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Homework Help: Uncertainty relation

  1. Jan 20, 2009 #1
    1. The problem statement, all variables and given/known data
    Could someone please have a look at this?
    I am to show that from the inequation
    [tex]\langle\left \psi | \hbar^2D^2 | \psi\right\rangle + mk\langle \left\psi | x^2 | \psi\right\rangle\geq\hbar\sqrt{mk}[/tex]
    you can get the Heisenberg uncertainty relation
    [tex]
    \langle\psi | \hbar^2D^2 | \psi\rangle\langle \left\psi | x^2 | \psi\right\rangle\geq\frac{1}{4}\hbar^2[/tex]
    for all normalized functions [tex]\psi \in S(\mathbb{R})[/tex]


    2. Relevant equations
    I know that
    [tex]H_0=\frac{\hbar^2}{2m}D^2+\frac{1}{2}kx^2[/tex]
    and that
    [tex]H\psi=\hbar\omega\sum_{n=0}^{\infty}(n+1/2)\langle\Omega_n | \psi\rangle\Omega_n[/tex]
    and that [tex]H_0\psi=H\psi[/tex] for [tex]\omega=\sqrt{k/m}[/tex]


    3. The attempt at a solution
    I tried to square on both sides:
    [tex]\langle\psi | \hbar^2D^2 | \psi\rangle^2+m^2k^2\langle \left\psi | x^2 | \psi\right\rangle^2+2mk\langle \left\psi | x^2 | \psi\right\rangle\langle\psi | \hbar^2D^2 | \psi\rangle \geq\hbar^2 mk[/tex]

    But that doesn't seem to work right
     
  2. jcsd
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