(adsbygoogle = window.adsbygoogle || []).push({}); 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

why?

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!

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# Quantum Physics Problem

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