We now assume that the probability amplitudes are time-dependent, though the basis states are not. The Time-dependent Schrödinger equation states i ℏ ∂ ∂ t | ψ ⟩ = H | ψ ⟩ {\displaystyle i\hbar {\frac {\partial }{\partial t}}|\psi \rangle =H|\psi \rangle } , and proceeding as before (substituting for | ψ ⟩ {\displaystyle |\psi \rangle } and premultiplying by ⟨ 1 | , ⟨ 2 | {\displaystyle \langle 1|,\langle 2|} again produces a pair of coupled linear equations, but this time they are first order partial differential equations: i ℏ ∂ ∂ t c = H c {\displaystyle i\hbar {\frac {\partial }{\partial t}}\mathbf {c} =\mathbf {Hc} } . If H {\displaystyle \mathbf {H} } is time independent there are several approaches, to find the time dependence of c 1 , c 2 {\displaystyle c_{1},c_{2}} , such as normal modes.