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One of the simplest examples of non-trivial time-dependent states is that of an equal mix, two-state superposition in the infinite square well.⁵  The position- and momentum-space wave functions are just

Ψ_n1n2(x,t) = 2^−1/2 [ψ_n1(x,t) + ψ_n2(x,t)] ,

and

Φ_n1n2(p,t) = 2^−1/2 [φ_n1(p,t) + φ_n2(p,t)] ,

where ψ_n(x,t) = exp(−iE_nt/ħ) ψ_n(x) and φ_n2(p,t) = exp(−iE_nt/ħ) φ_n(p). We can write these wave functions in a way that stresses their relative phases:

Ψ_n1n2(x,t) = 2^−1/2 exp(−iE_n1t/ħ) [ψ_n1(x,t) + exp(−i(E_n2 − E_n1)t/ħ) ψ_n2 (x,t)] , (10.11)

and

Φ_n1n2(p,t) = 2^−1/2 exp(−iE_n1t/ħ) [φ_n1(p,t) + exp(−i(E_n2 − E_n1)t/ħ) φ_n2(p,t)] . (10.12)

In this case there is a time-dependent relative phase that depends on the difference in energy eigenvalues and an overall time-dependent phase in Eqs. (10.11) and (10.12).

The animation depicts the time dependence of an arbitrary equal-mix two-state superposition.   Restart. The time is given in terms of the time it takes the ground-state wave function to return to its original phase, i.e., Δt = 1 corresponds to an elapsed time of 2πħ/E₁. You can change n₁ and n₂, the default values, n₁ = 1 and n₂ = 2 represent the standard case treated in almost every textbook. Explore the time-dependent form of the position-space and momentum-space wave functions for other n₁ and n₂.

⁵One of the earliest pedagogical visualizations of the time dependence of such a two-state system is by C. Dean, "Simple Schrödinger Wave Functions Which Simulate Classical Radiating Systems," Am. J. Phys. 27, 161-163 (1959).