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 differentiable finite-dimensional Hamiltonian, the 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:

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

For a time-independent Hamiltonian, Stone’s theorem relates strongly continuous one-parameter unitary groups to self-adjoint generators. On an infinite-dimensional space, the differential equation holds for states in $D(H)$; unitarity extends to every Hilbert-space vector. A time-dependent Hamiltonian generates a two-time propagator, and its existence needs suitable time regularity and domain hypotheses in addition to instantaneous self-adjointness.

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.

Example: phase rate measures an energy difference

For $H=\operatorname{diag}(E_1,E_2)$, the two components acquire phases $e^{-iE_1t/\hbar}$ and $e^{-iE_2t/\hbar}$. Their relative phase changes at angular rate $(E_2-E_1)/\hbar$. This is why an energy difference, rather than an absolute energy zero, sets a quantum oscillation frequency. Adding the same constant to both diagonal entries produces only a common phase.

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.

References

• M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, revised and enlarged 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.

Exercises

1. For a finite-dimensional matrix $H$, use $U(dt)=I-iHdt/\hbar+O(dt^2)$ to find the condition for first-order norm preservation. Explain why this calculation alone does not establish self-adjointness of an unbounded operator.

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 vectors in finite dimensions, one needs $H^\dagger=H$. For an unbounded operator, the analogous inner-product calculation establishes symmetry on its chosen domain. Self-adjointness also requires equality with the adjoint domain; first-order norm preservation does not prove that extra condition.