Bad Qubits — Learn Quantum by Doing

Multiple choice

In the vector model, an angular momentum state has length

|L| = \hbar\sqrt{\ell(\ell+1)}

and maximum z-projection

\hbar\ell

(the

m = \ell

state). Why can the vector never point exactly along the z-axis?

Because $L_z$ is not quantized and so cannot equal the full length of the vector
Because the maximum projection $\hbar\ell$ is actually larger than the length $\hbar\sqrt{\ell(\ell+1)}$
Because the z-axis is not a physically allowed direction for any angular momentum
Because $\hbar\ell$ is strictly less than $\hbar\sqrt{\ell(\ell+1)}$ ; perfect alignment would force $L_x = L_y = 0$ , violating the uncertainty from $[L_x, L_y] = i\hbar L_z$