Can the Hamiltonian be interpreted as the "speed" of unitary evolution?

Question

The Schrodinger equation

$$i\hslash \frac{d}{dt} \psi = H \psi$$

means a quantum state $\psi(t)$ evolves unitarily, that is,

$$\psi(t) = \exp(-\frac{i}{\hslash} H t) \psi(0)$$

where $\psi(0)$ is the initial state at time $t = 0$.

Suppose if we scale the energy levels of the Hamiltonian by some factor $0 < \zeta$, let

$$\tilde{H} = \frac{1}{\zeta} H$$

then for the evolution from $\psi(0)$ to a target state $\phi$,

$$\phi \equiv \psi(\tilde t = \zeta t) = \exp(-\frac{i}{\hslash} \frac{1}{\zeta} H \zeta t) \psi(0)$$

where the time $\tilde t$ to reach $\phi$ has to scale contravariantly to complement the change in $H$.

So can it be said the Hamiltonian $H$ is the "speed" of unitary evolution?

Answer 1

Sure, the Hamiltonian operator is the generator of 1-parameter unitary time flow in the sense of Stone's theorem.

Answer 2

To paraphrase @Qmechanic in more mundane language: Hamiltonian is the generator of unitary evolution, i.e., the rate of infinitesimal change. In this sense it can be called speed.

Note however, that the Hamiltonian appears in the exponent, and we do not have here a physical quantity, the derivative of which we could call speed. In other words, whether we call Hamiltonian speed has to do more with semantics rather than physics or math.

Answer 3

The Hamiltonian itself is not a speed, but you're right that evolution speed is proportional to the energy scale that defines the dynamics.

To define quantum evolution speed, consider first a classical signal with a finite range of frequencies in its Fourier spectrum. The width of the range tightly bounds the number of distinct values that can occur per unit time---this is Nyquist's signaling rate bound. Intuitively, doubling the width lets you double all frequencies in a Fourier sum, making everything happen twice as fast.

Time evolution of a quantum wavefunction is similar, with energy playing the role of frequency in determining how fast the state can change. Any well-defined energy width of a wavefunction bounds the rate at which distinct (orthogonal) states can occur in its time evolution. In the quantum literature, these bounds are called quantum speed limits.