Following from the [Born rule](Born%20rule.md) and the full time-dependent [wavefunction](Stationary%20quantum%20harmonic%20oscillator%20wave%20function.md) for a [quantum harmonic oscillator,](Quantum%20Harmonic%20Oscillator%20(index).md)  the probability of measuring the Harmonic oscillator at position $x$ in the $nth$ energy level is expressed as $|\psi_n(x)|^2=\sqrt{\frac{m\omega}{2^{2n}\pi\hbar(n!)^2}}H^2_n\bigg(\sqrt{\frac{m\omega}{\hbar}x}\bigg)e^{-2m\omega x^2/2\hbar}$ Where $H_n$ is the $nth$ [Hermite polynomial](Hermite%20polynomial.md). Notice that the probabilities are _not_ time dependent since the quantum harmonic oscillator is composed of [standing wave](Standing%20waves.md) modes and we could've also started with the [time independent expression:](Stationary%20quantum%20harmonic%20oscillator%20wave%20function.md)  Below we plot the resulting probability distributions in the first six energy levels where we see how each probability distribution is confined within with the harmonic oscillator potential, which in the [hamiltonian](quantum%20harmonic%20oscillator%20Hamiltonian.md) is the term $\hat{V}=\frac{m\omega^2}{2}\hat{x}^2$  # Comparison with the classical prediction Notice that at low energies the classical prediction differs greatly from the quantum mechanical result in its shape. This is most apparent at the ground state energy. If we consider the [classical prediction](1D%20Harmonic%20Oscillator.md), the position probability distribution is its [classical time average](Average%20position%20over%20time%20of%20a%20Harmonic%20Oscillator.md), which is $\sim \frac{1}{\sqrt{x_0^2-x^2}}$ regardless of the energy for a given initial displacement, $x_0.$ The corresponding energy at an initial displacement for a classical harmonic oscillator is  However, the ground state probability distribution in the quantum mechanical case is proportional to [Hermite polynomials](Hermite%20polynomial.md) along with being notably non-zero outside of the potential (as shown [above](Quantum%20Harmonic%20oscillator%20Probability%20distributions.md)). Shown below, for example, is a comparison between the classical prediction (orange) and the quantum mechanical result (blue) at the ground state energy, which is a [[Gaussian distribution]] in the quantum mechanical case.  ## Classical Limit Shown below is the probability distribution of the quantum harmonic oscillator at an [energy](quantum%20harmonic%20oscillator%20Hamiltonian.md#The%20energy%20spectrum) corresponding with $n=100$ (blue) overlaid with the [classical time average](Average%20position%20over%20time%20of%20a%20Harmonic%20Oscillator.md) for a classical [1D harmonic oscillator](1D%20Harmonic%20Oscillator.md) at the same energy (orange). In accordance with the [correspondence principle,](Correspondence%20Principle.md) the quantum harmonic oscillator begins to behave more like the classic oscillator as indicated below as $n$ increases to a large number.  #QuantumMechanics/QuantumHarmonicOscillators #QuantumMechanics/QuantumStateRepresentations/Wavefunctions