In position space, the [Wavefunction](Wavefunction.md) of a quantum harmonic oscillator is given in terms of [Hermite polynomial](Hermite%20polynomial.md)s as
$\psi_n(x,t)=\sqrt[4]{\frac{m\omega}{2^{2n}\pi\hbar(n!)^2}}H_n\bigg(\sqrt{\frac{m\omega}{\hbar}x}\bigg)e^{-m\omega x^2/2\hbar}e^{-iE_nt/\hbar}$ ^5c1135
where $E_n$ are the energy [eigenvalues](quantum%20harmonic%20oscillator%20Hamiltonian.md#The%20energy%20spectrum) of the [harmonic oscillator Hamiltonian](quantum%20harmonic%20oscillator%20Hamiltonian.md). These wave functions are exact solutions to the [Schrödinger equation.](The%20time%20dependent%20Schrödinger%20equation%20for%20the%20quantum%20harmonic%20oscillator.md)
#QuantumMechanics/QuantumDynamics/QuantumHarmonicOscillators
#QuantumMechanics/QuantumHarmonicOscillators