/\hbar) =\sum_{n=0}^\infty\frac{(-i\theta \hat J_z/\hbar)^n}{n!} = \mathbb 1 -\frac{i\sin(\theta)}{\hbar} \hat J_z +\frac{\cos(\theta)-1}{\hbar^2}\hat J_z^2. $$ This is similar, but not quite the same, as the original $(\ast)$. It does yield the desired invariance after $2\pi$ rotations, but it required a lot of work for a single $j$.
This does leave you with the need to prove that for general, integral $j$ all states will return to themselves after a $2\pi$ rotation. The proof is somewhat different, though, and it relies on the fact that the eigenvalues of $\mathbf J$ are all integers. In particular, for any given unit vector $\hat{\mathbf n}$ there will be a basis $\{|m⟩=|j,m,\hat{\mathbf n}⟩\}_m$ of eigenstates of $\hat{\mathbf n}\cdot\hat{\mathbf J}$ with integer eigenvalues: $$ \hat{\mathbf n}\cdot\hat{\mathbf J}|m⟩=\hbar m|m⟩,\quad m=-j,\ldots,j. $$ This lets you calculate the action of $\hat U(\vec \alpha)$ on each eigenstate: $$ \hat U(\vec \alpha)|m⟩ =\exp(-i\vec \alpha\cdot\hat{\mathbf J}/\hbar)|m⟩ =\exp(-i\alpha\hat{\mathbf n}\cdot\hat{\mathbf J}/\hbar)|m⟩ =\exp(-im\alpha)|m⟩. $$ For $\alpha=2\pi$ and integer $m$, this is exactly $|m⟩$, with no added phase. Since any arbitrary state $|\psi⟩$ can be expressed as a linear combination of the $|m⟩$ and those are unchanged by $\hat U(\vec \alpha)$, it follows that $|\psi⟩$ itself is also unchanged by $\hat U(\vec \alpha)$.
Finally, you should note that this proof also works to show that half-integral $j$s produce $\pi$ phases upon $2\pi$ rotations, since then every $m$ is half-integral, and $e^{-i2\pi m}\equiv -1$ regardless of $m$. Your original proof, though, also fails for $j\geq 3/2$, since the idempotency condition is also not fulfilled.