Stationary states are eigenstates of the Hamiltonian. This means that they are states with definite energy.

H^∣ϕn⟩=En∣ϕn⟩\hat{H}\ket{\phi_n} = E_n \ket{\phi_n}

Stationary states are in only a global phase. This means that all physical quantities stay constant in time. Note from unitary evolution-- a single eigenstate evolves as

∣ϕn(t)⟩=e−iEnt/ℏ∣ϕn⟩\ket{\phi_n(t)} = e^{-iE_n t/\hbar}\ket{\phi_n}

This matters because

∣ψ(0)⟩=∑ncn∣ϕn⟩\ket{\psi(0)} = \sum_n c_n \ket{\phi_n}

Note that cnc_n is a Probability amplitude and ϕn\phi_n is a Basis State and ∣ψ(0)⟩\ket{\psi(0)} is a state. Note

each component has its own phase so

∣ψ(t)⟩=∑ncn e−iEnt/ℏ∣ϕn⟩\ket{\psi(t)} = \sum_n c_n\, e^{-iE_n t/\hbar}\ket{\phi_n} ⟨x∣ψ(t)⟩=∑ncn e−iEnt/ℏ⟨x∣ϕn⟩\braket{x|\psi(t)} = \sum_n c_n\, e^{-iE_n t/\hbar}\braket{x|\phi_n} ψ(x,t)=∑ncn e−iEnt/ℏϕn(x)\psi(x,t) = \sum_n c_n\, e^{-iE_n t/\hbar}\phi_n(x)

A superposition therefore is not stationary. The differing phases produces Interference which is where time-dependence (EVs that oscillate) come from.

Note that In a single starionary state, EV are constant

E[A^]=⟨ϕn∣e+iEnt/ℏ A^ e−iEnt/ℏ∣ϕn⟩=⟨ϕn∣A^∣ϕn⟩\mathbb{E}[\hat A] = \braket{\phi_n|e^{+iE_nt/\hbar}\,\hat A\,e^{-iE_nt/\hbar}|\phi_n} = \braket{\phi_n|\hat A|\phi_n}

But in super position, EV can oscillate.

E[A^](t)=(∑ncn∗e+iEnt/ℏ⟨ϕn∣)A^(∑mcme−iEmt/ℏ∣ϕm⟩)\mathbb{E}[\hat A](t) = \Big(\sum_n c_n^* e^{+iE_n t/\hbar}\bra{\phi_n}\Big)\hat A\Big(\sum_m c_m e^{-iE_m t/\hbar}\ket{\phi_m}\Big) =∑n,mcn∗cm e−i(Em−En)t/ℏ⟨ϕn∣A^∣ϕm⟩= \sum_{n,m} c_n^* c_m\, e^{-i(E_m-E_n)t/\hbar}\braket{\phi_n|\hat A|\phi_m}