In 1 dimension what is the solution of the Schrödinger equation with potential
$$ V(x) = V_r + i V_i $$
Potentials are constant.
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The Hamiltonian $$H=T_{\text{kin}}+V_r+iV_i$$ will not be hermitian as $$(iV_i)^*=-iV_i.$$ Technically, you can make an ansatz $$\Psi(x,t)=A\int\text{d}k\ \hat\Psi(k)\ \text e^{-i(kx-\omega(k)t)}.$$ plug it into the differential equation and find $$\omega(k)=\frac{\hbar^2}{2m}+V_r+iV_i,$$ or $$\hbar k=\pm\sqrt{2m(E-iV_i)},$$ where $E$ is some real number/numbers. You also want a boundary condition. (As a vague remark, modelings of complex energies, which necessarily turn a phase like $\text e^{-i\omega}$ into somthing like the decending expression $\text e^{-\omega'}$, are associated with decay. But again, a particle that vanishes in time like that is probably not what you want to talk about.) |
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