Answer to Question #164565 in Quantum Mechanics for Hima bindu

Question #164565

Equation of Quantum hormonic oscillator


1
Expert's answer
2021-02-17T12:06:42-0500

An energy of a classical harmonic oscillator is given by

"E = \\frac{mv^2}{2}+\\frac{kx^2}{2} = \\frac{p^2}{2m}+\\frac{m\\omega^2 x^2}{2}"

Therefore by analogy we construct a hamiltonian of a QHO by associating the corresponding operators to the physical quantities :

"\\hat H = -\\frac{\\hbar^2}{2m} \\frac{\\partial^2}{\\partial x^2} +\\frac{1}{2}m \\omega^2 \\hat x^2" and thus from the Schrodinger equation we get

"\\hat H \\psi = \\frac{\\partial}{\\partial t} \\psi"

"-\\frac{\\hbar^2}{2m} \\frac{\\partial^2}{\\partial x^2} \\psi(x,t) +\\frac{1}{2}m \\omega^2 \\hat x^2 \\psi(x,t) = \\frac{\\partial}{\\partial t}\\psi(x,t)" which can be generalized to n-dimensions in the same way as we generalize it in the classical case.



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