Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted