Quantum Mechanics · Chapter 15

Time-dependent perturbations

A Hamiltonian that changes in time makes a system jump between levels. The interaction picture, transition amplitudes in first order, the response to a constant and to a periodic push, Fermi’s golden rule for decay into a continuum, and the two extremes of a changing Hamiltonian: the sudden change that leaves the state behind and the slow one that carries it along, with the geometric phase it leaves.

V^_{I} (t) = e^{\frac{i H ^ _{0} t}{ℏ}} V^e^{- \frac{i H ^ _{0} t}{ℏ}}

P_{i \to f} = ∣ c_{f} (t) ∣^{2}

ω_{f i} = \frac{E _{f} - E _{i}}{ℏ}

c_{f} (t) \propto 0 \int t V_{f i} (t^{'}) e^{i ω_{f i} t^{'}} d t^{'}

Lessons

Build the foundation

04Fermi’s golden ruleOften the final levels are not one but a continuum — an electron knocked out of an atom can leave with any energy, a photon can go off in any direction. Then the resonance peak of the last lesson sweeps over many final levels at once. Its height grows as the square of the time and its width shrinks as one over the time, so the area under it, the probability summed over the levels, grows in proportion to the time: a constant rate. That rate is Fermi’s golden rule. A constant rate of leaving empties a level exponentially, and a level that empties in a time $\tau$ has an energy blurred by $\hbar/\tau$.Read lesson →

05The sudden approximationIf the Hamiltonian changes abruptly — in a time short against $\hbar$ over the energies involved — the state has no time to respond. Just after the change it is the same vector as just before, but it is no longer an eigenstate: spread over the eigenstates of the new Hamiltonian, it is found in each with the squared overlap. When tritium’s nucleus emits a fast electron and becomes helium-3, the atomic electron, still in hydrogen’s ground state, suddenly finds a doubly charged nucleus, and ends in the ion’s ground state only seven times in ten.Read lesson →

06The adiabatic theoremThe opposite extreme: change the Hamiltonian slowly. Then a system that starts in the $n$-th eigenstate stays in the $n$-th eigenstate of the Hamiltonian of each moment, following it continuously, as long as the level never meets another. A box whose wall is moved out slowly keeps its particle in the ground state, whose energy falls smoothly; an oscillator whose spring slowly stiffens keeps its quantum number, its energy growing with the frequency. The only record of the journey is a phase.Read lesson →

07The Berry phaseCarry a Hamiltonian slowly round a closed loop of its parameters, back to where it started. The state returns to the same eigenstate, with two phases: the dynamical one, which depends on how long the journey took, and a geometric one, which depends only on the loop. Berry showed in 1984 that the geometric phase cannot be removed by choosing the phases of the eigenstates differently, and that interference measures it. For a spin that follows a magnetic field whose direction goes round a loop, it is minus half the solid angle the loop encloses.Read lesson →

08Chapter practiceBring the chapter together: the interaction picture, first-order transition amplitudes, constant and periodic perturbations, the golden rule, and the two extremes of a changing Hamiltonian with the geometric phase the slow one leaves. No worked example sits above the answers.Read lesson →

Mapping the lesson.

Time-dependent perturbations

Build the foundation