10.2: Superposition of time-dependent solutions
( \newcommand{\kernel}{\mathrm{null}\,}\)
There has been an example problem, where I asked you to show "that if ψ1(x,t) and ψ2(x,t) are both solutions of the time-dependent Schrödinger equation, than ψ1(x,t)+ψ2(x,t) is a solution as well." Let me review this problem
−ℏ22m∂2∂x2ψ1(x,t)+V(x)ψ1(x,t)=ℏi∂∂tψ1(x,t)−ℏ22m∂2∂x2ψ2(x,t)+V(x)ψ2(x,t)=ℏi∂∂tψ2(x,t)−ℏ22m∂2∂x2[ψ1(x,t)+ψ2(x,t)]+V(x)[ψ1(x,t)+ψ2(x,t)]=ℏi∂∂t[ψ1(x,t)+ψ2(x,t)]
where in the last line I have use the sum rule for derivatives. This is called the superposition of solutions, and holds for any two solutions to the same Schrödinger equation!