Advanced Quantum Mechanics · Tutorial 2c
The Josephson Hamiltonian H = p²/2m − λ cos q, solved exactly and compared with the semiclassical picture. The first tab places Bohr–Sommerfeld orbits next to the exact eigenstates; the second follows a coherent state through collapse and revival, long after the classical description has failed.
Orbits. Each coloured curve is a classical orbit picked by Bohr–Sommerfeld: area (n + ½)h inside the separatrix, and nh under a rotating orbit, which has no turning points. Rotating orbits come in pairs, one for each direction, at the same energy.
Spacing. Dots are exact gaps Ej+1 − Ej. Below the separatrix they follow ħωcl(E) and dip where the period diverges. Above it they split into near-zero gaps inside a pair and gaps of ħωcl between pairs.
Eigenstate. The exact density oscillates around the WKB density 2/(Tcl v(q)), the fraction of time the classical particle spends at q. Its Husimi function sits on the quantized orbit.
Collapse. The packet smears around its orbit because neighbouring levels are not equally spaced. The clock hands fan out at the same time.
Revival. At Trev/2 every phase is a multiple of π: hands near n̄ alternate up and down, and the packet re-forms half an orbit away. At Trev/4 the hands form two groups: a superposition of two packets. Hands far from n̄ drift off because of higher-order terms in En.
Markers. The dashed lines use the quartic spectrum of B1(b). The actual revivals come slightly earlier, because the levels near n̄ are more curved than the quartic estimate.