(Solution) - Consider the time independent one dimensional Schr dinger equati -(2025 Original AI-Free Solution)

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Consider the time-independent one-dimensional Schrödinger equation when the potential function is symmetric about the origin, i.e., when U(x) = U(?x).
(a) Show that if ?(x) is a solution of the Schrödinger equation with energy E, then ?(?x) is also a solution with the same energy E, and that, therefore, ?(x) and ?(?x) can differ by only a multiplicative constant.
(b) Write ?(x) = C?(?x), and show that C = ±1. Note that C = +1 means that ?(x) is an even function of x, and C = ?1 means that ?(x) is an odd function of x.