Hamiltonians as Generators

A Hamiltonian generates a continuous motion of state vectors, but the generator is not an everywhere-defined infinitesimal matrix. Its domain identifies the states whose motion has a Hilbert-space derivative. We will interpret that statement and its dependence on clocks and phases; Stone’s Theorem gives the precise correspondence and proof. For the physical introduction, see Hamiltonians as Generators.

Helpful background. Self-Adjoint Operators explains the domain requirement. Functions of Observables explains spectral functions and their probability distributions.

Hamiltonian and infinitesimal time generator

For an autonomous closed system, let $H$ be self-adjoint on $D(H)\subset\mathcal H$. The physical evolution convention is

$$U(t)=e^{-itH/\hbar}.$$

The exponential is a bounded spectral function of $H$, defined on all of $\mathcal H$. The corresponding generator in the mathematical convention is the skew-adjoint operator

$$G=-\frac{i}{\hbar}H,\qquad D(G)=D(H).$$

On this domain,

$$U(t)\psi=\psi-\frac{it}{\hbar}H\psi+o(t)$$

in Hilbert-space norm. The remainder is a statement about the chosen vector. It does not generally give an operator-norm estimate for the entire unit ball. A second-order vector expansion requires additional regularity, such as $\psi\in D(H^2)$.

The Hamiltonian is recovered from the derivative, including its domain. A formal differential expression without boundary conditions therefore does not specify a generator. Different self-adjoint realizations can produce different evolutions even when their local differential formulas agree; see Domains of Hamiltonians.

Self-adjointness also does not require a lower spectral bound. The momentum operator generates translations despite having spectrum equal to the whole real line. Lower boundedness is an additional condition relevant to energetic stability, discussed in Semibounded Hamiltonians.

Clocks and the energy origin

Suppose a new time parameter $\tau$ is related by $t=a\tau$ with a nonzero real constant $a$. Then

$$U(a\tau)=e^{-i\tau(aH)/\hbar}.$$

The generator relative to $\tau$ is $aH$, with the same domain. Rescaling the time parameter changes the numerical generator; reversing the parameter changes its sign. The units of $H/\hbar$ are inverse time.

An additive real constant changes the vector evolution by a scalar phase:

$$H_c=H+cI,\qquad U_c(t)=e^{-ict/\hbar}U(t).$$

All pure-state projectors and all conjugated observables evolve identically under $H$ and $H_c$. The two unitary groups are nonetheless different. There is no conflict with uniqueness in Stone’s theorem: that uniqueness concerns a specified group of unitaries, including its phases. Ray trajectories alone do not determine the energy origin.

For example, $H=(\hbar\omega/2)\sigma_z$ and $H_c=H+(\hbar\omega/2)I$ have spectra $\{-\hbar\omega/2,\hbar\omega/2\}$ and $\{0,\hbar\omega\}$. Their relative phase evolves at the same angular frequency $\omega$. Interference between the two levels measures the energy difference, while the common shift cancels.

Finite mean energy does not ensure differentiability

Consider $\mathcal H=\ell^2(\mathbb N)$ and the positive diagonal Hamiltonian

$$He_n=E_*n e_n,\qquad E_*>0.$$

Its domain is the set of sequences $c=(c_n)$ for which $\sum_n n^2|c_n|^2<\infty$. Every square-summable sequence still evolves by

$$(U(t)c)_n=e^{-itE_*n/\hbar}c_n.$$

Take $c_n=Cn^{-3/2}$, where $C$ normalizes the state. The spectral mean energy exists because

$$\int\lambda\,d\mu_c^H(\lambda) =E_*C^2\sum_{n=1}^{\infty}n^{-2}<\infty.$$

But the second moment is infinite:

$$\int\lambda^2\,d\mu_c^H(\lambda) =E_*^2C^2\sum_{n=1}^{\infty}n^{-1}=\infty.$$

Thus $c\in D(H^{1/2})$ but $c\notin D(H)$. Its mean energy is a form expectation, not an inner product with a Hilbert-space vector $Hc$. Its unitary trajectory is continuous and lacks a strong time derivative. This is a regularity distinction, not a failure of the state or of its evolution.

For a normalized $\psi\in D(H)$, the vector speed at zero is $\|H\psi\|/\hbar$. Subtracting the mean energy removes the infinitesimal phase component: the speed of $e^{it\langle H\rangle/\hbar}U(t)\psi$ at zero is $\Delta H/\hbar$. This explains why energy differences and fluctuations, rather than the arbitrary energy origin, control motion of a ray. The geometric formulation belongs to Projective Hilbert Space.

Spectral cutoffs and finite computations

A finite matrix can approximate selected trajectories while failing to approximate the full generator uniformly. In the diagonal example let $P_N$ project onto the first $N$ basis vectors and let $H_N$ agree with $H$ there and vanish on the complement. Then

$$\sup_{t\in\mathbb R}\|(e^{-itH_N/\hbar}-U(t))\psi\| \leq2\|(I-P_N)\psi\|\longrightarrow0.$$

The estimate is state-dependent. It supplies strong convergence of evolutions without an operator-norm approximation of the unbounded Hamiltonian. A numerical truncation should therefore be assessed on the state family and observables under study, with control of discarded spectral weight.

Exercises

1. Show directly that adding $cI$ leaves the evolution of every density operator unchanged.

Solution

The scalar phases cancel in $U_c(t)\rho U_c(t)^*=U(t)\rho U(t)^*$. Consequently every bounded-observable expectation is unchanged. This argument requires no differentiability of $\rho$ and no finite energy moment.

2. In the cutoff example choose $t_N=\pi\hbar/((N+1)E_*)$. Show why the smallness of $t_N$ does not make the cutoff error uniformly small over normalized states.

Solution

On $e_{N+1}$ the cutoff evolution is the identity, whereas $U(t_N)e_{N+1}=-e_{N+1}$. Their difference has norm two, so the operator-norm error is two although $t_N\to0$. The offending state changes with $N$; this does not contradict strong convergence on each fixed state.

References

• B. C. Hall, Quantum Theory for Mathematicians, Graduate Texts in Mathematics 267, Springer, 2013, publisher edition — quantum dynamics and unbounded observables.
• G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, 2nd ed., Graduate Studies in Mathematics 157, American Mathematical Society, 2014, Section 5.1, author’s text — self-adjoint generators and time evolution.