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Maximization of an Uncertainty Product

  1. Apr 7, 2015 #1
    1. The problem statement, all variables and given/known data

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

    2. Relevant equations

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

    3. The attempt at a solution

    Since the Pauli sigma matrices are involuntary and the ket is normalized we see immediately that [tex]\langle S_x^2\rangle=\langle S_y^2\rangle =\frac{\hbar^2}4.[/tex] Using elementary trig we can also compute [tex]\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.[/tex] The uncertainty product is then a function F of two variables that should be maximized, and simplification yields [tex]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).[/tex] 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?
     
  2. jcsd
  3. Apr 7, 2015 #2

    TSny

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    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.
     
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