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### Why do you need symmetric and antisymmetric solutions of the time-independent Schrödinger Equation by a given potential $V(x)$? [duplicate]

I've calculated many symmetric and antisymmetric solutions of the time-independent Schrödinger Euqation by a given square potentials $V(x)$. Just for practice etc., but honestly I do not understand ...
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### Schrodinger equation: If $V(x)=V(-x)$ then prove that $\psi(x)=\psi(-x)$ or $\psi(x)=-\psi(-x)$ [duplicate]

The title explains itself. If the potential is an even function then prove that wave function is either odd or even. I set $-x$ in Schrodinger equation and find out that $\psi(-x)$ is also a solution ...
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### Solutions to time-independent Schrödinger's equation with symmetrical (even) potential [duplicate]

A problem from Griffith's Introduction to Quantum Mechanics asks to prove the following: Given a symmetric potential $V(x)$ $(=V(-x))$, the solutions to the time-independent Schrödinger's equation ...
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