# Can the Hamiltonian be interpreted as the “speed” of unitary evolution?

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?