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Quantum Physics Problem

  1. Sep 11, 2008 #1
    1. The problem statement, all variables and given/known data

    suppose V(x) is complex, obtain an expression for

    ∂/∂t P(x,t) and
    d/dt ∫-∞dxP(x,t)

    for absorption of particles the last quantity must be negative (since particles disappear, the probability of their being anywhere decreases). What does this tell us about the imaginary part of V(x)? (ch 2, problem 11 gasiorowicz)

    2. Relevant equations

    V(x) is the potential energy

    schrodinger equation

    ∂/∂t P(x,t)= (∂ψ*)/∂t ψ+ψ*∂ψ/∂t

    ∂/∂t ∫-∞dxP(x,t)=-∫-∞dx ∂/∂x j(x,t)= 0

    where j(x,t) is the probability current

    but these may only be valid if V(x) is real


    3. The attempt at a solution

    I see how one can calculate

    ∂/∂t P(x,t)= (∂ψ*)/∂t ψ+ψ*∂ψ/∂t

    by plugging in the general schrodinger equation and its complex conjugate but in this situation V(x) must be real

    Why does the potential energy V(x) have to be real though?

    How would you find ∂/∂t P(x,t) if V(x) were complex?

    I have not yet taken a complex analysis class so any recommendations of topics in complex analysis to look up would be appreciated

    Any help would be greatly appreciated!
  2. jcsd
  3. Sep 14, 2008 #2
    First, write down the Schrodinger equation:

    [tex]i\hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m} \nabla^2 \Psi + V\Psi[/tex]

    and then it's complex conjugate (the same equation, only every imaginary unit must be added a minus sign). Then multiply each of them by [tex]\Psi[/tex] and [tex]\Psi^*[/tex] correspondingly and subtract. You'll see that if potential V has an imaginary part, it doesn't cancel and thus

    [tex]\frac{\partial P}{\partial t} \neq 0[/tex]

    so the probability for finding the particle decreases in time.
  4. Sep 14, 2008 #3
    Thank you so much! I guess I just didn't see that making V(x) complex is as simple as turning it into (V(x) + some imaginary part)
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