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5. [8 points] Consider the 1D time-independent Schrödinger equation ħ² d² 2m dx² + V (x) y = Εψ with the bi-cubic potent
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5. [8 points] Consider the 1D time-independent Schrödinger equation ħ² d² 2m dx² + V (x) y = Εψ with the bi-cubic potent
5. [8 points] Consider the 1D time-independent Schrödinger equation ħ² d² 2m dx² + V (x) y = Εψ with the bi-cubic potential 2ħ² V(x): 22 [+ (-²)* - ³(²)]. = where to is a parameter. Note that there are no scattering states, only bound states, and that the potential is symmetric. (a) The wavefunction VG = e -(x/x0)4 is an unnormalized ground state solution of the Schrödinger equation. Determine the ground state energy EG- (b) Sketch V(x), g(x), and Ec in the usual manner. (c) Determine formulas for the locations xt of the classical turning points, and denote them on your sketches in part (b). (d) Extra credit: Check that c has inflection points at those classical turning points.
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