- #1
Felicity
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Homework Statement
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)
Homework 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
why?
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!