A question regarding a Hamiltonian.

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Discussion Overview

The discussion revolves around the transformation of a Hamiltonian in classical mechanics, specifically the transformation from H = xp to H = p² - x². Participants explore the implications of this transformation in both classical and quantum mechanics contexts.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant references a paper that suggests a rotation transformation can change the Hamiltonian from H = xp to H = p² - x².
  • Another participant proposes specific transformations in phase space, defining new variables p' and x' and questions the form of H in these new variables.
  • A participant raises a concern regarding the commutation relation between x and p, noting that they do not commute and questioning the implications for the transformation.
  • It is mentioned that the Hamiltonian H = xp is classical and that in quantum mechanics, it is not Hermitian until symmetrized.
  • Participants agree that in quantum mechanics, the symmetrized form (xp + px)/2 is used, with one participant elaborating on the notation for clarity.

Areas of Agreement / Disagreement

There is some agreement on the need for symmetrization in quantum mechanics, but the discussion includes differing views on the implications of the transformations and the nature of the Hamiltonian in classical versus quantum contexts. The overall discussion remains unresolved regarding the specific transformation details and their consequences.

Contextual Notes

Participants express uncertainty about the exact nature of the transformation and its implications, particularly in relation to the non-commutativity of x and p. The discussion also highlights the dependence on definitions of the Hamiltonian in different contexts.

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In the phase space (x,p), consider the transformations/rotations

[tex]2p'=p+x[/tex]

[tex]2x'=p-x[/tex]

What is H(x(x',p'),p(x',p')) equal to ?
 
But aren't x and p not commutable? ([tex][x,p]=i\hbar[/tex]).

I mean [tex]p'^2-x'^2=1/4 (p^2+x^2+px+xp- p^2-x^2 +px+xp)=1/2 \{x,p\}[/tex]
 
When he writes H = xp, he means a classical Hamiltonian. In QM it's not Hermitian until you symmetrize it.

"... its quantum counterpart (obtained by symmetrization)..."
 
Ok, thanks.
So in QM we would take (xp+px)/2.
 
MathematicalPhysicist said:
Ok, thanks.
So in QM we would take (xp+px)/2.

Actually [itex]\frac{1}{2}\left(\bar{\displaystyle{\hat{x}\hat{p}+\hat{p}\hat{x}}}\right)[/itex] (the bar should extend on both terms in the bracket), but for practical purposes, the operator without the bar is enough.
 
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