Minimizing angular momentum uncertainties

[mrbetadine]

Homework Statement

Consider a physical system of fixed angular momentum $l$. The state of the system is in the subspace spanned by $2l+1$ eigenvectors $|l,m\rangle$ of $L_z$ ($-l\leq m\leq +l$). Find the state $|\psi_0\rangle$ of the system for which $(\Delta L_x)^2+(\Delta L_y)^2+(\Delta L_z)^2$ is minimal.

Relevant Equations

n/a

\begin{align*}
&(\Delta L_x)^2+(\Delta L_y)^2+(\Delta L_z)^2\\
={}&\langle L_x^2\rangle-\langle L_x \rangle^2+\langle L_y^2\rangle-\langle L_y \rangle^2+\langle L_z^2\rangle-\langle L_z \rangle^2\\
={}&\langle L_x^2+L_y^2+L_z^2 \rangle-(\langle L_x \rangle^2+\langle L_y \rangle^2+\langle L_z \rangle^2)\\
={}&l(l+1)\hbar^2-(\langle L_x \rangle^2+\langle L_y \rangle^2+\langle L_z \rangle^2).
\end{align*}
To minimize $(\Delta L_x)^2+(\Delta L_y)^2+(\Delta L_z)^2$ is equivalent to maximizing
$$ \langle L_x \rangle^2+\langle L_y \rangle^2+\langle L_z \rangle^2$$

How should I proceed?
I have also obtained an alternative expression
$$\langle L_x \rangle^2+\langle L_y \rangle^2+\langle L_z \rangle^2=\langle L_+\rangle\langle L_-\rangle+\langle L_z\rangle^2 $$
where
$$ L_\pm=L_x\pm i L_y$$
but it does not help much. It would be easy if we restrict ourselves to finding states of the form $|l,m\rangle$.

*The hint I got indicates that $|\psi_0\rangle$ is the solution to the equations
\begin{align*}
(L_x+iL_y)|\psi_0\rangle&=0\\
L_z|\psi_0\rangle&=l\hbar|\psi_0\rangle.
\end{align*}

 

[anuttarasammyak]

By the hint
$$<L_x>=<L_y>=0$$
$$<L_z>=l\hbar$$

 

[JimWhoKnew]

There may be a simple and elegant solution to the problem, but I can't see it currently. So I'll suggest a brute force approach.

Note that if $~|\psi_0\rangle~$ minimizes the uncertainty, then any state derived from it by a rotation is just as good.

Any normalized state in the $l$ subspace can be spanned as$$|\psi\rangle=\sum^l_{m=-l}\left(\alpha_m+i\beta_m\right)|l,m\rangle\quad,$$where $~\{\alpha_m,\beta_m\}~$ are $~4l+2~$ real coefficients.
As you concluded in post #1, minimizing the uncertainty is the same as maximizing$$f:=\langle L_+\rangle\langle L_-\rangle+\langle L_z\rangle^2 $$(which can now be expressed explicitly as a real function of the real $~\{\alpha_m,\beta_m\}~$), subjected to the normalization constraint$$\chi:=\langle\psi|\psi\rangle-1=\sum\left(\alpha_m^2+\beta_m^2\right)-1=0\quad.$$The method of Lagrange multipliers is usually applicable in cases like this.