Question

Equation of Quantum hormonic oscillator

Answer 1

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.