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Uncertainty principle problem

  1. Nov 9, 2009 #1
    1. The problem statement, all variables and given/known data


    The heisenberg uncertainty principle can be derived by operator algebra , as follows. Consider a one-dimensional system, with position and momentum observables x and p. The goal is to find the minimum possible uncertainties in the predicted values of the position and momentum in any state [tex]|\varphi>[/tex] of the system. We need the following preliminaries.
    2. Relevant equations



    3. The attempt at a solution

    a) Suppose the self-adjoint observables q and r satisfy the commutation relation

    [r,q]=iq, where c is a constant(not an operator). Show c is real.

    should I take the self-adjoint of [r,q], i.e.[tex][r,q]^{\dagger}[/tex] ?

    b) Let the system have the normalized state vector [tex]|\varphi>[/tex] and define the ket vector

    [tex] |\phi>=(\alpha*r+iq)|\varphi>[/tex] where [tex] \alpha[/tex] is a real constant(again, a number , not an operator). used equations [tex]<\phi|\phi> >=0[/tex] and [r,q]=ic to show that

    [tex]\alpha^2<r^2>-\alpha*c+<q^2>>=0[/tex], where [tex]<r^2>=<\varphi|r^2|\varphi>[/tex] and [tex]q^2[/tex]

    Should I begin by finding [tex] <\phi|\phi>[/tex]?

    Since, [tex] |\phi>=(\alpha*r+iq)|\varphi>[/tex] would that mean [tex] <\phi|=<|\varphi(\alpha*r-iq)[/tex]

    c) By seeking the value of [tex]\alpha[/tex] that minimizes the left side of the equation [tex]\alpha^2<r^2>-\alpha*c+<q^2>>=0[/tex], show

    [tex] <r^2><q^2>=c^2/4[/tex]

    should I multiply the expectation value [tex]<r^2>[/tex] to the equation [tex]\alpha^2<r^2> -\alpha*c+<q^2>>=0[/tex]
     
  2. jcsd
  3. Nov 9, 2009 #2
    No one understood my question and/or solution?
     
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