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10.2 Superposition of time-dependent solutions

There has been an example problem, where I asked you to show “that if {ψ}_{1}(x,t) and {ψ}_{2}(x,t) are both solutions of the time-dependent Schrödinger equation, than {ψ}_{1}(x,t) + {ψ}_{2}(x,t) is a solution as well.” Let me review this problem

\begin{eqnarray} −{{ℏ}^{2}\over 2m} {{∂}^{2}\over ∂{x}^{2}}{ψ}_{1}(x,t) + V (x){ψ}_{1}(x,t)& =& {ℏi∂\over ∂t} {ψ}_{1}(x,t) %& \\ −{{ℏ}^{2}\over 2m} {{∂}^{2}\over ∂{x}^{2}}{ψ}_{2}(x,t) + V (x){ψ}_{2}(x,t)& =& {ℏi∂\over ∂t} {ψ}_{2}(x,t) %& \\ −{{ℏ}^{2}\over 2m} {{∂}^{2}\over ∂{x}^{2}}[{ψ}_{1}(x,t) + {ψ}_{2}(x,t)] + V (x)[{ψ}_{1}(x,t) + {ψ}_{2}(x,t)]& =& {ℏi∂\over ∂t} [{ψ}_{1}(x,t) + {ψ}_{2}(x,t)],%&(10.4) \\ \end{eqnarray}

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?

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