Maximization of an Uncertainty Product

[Theage]

Homework Statement

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Sakurai problem 1.20: find the linear combination of spin-up and spin-down S_z eigenkets that maximizes the uncertainty product $\langle(\Delta S_x)^2\rangle\langle(\Delta S_y)^2\rangle.$

Homework Equations

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In general, we can write a normalized spin-space ket as $$\vert\alpha\rangle = \cos\phi\vert +\rangle+\sin\phi e^{i\theta}\vert -\rangle.$$ Various trig identities are probably relevant (sums and differences of similar complex exponentials and the like). Also, the spin-1/2 representation $S_i = \frac\hbar 2\sigma_i$ is certainly relevant.

The Attempt at a Solution

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Since the Pauli sigma matrices are involuntary and the ket is normalized we see immediately that $$\langle S_x^2\rangle=\langle S_y^2\rangle =\frac{\hbar^2}4.$$ Using elementary trig we can also compute $$\langle S_x\rangle^2 = \hbar^2\sin^2\phi\cos^2\phi\cos^2\theta,\qquad\langle S_y\rangle = \hbar^2\sin^2\phi\cos^2\phi\sin^2\theta.$$ The uncertainty product is then a function F of two variables that should be maximized, and simplification yields $$F(\phi,\theta) = \hbar^4(\frac 1 4-\sin^2\phi\cos^2\phi\cos^2\theta)(\frac 1 4-\sin^2\phi\cos^2\phi\sin^2\theta) = \frac{\hbar^4}{16}(1-\sin^2(2\phi)+\frac 1 4\sin^4(2\phi)\sin^2(2\theta).$$ But this has degenerate maxima at phi = nπ/2 for any theta, which is definitely not the expected answer for the problem. Is this due to an algebraic mistake on my part or a conceptual mishap?

 

[TSny]

I think your answer is correct. Can you describe the physical interpretation of your angles $\phi$ and $\theta$ (or better, $2\phi$ and $\theta$ )? If so, your answer shouldn't be too surprising.