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Probability for a non-hermitian hamiltonian

  1. Oct 25, 2015 #1
    1. The problem statement, all variables and given/known data
    Given a non-hermitian hamiltonian with V = (Re)V -i(Im)V. By deriving the conservation of probability, it can be shown that the total probability of finding a system/particle decreases exponentially as e(-2*ImV*t)/ħ

    2. Relevant equations
    Schrodinger Eqn, conservation of probability (∂P/∂t = -∇⋅j), P is the probability and j is the probability current density, Re V is the real part of V while Im V is the imaginary part of V

    3. The attempt at a solution
    It is known that ∫∫∫ P d3r = constant, that is ,the total probability for finding the particle should be constant throughout the universe.

    iħ ∂ψ/∂t = -ħ2/2m ∇2ψ + Vψ = -ħ2/2m ∇2ψ + (Re)Vψ -i(Im)Vψ
    By taking the complex conjugate, we have
    -iħ ∂ψ*/∂t = -ħ2/2m ∇2ψ* +V*ψ* = -ħ2/2m ∇2ψ* + (Re)Vψ* +i(Im)Vψ*
    By multiplying by ψ* the first equation and by ψ the second equation then subtracting the second from the first, we have
    iħ ∂(ψψ*)/∂t = -ħ2/2m (ψ*∇2ψ-ψ∇2ψ*) - 2i (Im)V ψψ* = -∇⋅j - 2i (Im)V ψψ*

    I don't know where to go already from here. Any suggestions or hints?
     
  2. jcsd
  3. Oct 25, 2015 #2

    blue_leaf77

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    Just use the definition of total probability in quantum mechanics: ##\langle \psi(t) | \psi(t) \rangle##. Now ##|\psi(t)\rangle = e^{-iVt/\hbar} |\psi(0)\rangle##. What will you get then when calculating ##\langle \psi(t) | \psi(t) \rangle##?
     
  4. Oct 25, 2015 #3
    I will get ⟨ψ(t)|ψ(t)⟩ = ⟨ψ(0)|ψ(0)⟩ but what does this tell me? Also, I need to use the conservation of probability equation to derive the question.
     
  5. Oct 25, 2015 #4

    blue_leaf77

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    No, you won't because in this problem ##V## is not Hermitian. Try plugging in ##V = \textrm{Re}[V] - i \textrm{Im}[V]##.
     
  6. Oct 25, 2015 #5
    Sorry, I forgot that it is not hermitian, so we will get e2i(Im)Vt
     
  7. Oct 25, 2015 #6

    blue_leaf77

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    It will be ##\langle \psi(0) |e^{-i\textrm{Im}[V]t/\hbar}| \psi(0) \rangle##, at this point we should know how ##\textrm{Im}[V]## acts on ##| \psi(0) \rangle##, i.e. whether it is a function of space at all. But since the question asks you to prove that "decreases exponentially as e(-2*ImV*t)/ħ", I think it is implied that ##\textrm{Im}[V]## is a constant and thus the sandwiched exponential operator may be taken out.
     
    Last edited: Oct 25, 2015
  8. Oct 25, 2015 #7
    Oh! I did something wrong when taking the complex conjugate of V, but anyways, you really cleared everything. Thanks!
     
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