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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?

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

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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.

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