The time-independent Schrodinger equation is an eigenvalue problem. Consider a mass m in one dimensi ...

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The time-independent Schrodinger equation is an eigenvalue problem. Consider a mass m in one dimension under the influence of a restoring force, and thus a parabolic potential energy function of the form U(x) = (1/2) m omega^2 x^2 (the 1D harmonic oscillator). (a) Write down the time-independent Schrodinger equation. (b) Show explicitly that psi(x) = A exp(-m omega x^2 / (2 hbar)) is a solution (eigenfunction) of the time-independent Schrodinger equation, and find the corresponding energy E. Here A is a normalization constant that does not depend on x; we do not need its explicit form.