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Quantization Postulates for a Particle

  1. Jan 23, 2014 #1
    Show that the operators x^2 p_x^2+p_x^2 x^2 and 〖 (xp_x+p_x x)〗^2/2 differ only by terms of order ℏ^2.






    The attempt at a solution is attached (Postulates.pdf)
     

    Attached Files:

  2. jcsd
  3. Jan 23, 2014 #2
    I don't know what you are trying to do in your solution ,you should explain it better. The first line equality is not correct,keep in mind that x,p momentum do not commute. I suggest expanding the second term first and see how it differs from the first.
     
  4. Jan 24, 2014 #3
    I first found the quantum mechanical operator corresponding to the classical quantities xP_x, and according to the information I found on a downloaded file ("Dry2ans.pdf"), can't remember the source, I found that:
    xP_x → xP_x + P_x x

    As per your suggestion, bp_psy, I don't know which second term you are talking about. Is it of the first expansion or which one?
     
  5. Jan 26, 2014 #4
    You initial post does not say that x,[itex]p_x[/itex] are classical observables but operators.Which one is it?
    The classical observable [itex]xp_x[/itex] is represented by hermitian operator [itex]\hat{x}\hat{p}_{x}+\hat{p}_{x} \hat{x}[/itex] as they say in that document but the operator [itex]\hat{x}\hat{p}_{x}[/itex] is very different from [itex]\hat{x}\hat{p}_{x}+\hat{p}_{x}\hat{x}[/itex]. Sometime people do not hat their operators so you shouldn't always assume that no hats mean classical observables.
    What I meant by the second term is [itex]\frac{(\hat{x}\hat{p}_{x}+\hat{p}_{x}\hat{x})^2}{2}[/itex].
     
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