Consider the one-dimensional harmonic oscillator with potential energy V(x) = mwr, and the spatial part of...

 

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Consider the one-dimensional harmonic oscillator with potential energy V(x) = mw²r², and the spatial part of the ground state energy eigenfunction uo(x) = Ne-²² where N is a normalization constant that you do not need to determine. (i) Make a sketch of V(z) and a sketch of up (x). (ii) Show by direct substitution into the time-independent Schrödinger equation for the one-dimensional harmonic oscillator that uo(r) is a solution, and determine the energy eigenvalue. Show your working. Consider the ladder operators A and A+ for the one-dimensional harmonic oscillator acting on the spatial parts of the energy eigenfunctions un. Using A+ un= √n +1 un+1 and Aun = √√nun-1, determine  Âun and ÂÂ+un (assume n > 1). Explain how these results agree with the commutator [Â, Â+] = 1. Consider the one-dimensional harmonic oscillator with potential energy V(x) = mw²r², and the spatial part of the ground state energy eigenfunction uo(x) = Ne-²² where N is a normalization constant that you do not need to determine. (i) Make a sketch of V(z) and a sketch of up (x). (ii) Show by direct substitution into the time-independent Schrödinger equation for the one-dimensional harmonic oscillator that uo(r) is a solution, and determine the energy eigenvalue. Show your working. Consider the ladder operators A and A+ for the one-dimensional harmonic oscillator acting on the spatial parts of the energy eigenfunctions un. Using A+ un= √n +1 un+1 and Aun = √√nun-1, determine  Âun and ÂÂ+un (assume n > 1). Explain how these results agree with the commutator [Â, Â+] = 1.

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i Sketch of Vx The potential energy function Vx for the onedimensional harmonic oscillator is given ... View the full answer