4.6: Radioactive decay and imaginary potentials

If the number of particles in a given state is reduced in time, then the total intensity of that state is reduced. Consider a particle moving in a region of imaginary potential $V (r) = −iV_0$. The TDSE is:

$$i\hbar \frac{ \partial}{\partial t}|\Phi , t \rangle = [H_0 − iV_0]|\Phi , t \rangle \nonumber$$

Assume that the time independent part of the state is an combination of eigenstates of the real part of the Hamiltonian:

$$|\Phi , t \rangle = \sum_n c_n(t) \text{ exp}(−iE_nt/\hbar )|\Phi_n \rangle; \quad \text{ where } \quad H_0|\Phi_n \rangle = E_n|\Phi_n \rangle \nonumber$$

Following the same analysis as for TDSE, premultiplying by $\langle m|$, and for constant $V_0$, $V_{mn} = \delta_{mn}V_0$ we obtain:

$$i\hbar \dot{c}_m = −iV_0c_m \quad \Rightarrow \quad |c_m(t)|^2 = |c_m(0)|^2 e^{−2V_0t/\hbar} \nonumber$$

Thus the probability amplitude of the state decreases in time. An imaginary potential can be used to represent destruction of particles, either by absorption (in a scattering process, perhaps) or by radioactive decay. Obviously the ket is not a full description of the system, since that should include information about the decay products. The lifetime of the state is $\tau = \hbar /2V_0$.

Notice that $−iV_0$ is not a Hermitian operator, and so it is not possible to perform a single measurement of half life.