= Energy eigenspace
The energy eigenspace at $E$ is the <vector subspace>
$$
\mathcal E_E=\ker(H-EI)=\{\psi\in D(H):H\psi=E\psi\},
$$
where $H$ is the <Hamiltonian operator> and $I$ is the <identity operator>. It is nonzero exactly when $E$ is an <energy eigenvalue>. Its <dimension> is the multiplicity of that <energy eigenvalue>; <degenerate energy levels> have multiplicity greater than one. The <ground-state subspace> is $\mathcal E_{E_0}$, and its <dimension> is the <ground-state degeneracy>. A vector in an <energy eigenspace> need not be one of a specially chosen <orthonormal basis> of <energy eigenstates>.
<Linearity> makes the useful point immediate: if $H\phi_1=E\phi_1$ and $H\phi_2=E\phi_2$, then $H(a\phi_1+b\phi_2)=E(a\phi_1+b\phi_2)$. Every normalized nonzero combination stays in the same <energy eigenspace>. For a time-independent <Hamiltonian operator>, its evolution is multiplication by the common <global phase> $e^{-iEt/\hbar}$, so it is a <stationary state>. A <quantum superposition> within a degenerate <energy eigenspace> can change spatial shape when its coefficients are chosen differently, but its density does not subsequently beat under this unchanged <Hamiltonian operator>.
Across distinct <energy eigenspaces>, relative <quantum phases> evolve. For orthonormal <eigenfunctions> $\phi_1,\phi_2$ of the <Hamiltonian operator> with <energy eigenvalues> $E_1\ne E_2$, an equal-weight <quantum superposition> has
$$
\psi(x,t)=\frac{e^{-iE_1t/\hbar}\phi_1(x)+e^{-iE_2t/\hbar}\phi_2(x)}{\sqrt2},
$$
and position <probability density>
$$
|\psi(x,t)|^2=\frac{|\phi_1(x)|^2+|\phi_2(x)|^2}{2}+\operatorname{Re}\!\left[\phi_1(x)^*\phi_2(x)e^{-i(E_2-E_1)t/\hbar}\right].
$$
The interference term can change local probabilities even though the <Born rule> assigns the same time-independent probability $1/2$ to each <energy measurement> outcome. <Orthogonality> makes its integral zero, preserving <wavefunction normalization>. This <pure state> is not a <stationary state>; some particular measurement distributions may nevertheless remain constant. An incoherent <mixed state> with the same energy populations has no such interference term.
\Image[/ground-excited-states-stationary-density.png]
{title=An excited energy eigenstate can be stationary while a two-energy superposition beats}
{description=Exact <infinite square well> densities, not measured data, with $u=x/L$ and $\theta=(E_2-E_1)t/\hbar$. The left panel shows the unchanged density of the second <energy eigenstate> at three times. The right panel shows a coherent equal-weight <quantum superposition> at $\theta=0,\pi/2,\pi$; its energy populations remain $1/2,1/2$. The dashed curve is the density of the corresponding incoherent <mixed state>.}
For the pictured <infinite square well>, $\theta=3E_1t/\hbar$ and the interference repeats after $T=2\pi\hbar/(3E_1)$. At $t=0$, direct integration gives $\langle x\rangle=L/2-16L/(9\pi^2)$; halfway through the period it is $L/2+16L/(9\pi^2)$. This is a moving <expectation value>, rather than a trajectory of an individual particle. The distinction between an <energy eigenspace> and a combination of different <energy eigenspaces> is what makes the different behavior possible.
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