Question 705028

The quantum mechanical system is given to be in the angular momentum state $|\psi\rangle = |l=4, m=4\rangle$. We need to find the uncertainty in the x-component of angular momentum ($L_x$), which is given by the standard formula: $\Delta L_x = \sqrt{\langle L_x^2 \rangle - \langle L_x \rangle^2}$

Step 1: Find the expectation value $\langle L_x \rangle$ For any eigenstate of $L_z$ (like the given state $|l, m\rangle$), the expectation values of the transverse components $L_x$ and $L_y$ are always zero. $\langle L_x \rangle = 0$

Step 2: Find the expectation value $\langle L_x^2 \rangle$ We know the relation for the total angular momentum squared operator: $L^2 = L_x^2 + L_y^2 + L_z^2$

Taking the expectation value of this operator for the state $|l, m\rangle$: $\langle L^2 \rangle = \langle L_x^2 \rangle + \langle L_y^2 \rangle + \langle L_z^2 \rangle$

Due to the symmetry of the $L_x$ and $L_y$ operators in an $L_z$ eigenstate, their expectation values squared are equal: $\langle L_x^2 \rangle = \langle L_y^2 \rangle$

Substituting this into the expectation value equation: $\langle L^2 \rangle = 2\langle L_x^2 \rangle + \langle L_z^2 \rangle$ $2\langle L_x^2 \rangle = \langle L^2 \rangle - \langle L_z^2 \rangle$ $\langle L_x^2 \rangle = \frac{\langle L^2 \rangle - \langle L_z^2 \rangle}{2}$

We know the standard eigenvalue results for these operators: $\langle L^2 \rangle = l(l+1)\hbar^2$ $\langle L_z^2 \rangle = m^2\hbar^2$

For the given state $l=4$ and $m=4$: $\langle L^2 \rangle = 4(4+1)\hbar^2 = 20\hbar^2$ $\langle L_z^2 \rangle = (4)^2\hbar^2 = 16\hbar^2$

Now, substitute these values to calculate $\langle L_x^2 \rangle$: $\langle L_x^2 \rangle = \frac{20\hbar^2 - 16\hbar^2}{2} = \frac{4\hbar^2}{2} = 2\hbar^2$

Step 3: Calculate the Final Uncertainty $\Delta L_x$ Substitute the values back into the uncertainty formula: $\Delta L_x = \sqrt{\langle L_x^2 \rangle - \langle L_x \rangle^2}$ $\Delta L_x = \sqrt{2\hbar^2 - 0}$ $\Delta L_x = \sqrt{2}\hbar$

Final Answer: The uncertainty in $L_x$ is $\hbar\sqrt{2}$. This corresponds to Option (A).