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Hamiltonian of oscillators quantized proof

  1. Sep 30, 2012 #1
    A matrix [itex]N[/itex] has all entries except diagnoal ones zero, and first row, first column also zero, other diagonal entries depending on its column, row place; for example, second row, second column has one as its entry, third row, third column has two as its entry and so on.

    It is known that [itex]H = \hbar\omega (N + \frac{1}{2}) [/itex].

    The question is, my textbook says that if there are a continuous range of possible values, then for some states, [itex]\langle (N - \langle N \rangle )^2 \rangle \leq [\frac{1}{2} (n - \langle N \rangle)]^2[/itex] , where [itex]n = 0,1,2,3,...,[/itex] then progresses to say that since [itex]\langle (N - \langle N \rangle )^2 \rangle \geq (n - \langle N \rangle)^2 (\langle I_0 \rangle + \langle I_1\rangle + ...)[/itex] then [itex]\langle (N - \langle N \rangle )^2 \rangle \geq (n - \langle N \rangle)^2[/itex].

    Then it says that this proves that there cannot be a continuous range of possible values, as [itex](0-\langle N \rangle)^2[/itex], [itex](1 - \langle N \rangle)^2[/itex] and so on cannot be smaller than [itex](n - \langle N \rangle)^2[/itex].

    I am not sure how one gets [itex]\langle (N - \langle N \rangle )^2 \rangle \geq (n - \langle N \rangle)^2 (\langle I_0 \rangle + \langle I_1\rangle + ...)[/itex]. Can anyone show me the proof?

    Also, how does one get the part - [itex](0-\langle N \rangle)^2[/itex], [itex](1 - \langle N \rangle)^2[/itex] and so on cannot be smaller than [itex](n - \langle N \rangle)^2[/itex]?
     
  2. jcsd
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