The harmonic oscillator wave functions are composed of [standing wave](Standing%20waves.md) modes that exist at discrete energy levels. Therefore, since we're looking at standing waves here, we ignore [time dependence](Dynamics%20of%20the%20quantum%20harmonic%20oscillator.md) and consider the following stationary wave functions. # Position space wavefunction The stationary position space wave function is as follows: $\psi_n(x)=\sqrt[4]{\frac{m\omega}{2^{2n}\pi\hbar(n!)^2}}H_n\bigg(\sqrt{\frac{m\omega}{\hbar}x}\bigg)e^{-m\omega x^2/2\hbar},$ ^813575 which are naturally solutions to the [stationary Schrödinger equation](The%20stationary%20Schr%C3%B6dinger%20equation%20for%20the%20quantum%20harmonic%20oscillator.md#Stationary%20equation). Here $H_n\bigg(\sqrt{\frac{m\omega}{\hbar}x}\bigg)$ are [Hermite polynomials.](Hermite%20polynomial.md) These wave functions are also built out of [Fock states](Harmonic%20Oscillator%20State%20Vector.md) such that $\psi_n(x)=\langle x|n\rangle.$ Here we show the plots for the first four wave functions in position.  Image adapted from Griffiths D., _Introduction to Quantum Mechanics._ # Momentum space wavefunction # Raising and lowering the energy states Given a [stationary wavefunction](Stationary%20quantum%20harmonic%20oscillator%20wave%20function.md#Stationary%20wave-functions), $\psi_n$ that is a solution to the [stationary Schrödinger equation](The%20stationary%20Schr%C3%B6dinger%20equation%20for%20the%20quantum%20harmonic%20oscillator.md#Stationary%20equation), $\hat{a}^\dagger\psi_n$ and $\hat{a}\psi_n$ are also solutions. Thus we can use the [ladder operators](Harmonic%20Oscillator%20Ladder%20Operators.md) to transform between harmonic oscillator wavefunctions. This is shown [here](The%20stationary%20Schr%C3%B6dinger%20equation%20for%20the%20quantum%20harmonic%20oscillator.md#Proof). # Distribution in space ## The Ground state ## Higher energy levels You may notice that the non-zero region expands in $x$ with energy level. We find that this widening of the wave function follows the contour of the harmonic oscillator [potential](quantum%20harmonic%20oscillator%20Hamiltonian.md) ($\sim x^2/2$). Thus if we plot the wave functions at discrete intervals corresponding to energy levels $n$, we can trace out the contour of the potential as shown below, where the [potential](Potential%20Barrier.md) boundary _intersects_ the wave functions close to its edges. (more on that [below](Stationary%20quantum%20harmonic%20oscillator%20wave%20function.md#Correspondence%20with%20the%20classical%20limit))  ## Comparison with the classical case Look again at the [wave function plots](Stationary%20quantum%20harmonic%20oscillator%20wave%20function.md#Higher%20energy%20levels). It is important to notice also that, unlike what would be expected for [classical harmonic oscillator position functions](1D%20Harmonic%20Oscillator.md#Solution), the wave-function extends outside of the [potential barrier](Potential%20Barrier.md) - indicating a non-zero [probability](Quantum%20Harmonic%20oscillator%20Probability%20distributions.md) that an oscillating particle would be found outside of the boundaries of the potential. ## The wavefunction at the potential barrier The effect on the [wave function](Stationary%20quantum%20harmonic%20oscillator%20wave%20function.md) is that it changes from being sinusoidal within the potential to being an exponential decay outside of the potential. See a detailed discussion of the quantum harmonic oscillator at the potential boundary [here.](The%20quantum%20harmonic%20oscillator%20at%20the%20classical%20potential%20boundary.md) # Derivation of normalization constant --- # Recommended Reading Derivations and discussions surrounding the harmonic oscillator wavefunction are a key component of any introduction to quantum mechanics. Examples of such introductions are: * [Griffiths D. J., _Introduction to Quantum Mechanics_, Pearson Prentice Hall, 2nd edition, 2005.](Griffiths%20D.%20J.,%20Introduction%20to%20Quantum%20Mechanics,%20Pearson%20Prentice%20Hall,%202nd%20edition,%202005..md) pgs. 51-59. Here the wavefunctions for stationary quantum mechanical harmonic oscillators are derived as solutions to the [the time independent Schrödinger equation.](The%20stationary%20Schrödinger%20equation%20for%20the%20quantum%20harmonic%20oscillator.md) Here the Hermite polynomials are shown to arise from an analytic solution to the differential equation. Note page 58 includes figures similar to those presented here. #QuantumMechanics/QuantumHarmonicOscillators #QuantumMechanics/StationaryStateQuantumSystems/QuantumHarmonicOscillators #QuantumMechanics/QuantumStateRepresentations/StateVectors