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- The dynamics of a particle moving one-dimensionally in a potential V (x) is governed by the Hamiltonian Ho = p²/2m + V (x), where p = is the momentuin operator. Let E, n = of Ho. Now consider a new Hamiltonian H given parameter. Given A, m and E, find the eigenvalues of H. -ih d/dx 1, 2, 3, ... , be the eigenvalues Ho + Ap/m, where A is a %3|Show that the function \[ S=S(q, \beta, t)=\frac{m \omega}{2}\left(q^{2}+\beta^{2}\right) \cot \omega t-m \omega q \beta \csc \omega t \] is a solution to the Hamilton-Jacobi equation for Hamilton's principal function for the linear harmonic oscillator with \[ H=\frac{1}{2 m}\left(p^{2}+m^{2} \omega^{2} q^{2}\right) \] Show that this function generates a correct solution to the motion of the harmonic oscillator.