The Heisenberg picture and time-dependent perturbation theory
The Schrödinger picture puts all of a quantum system's time dependence into the evolving state, with fixed operators; the Heisenberg picture is a physically equivalent bookkeeping choice that instead evolves the operators and leaves the state fixed. Time-dependent perturbation theory uses this same machinery to answer a different question — given a small time-dependent nudge to a system already sitting in an energy eigenstate, at what rate does it transition to another eigenstate — with Fermi's golden rule giving that rate to leading order.
Practise this conceptThe Schrödinger equation puts time evolution into the state: the wavefunction changes, while an operator like position or momentum stays the same fixed mathematical object at every instant. The Heisenberg picture repackages the exact same physics by doing the opposite — it freezes the state vector at some reference time and instead lets the operators evolve, governed by an equation built from the commutator with the Hamiltonian. Every measurable prediction, every expectation value, comes out identical either way; the split between "what evolves" and "what's fixed" is a choice of bookkeeping, not a fact about nature.
The Heisenberg picture earns its keep in a specific situation: when you want to track how an operator's own dynamics compare to its classical counterpart, since the Heisenberg equation of motion looks structurally like Hamilton's equations from classical mechanics, just with commutators standing in for Poisson brackets.
Time-dependent perturbation theory asks a related but different question, usually in the Schrödinger picture: a system sits in an eigenstate of some Hamiltonian, then a small time-dependent term gets switched on. The unperturbed eigenstates don't change shape — they remain a fixed basis throughout — but the system's state becomes a mixture of them, with amplitudes that shift over time according to the perturbation's matrix elements between states. Fermi's golden rule packages the leading-order result of this calculation into a transition rate, one of the most-used formulas in atomic, nuclear and particle physics for "how fast does this decay or transition happen."
Key ideas
Requires: The Schrödinger equation, Operators and observables, Expectation values and measurement
Common misconceptions
- Believing the Heisenberg picture describes different physics from the Schrödinger picture — they are mathematically equivalent bookkeeping schemes that give identical predictions for every measurable expectation value; only where the time dependence is written down differs.
- Believing the state vector still evolves in time in the Heisenberg picture, just described differently — by definition the Heisenberg-picture state vector is fixed once and for all at the reference time; it is the operators that carry all the time dependence.
- Believing time-dependent perturbation theory means the unperturbed energy eigenstates themselves gradually change shape — the eigenstates stay fixed as a basis throughout; what evolves is the probability amplitude of finding the system in each one, driven by the perturbation's matrix elements between them.
Resources
Part of these UT courses
This concept is mapped as covering part of the following course(s) in the University of Tartu curriculum.
- Quantum mechanicsLOFY.04.073