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Hamiltonian which is invariant under time reversal question.

  1. Nov 5, 2009 #1

    MathematicalPhysicist

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    Gold Member

    1. The problem statement, all variables and given/known data
    Assuming that the Hamiltonian is invariant under time reversal, prove that the wave function for a spinless nondegnerate system at any given instant of time can always be chosen to be real.



    2. Relevant equations
    [tex]\psi(x,t)=<x|e^{-iHt/\hbar}|\psi_0>[/tex]
    The Time-Reversal operator: [tex]T^2=-1 \\ T|x>=|x> [/tex]
    HT=TH.

    3. The attempt at a solution
    [tex]\psi(x,t)=<x|e^{-iHt/\hbar}|\psi_0>=<x|e^{-iHt/\hbar}TT^-1|\psi_0>=<x|Te^{iHt/\hbar}T^{-1}|\psi_0>=<x'|exp(-iHt/\hbar}|\psi
    _0'>=<x|Texp(-iHt/\hbar)T^-1|\psi_0>=(1)[/tex]
    [tex](1)=<x|exp(iHt/\hbar)|\psi_0>=\psi(x,t)*[/tex]
    where the last line is assured because |x'>=T|x>=|x>, |\psi_0'>=T|psi_0>.
    Is this fine as a proof or not?
     
  2. jcsd
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