Advanced Quantum Mechanics · Tutorial 3
Two scattering problems solved exactly. The first tab sends wave packets through a double barrier, built as superpositions of the stationary scattering states of the lecture. The second tab scatters a plane wave off a Yukawa potential in three dimensions and compares the exact partial-wave result with the first Born approximation.
Packet. The packet is the superposition ∫dk c(k) ψk(x) e−iEt/ħ of exact scattering states, Eq. (3.7) of the notes. Its energy distribution |c|² is the shaded curve in the spectrum: a long packet is sharp in energy, a short one is broad.
Resonance. On a resonance the wave between the barriers builds up to |ψ|² ≈ 4/τ1, and the reflected wave on the left vanishes: no standing-wave ripple in the dashed curve. The particle then stays between the barriers for a time of order ħ/Γ before leaking out on both sides.
Short packet. A packet shorter than the well bounces back and forth, and a small pulse leaks out at every hit. Its energy spread covers several resonances, so the transmitted fraction is close to the energy average of τ, not to τ(E0).
Born. In first order, f depends only on the momentum transfer ħq = 2k sin(θ/2), and is the same for g and −g. The exact result is not: compare the strong repulsive and attractive presets.
Forward peak. For k ≪ ħμ only the s-wave is scattered and the cross section is flat in θ. For k ≫ ħμ it is confined to θ ≲ ħμ/k, and at larger angles it follows the Rutherford curve.
Validity. Born works when |g| ≪ 1 at all energies, and also at high energy when |g|ħμ/k ≪ 1. For an attractive potential with g < −0.84 there is a bound state, and the low-energy cross section is far from Born.