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Physics LibreTexts

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

22m2x2ψ1(x,t)+V(x)ψ1(x,t)=itψ1(x,t)22m2x2ψ2(x,t)+V(x)ψ2(x,t)=itψ2(x,t)22m2x2[ψ1(x,t)+ψ2(x,t)]+V(x)[ψ1(x,t)+ψ2(x,t)]=it[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!

Question: Why doesn't it work for the time-independent Schrödinger equation?


This page titled 10.2: Superposition of time-dependent solutions is shared under a CC BY-NC-SA 2.0 license and was authored, remixed, and/or curated by Niels Walet via source content that was edited to the style and standards of the LibreTexts platform.

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