In classical mechanics, the orbital angular momentum of a particle is defined as $\textbf{L}=\textbf{r}\times\textbf{p}$. This is zero in the rest frame of the particle where $\textbf{p}=0$.

Quantum mechanically, $\textbf{p}$ is an operator. So putting $\hat{\textbf{p}}=0$ in $\hat{\textbf{L}}=\hat{\textbf{r}} \times\hat{\textbf{p}}$ and claim that the orbital angular momentum of a quantum particle is zero does not make sense. One must look at the value of $\hat{\textbf{L}}^2$ on the "wavefunction in the rest frame" of the particle.

How does one find the wavefunction of a particle in its rest frame?

In classical mechanics, the orbital angular momentum of a particle is defined as $\textbf{L}=\textbf{r}\times\textbf{p}$. This is zero in the rest frame of the particle where $\textbf{p}=0$.

Quantum mechanically, $\textbf{p}$ is an operator. So putting $\hat{\textbf{p}}=0$ in $\hat{\textbf{L}}=\hat{\textbf{r}} \times\hat{\textbf{p}}$ and claiming that the orbital angular momentum of a quantum particle is zero in its rest frame does not make sense. One must look at the value of $\hat{\textbf{L}}^2$ on the "wavefunction in the rest frame" of the particle.

How does one find the wavefunction of a particle in its rest frame?

In classical mechanics, the orbital angular momentum of a particle is defined as $\textbf{L}=\textbf{r}\times\textbf{p}$. This is zero in the rest frame of the particle where $\textbf{p}=0$. Quantum mechanically, $\textbf{p}$ is an operator. So putting $\hat{\textbf{p}}=0$ in $\hat{\textbf{L}}=\hat{\textbf{r}}\times\hat{\textbf{p}}$ and claim that the orbital angular momentum of a quantum particle is zero does not make sense. One must look at the value of $\hat{\textbf{L}}^2$ on the wavefunction in the rest frame of the particle. How does one do that in quantum mechanics? How should I start?

In classical mechanics, the orbital angular momentum of a particle is defined as $\textbf{L}=\textbf{r}\times\textbf{p}$. This is zero in the rest frame of the particle where $\textbf{p}=0$. 

Quantum mechanically, $\textbf{p}$ is an operator. So putting $\hat{\textbf{p}}=0$ in $\hat{\textbf{L}}=\hat{\textbf{r}} \times\hat{\textbf{p}}$ and claim that the orbital angular momentum of a quantum particle is zero does not make sense. One must look at the value of $\hat{\textbf{L}}^2$ on the "wavefunction in the rest frame" of the particle.

How does one find the wavefunction of a particle in its rest frame?