Show that ⟨j, j|Jx| j, j⟩=⟨j, j|Jy| j, j⟩=0 and that ⟨j, j|(Jx^2+. .Jy^2)|j, j⟩=j. Discuss the i

step 1 Are the possible values of the quantum number J associated with the vector J for the 3D state of

Show that ⟨j, j|Jx| j, j⟩=⟨j, j|Jy| j, j⟩=0 and that ⟨j,...

Show that $\left\langle j, j\left|J_{x}\right| j, j\right\rangle=\left\langle j, j\left|J_{y}\right| j, j\right\rangle=0$ and that $\langle j, j|\left(J_{x}^{2}+\right.$ $\left.J_{y}^{2}\right)|j, j\rangle=j$. Discuss the implications of these results for the uncertainty in the orientation of the classical angular momentum vector $\mathbf{J}$ for both small and large values of $j$.

Key Concepts

Semiclassical Limit

In the semiclassical limit, where the quantum numbers such as j become very large, the relative fluctuations in the components orthogonal to the main axis diminish compared to the magnitude of the angular momentum along the z-axis. This causes the classical picture of a well-defined angular momentum vector to emerge, as the fractional uncertainties decrease, even though the absolute quantum uncertainties remain nonzero.

Quantum Uncertainty

The results highlight a manifestation of the uncertainty principle. Despite having a precise value for the z-component of the angular momentum, the orthogonal components (Jx and Jy) exhibit inherent uncertainties, as evidenced by their zero mean but nonzero variance. This uncertainty is an intrinsic aspect of quantum systems, where the precise measurement of one observable induces indeterminacy in its complementary observables.

Angular Momentum Eigenstates

Angular momentum eigenstates, denoted by |j, m>, are the fundamental states in quantum mechanics that simultaneously diagonalize the total angular momentum operator J² and one of its components, typically chosen to be Jz. The state |j, j> is the highest weight state where the angular momentum is maximally aligned along the chosen quantization axis, which results in precise knowledge of the z-component while leaving the orthogonal components inherently uncertain.

Expectation Values

Expectation values in quantum mechanics represent the average measurement outcomes of observables in a given state. For the state |j, j>, the expectation values of the operators Jx and Jy are zero, reflecting the fact that the state is symmetrically aligned along the z-axis. However, because the squares of these components yield nonzero expectation values, they indicate the presence of quantum fluctuations even in a state with a well-defined total angular momentum.

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