Eikonal Blog


Harmonic oscillator

Filed under: Quantum mechanics — Tags: , , — sandokan65 @ 20:23

Potential function: U(x)=\frac{m\omega^2 x^2}{2}

Solutions of the Schr\”odinger equation:

  • States: \psi_n(x) = (\frac{m}{\hbar\omega})^{\frac14}  \frac{1}{\sqrt{2^n n! \pi}} H_n(\sqrt{\frac{m\omega}{\hbar}}x) e^{-\frac{m\omega x^2}{\hbar}}
  • Energies: E_n=\hbar\omega (n+\frac12)

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