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dingo_d
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Homework Statement
Let [tex]|\psi\rangle[/tex] and [tex]|\psi '\rangle[/tex] be solutions to the same Schrodinger equation. Show than, that [tex]c|\psi\rangle+c'|\psi '\rangle[/tex] is the solution, where c and c' are arbitrary complex coefficients, for which holds: [tex]|c|^2+|c'|^2=1[/tex]
The Attempt at a Solution
Now this follows from linearity of the Schrodinger equation (that every linear combination is the solution). But how to prove it directly?
I've started with:
[tex]i\hbar \frac{\partial}{\partial t}|\psi\rangle=\hat{H}|\psi\rangle[/tex]
[tex]i\hbar \frac{\partial}{\partial t}|\psi'\rangle=\hat{H}|\psi'\rangle[/tex]
And added them up:
[tex]i\hbar\left( \frac{\partial}{\partial t}|\psi\rangle+\frac{\partial}{\partial t}|\psi'\rangle\right)=\hat{H}(|\psi\rangle+|\psi'\rangle)[/tex]
And... now I don't know what to do next :\
I think I'm taking the superposition principle way to lightly :\