Time Evolution

The Hamiltonian generates the evolution of a closed quantum system. This chapter introduces the Schrödinger equation, unitary propagation, stationary states, and conservation laws. You will learn which quantities evolve, which stay fixed, and how to distinguish a state with definite energy from a superposition of energies.

Relative phases produce dynamics

For a time-independent self-adjoint Hamiltonian, evolution is

$$\lvert\psi(t)\rangle=e^{-iHt/\hbar}\lvert\psi(0)\rangle.$$

This unitary formula applies to every Hilbert-space state. Its differential form, $i\hbar\partial_t\lvert\psi\rangle=H\lvert\psi\rangle$, holds strongly for initial vectors in the domain of $H$.

Suppose $H\lvert0\rangle=E_0\lvert0\rangle$ and $H\lvert1\rangle=E_1\lvert1\rangle$. The initial state $\lvert+\rangle=(\lvert0\rangle+\lvert1\rangle)/\sqrt2$ becomes

$$\lvert\psi(t)\rangle= \frac{e^{-iE_0t/\hbar}\lvert0\rangle+e^{-iE_1t/\hbar}\lvert1\rangle}{\sqrt2}.$$

Its energy probabilities stay $1/2,1/2$, but its probability of returning to $\lvert+\rangle$ is

$$p_+(t)=\cos^2\!\left(\frac{(E_1-E_0)t}{2\hbar}\right).$$

A single energy eigenstate acquires only a global phase and gives time-independent probabilities for fixed measurements. In a superposition, changing relative phases can produce observable oscillations.

Reading sequence

Question	Article	What you will learn
What operator generates evolution?	Hamiltonians	energy observable and generator
What differential equation evolves a state?	Schrödinger Equation	initial-value evolution
Why are closed-system probabilities preserved?	Unitary Time Evolution	preserved norms and inner products
How is evolution mapped between two times?	Time-Evolution Operator	evolution between two times
Which states have time-independent predictions?	Stationary States	ray and density-operator criteria
What does definite energy mean?	Energy Eigenstates	spectral, measurement, and phase structure
What changes under external driving?	Time-Dependent Hamiltonians	external driving and time ordering
When is a quantity conserved?	Conservation Laws	commutator and explicit-time criteria
How do pictures redistribute time dependence?	Pictures of Motion Overview	consistent transformations of states and observables

Extend the calculation

Read Hamiltonians, the Schrödinger equation, and unitary evolution before stationary states and energy eigenstates. The dynamics volume develops the solution methods in detail.

An explicitly driven Hamiltonian needs a time-ordered propagator in general. A reduced subsystem may evolve nonunitarily even when the combined system evolves unitarily. These are different extensions of the simple closed, time-independent example.

For conservation, check both commutation with the evolution and any explicit time dependence of the observable. With unbounded observables, a formal commutator alone does not replace the necessary domain and spectral conditions.

Try it. What changes in $p_+(t)$ if the same constant $C$ is added to both energies?

Solution

Nothing: the energy difference stays fixed. The whole ket acquires the additional phase $e^{-iCt/\hbar}$, which cancels from probabilities.

References

• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford University Press, 1958.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. I: Functional Analysis, rev. ed., Academic Press, 1980.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955.