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Linearity

The Schrödinger equation is a linear differential equation, meaning that if two state vectors | ψ 1 ⟩ {\displaystyle |\psi _{1}\rangle } and | ψ 2 ⟩ {\displaystyle |\psi _{2}\rangle } are solutions, then so is any linear combination | ψ ⟩ = a | ψ 1 ⟩ + b | ψ 2 ⟩ {\displaystyle |\psi \rangle =a|\psi _{1}\rangle +b|\psi _{2}\rangle } of the two state vectors where a and b are any complex numbers. Moreover, the sum can be extended for any number of state vectors. This property allows superpositions of quantum states to be solutions of the Schrödinger equation. Even more generally, it holds that a general solution to the Schrödinger equation can be found by taking a weighted sum over a basis of states. A choice often employed is the basis of energy eigenstates, which are solutions of the time-independent Schrödinger equation. In this basis, a time-dependent state vector | Ψ (t) ⟩ {\displaystyle |\Psi (t)\rangle } can be written as the linear combination | Ψ (t) ⟩ = ∑ n A n e − i E n t / ℏ | ψ E n ⟩, {\displaystyle |\Psi (t)\rangle =\sum _{n}A_{n}e^{{-iE_{n}t}/\hbar }|\psi _{E_{n}}\rangle,} where A n {\displaystyle A_{n}} are complex numbers and the vectors | ψ E n ⟩ {\displaystyle |\psi _{E_{n}}\rangle } are solutions of the time-independent equation H ^ | ψ E n ⟩ = E n | ψ E n ⟩ {\displaystyle {\hat {H}}|\psi _{E_{n}}\rangle =E_{n}|\psi _{E_{n}}\rangle }.