4.6: Radioactive decay and imaginary potentials

Last updated
Save as PDF

Page ID: 28765

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

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.