Autumn · project 4 of 20

Resonances and quasi-bound states

Topic: Bound states and spectra

Goal: Study quasi-bound states and resonances in quantum mechanics, the relation between the width of a resonance and the lifetime, and the interpretation of resonances as poles of the \(S\)-matrix in the complex energy plane.

  1. Consider the one-dimensional potential \[V(x)= \begin{cases} -V_0, & |x|<a,\\ 0, & |x|>a. \end{cases}\]

  2. Find the reflection and transmission coefficients \(R(E)\), \(T(E)\).

  3. Show that there are resonance peaks with \(T(E)\approx1\) at \(E>0\).

  4. Continue the energy into the complex plane, \(E\to E-i\Gamma/2\), and find the conditions for the poles of the \(S\)-matrix.

  5. Relate the width of the resonance \(\Gamma\) to the lifetime of the state, \[\tau \sim \frac{1}{\Gamma}.\]

  6. Plot numerically:

    • \(T(E)\);

    • the trajectories of the poles as \(V_0\) and \(a\) vary.

  7. Discuss why resonances are more naturally interpreted in the language of scattering than in the language of “bound states”.

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