Unitary Time Evolution and the Heisenberg Picture
Time evolution is a unitary operator generated by the Hamiltonian, which is what keeps total probability fixed. Putting the time dependence on the operators instead of the state gives identical predictions and often less algebra.
What a learner can do afterwards
- Writes the evolution operator for a time-independent Hamiltonian and applies it to a superposition
- Explains why unitarity is exactly what conserves total probability
- Moves a problem between the Schrödinger and Heisenberg pictures and checks the answers agree
1 · Read
For a Hamiltonian that does not change with time, evolution is one operator: U of t equals the exponential of minus i times H times t. Feed it the starting state and it returns the state at time t. Time travel for states is a single multiplication.
Start with a superposition of two energy states. Each part spins its phase at its own energy rate while every probability sits still. The total stays exactly 1 at all times: phases turn, populations wait.
Unitarity is exactly what conserves total probability. A unitary operator keeps every length at 1, and lengths are probabilities waiting to be read. Because U never stretches a state, nothing leaks out of the total.
The Heisenberg picture freezes states and lets operators carry the time dependence instead. Every prediction matches the usual picture, often with less algebra. Ask about an observable over time: evolve the operator. Ask about the state itself: keep states moving.
Unitary evolution preserves probability; Heisenberg moves the clock onto operators.
2 · Watch
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.