Hamiltonians as Generators of Time Evolution

The Hamiltonian is the generator of time evolution: it determines the infinitesimal unitary transformation that moves a closed quantum state forward in time. For the one-parameter group structure, see One-Parameter Unitary Groups. For the classical phase-space version of the same generator idea, see Hamiltonian Mechanics Review.

Infinitesimal Time Translation

For a short interval $dt$, the time-dependent Schrödinger equation gives

$$\lvert\psi(t+dt)\rangle = \left(I-\frac{i}{\hbar}H(t)\,dt\right) \lvert\psi(t)\rangle +O(dt^2).$$

The coefficient of $dt$ is $-iH/\hbar$. This is the generator role of the Hamiltonian.

Finite Time Evolution

Infinitesimal evolution builds the time-evolution operator $U(t,t_0)$:

$$\lvert\psi(t)\rangle =U(t,t_0)\lvert\psi(t_0)\rangle.$$

For a time-independent Hamiltonian,

$$U(t,t_0)=e^{-iH(t-t_0)/\hbar}.$$

For a general time-dependent Hamiltonian, the finite operator is built from ordered products of infinitesimal evolutions.

Self-Adjointness and Unitarity

For closed systems, $H$ is represented by a self-adjoint operator. This condition is what makes the generated time evolution unitary:

$$U^\dagger(t,t_0)U(t,t_0)=I.$$

The rigorous theorem behind this statement is Stone’s theorem: strongly continuous one-parameter unitary groups are generated by self-adjoint operators. This page uses the physics version; rigorous pages should handle the domain hypotheses.

Relation to Energy

When the Hamiltonian is time independent and represents a time-translation symmetry, it is the energy observable. Energy eigenstates satisfy

$$H\lvert E\rangle=E\lvert E\rangle.$$

When the Hamiltonian depends explicitly on time, it still generates time evolution, but energy need not be conserved. In driven systems, the word “energy” must be used with care.

Hamiltonian Versus Hamiltonian Matrix

The same Hamiltonian can be represented in many ways:

• as a differential operator in position space;
• as a finite matrix in a truncated or finite-dimensional Hilbert space;
• as a diagonal operator in the energy basis;
• as a sum of creation and annihilation operators in Fock space;
• as a classical Hamiltonian plus quantization rule in semiclassical discussions.

Changing representation does not change the generator role.

Common Mistakes

• Saying “Hamiltonian” when only a particular matrix representation is meant.
• Assuming every Hermitian-looking formula is self-adjoint on the intended domain.
• Equating time-dependent Hamiltonians with conserved energy.
• Forgetting that effective non-Hermitian Hamiltonians describe reduced or conditional dynamics, not ordinary closed-system evolution.
• Treating the Hamiltonian only as an energy formula and not as the generator of motion.

Cross-Links

• Hamiltonian
• Hamiltonian Operator
• One-Parameter Unitary Groups
• Hamiltonian Mechanics Review
• Time-Dependent Schrödinger Equation
• Time-Evolution Operator
• Hermitian Operators

References

• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, revised and enlarged ed., Academic Press, 1980.

Exercises

1. Starting from $U(dt)=I-iHdt/\hbar+O(dt^2)$, explain why $H$ must be self-adjoint for norm preservation to first order.

Solution

To first order,

$$U^\dagger(dt)U(dt) = \left(I+\frac{iH^\dagger dt}{\hbar}\right) \left(I-\frac{iHdt}{\hbar}\right) +O(dt^2).$$

This equals

$$I+\frac{i}{\hbar}(H^\dagger-H)dt+O(dt^2).$$

For this to equal $I$ to first order for arbitrary states, one needs $H^\dagger=H$.