= Energy populations and evolving relative phases
{title2=$c_n(t)=c_n(0)e^{-iE_nt/\hbar}$}
Suppose a constant <Hamiltonian operator> has a complete discrete <orthonormal basis> of <energy eigenstates>, $H|n\rangle=E_n|n\rangle$. Expanding an initial <pure state> and applying the <time-evolution operator> term by term gives
$$
|\psi(0)\rangle=\sum_n c_n|n\rangle,\qquad
|\psi(t)\rangle=\sum_n c_ne^{-iE_nt/\hbar}|n\rangle.
$$
A <projective measurement> resolving these <basis> vectors therefore assigns the same <Born rule> <probability> $|c_n|^2$ to each label $n$ at every time. An <energy measurement> groups labels sharing the same <energy eigenvalue>: $P(E)=\sum_{n:E_n=E}|c_n|^2$ is also constant. If $c_m,c_n\ne0$, their <probability amplitude> ratio instead evolves as
$$
\frac{c_n(t)}{c_m(t)}=\frac{c_n}{c_m}
\exp\!\left[-\frac{i(E_n-E_m)t}{\hbar}\right].
$$
Only <energy> differences enter this relative <quantum phase>. Adding a constant $\varepsilon I$ to the <Hamiltonian operator> multiplies the entire <pure state> by $e^{-i\varepsilon t/\hbar}$, a <global phase>, and leaves the <density operator> unchanged.
For a fixed <observable> $A$, the <expectation value> can nevertheless vary:
$$
\langle A\rangle_t=\sum_{m,n}c_m^*c_n
e^{i(E_m-E_n)t/\hbar}\langle m|A|n\rangle.
$$
Terms joining distinct <energy eigenvalues> can oscillate; diagonal terms cannot. If $[A,H]=0$, these terms between different <energies> vanish, so the <expectation value> is constant. This distinguishes <conservation of energy> from the stronger claim that every <observable> is stationary.
An <energy eigenstate> has only a <global phase> evolution and is a <stationary state>. A <quantum superposition> entirely within one degenerate <energy eigenspace> is also an <energy eigenstate> and is stationary. For a general <density operator> in finite <dimension>, stationarity under a constant <Hamiltonian operator> means $[H,\rho]=0$: differentiating $U\rho U^\dagger$ proves necessity, and commuting $\rho$ through the exponential proves sufficiency. Merely fixing the diagonal energy populations does not fix the off-diagonal <quantum coherence>.
Back to article page
