Quantum Harmonic Oscillator Wave Function — Hermite Polynomials
Visualize the eigenfunctions ψ_n(x) and probability densities of the 1D quantum harmonic oscillator from quantum number, mass, angular frequency and ℏ, alongside energy levels and classical turning points.
Parameters
Quantum number n
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Mass m
a.u.
Angular frequency ω
a.u.
Reduced Planck constant ℏ
a.u.
All quantities are in atomic units. Changing n changes the energy level and the number of nodes.
Results
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Energy E_n = ℏω(n+1/2)
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Classical turning point x_c = √((2n+1)ℏ/mω)
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Normalization ∫|ψ|²dx
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Number of nodes (zeros of H_n)
Wave function ψ_n(x) and probability density |ψ_n(x)|²
Top: parabolic potential V(x)=½mω²x² (blue) and discrete energy levels E_0..E_n (black horizontal lines). Bottom: ψ_n(x) (green) and |ψ_n(x)|² (red), with classical turning points ±x_c marked by dashed lines.
Theory & Key Formulas
The 1D quantum harmonic oscillator Schrödinger equation has eigenfunctions that factor into a Hermite polynomial $H_n(\xi)$ and a Gaussian.
Energies are discrete and equally spaced by ℏω, and even the ground state retains the zero-point energy ℏω/2. The wave function tunnels exponentially beyond the classical turning points.
What is the Quantum Harmonic Oscillator Wave Function Simulator?
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A quantum harmonic oscillator is basically the quantum version of a mass on a spring, right? What is actually different from the classical case?
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The biggest difference, roughly speaking, is that the energy is not continuous but quantized into discrete steps. $E_n = \hbar\omega(n+1/2)$ gives equally spaced rungs at n=0,1,2,... In the simulator, raise the quantum number n and watch the black horizontal lines on the top plot move up by exactly ℏω each time. With the defaults (n=3, m=ω=ℏ=1) the energy comes out to E_3 = 3.5.
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Even at n=0 the energy isn't zero?
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Right — that is the famous "zero-point energy". Even the ground state keeps E_0 = ℏω/2. Classically a spring can sit still at zero energy, but in quantum mechanics the uncertainty principle forbids zero position and zero momentum at the same time. That is why crystal lattices keep on vibrating slightly even at absolute zero. This residual motion is the starting point for quantum corrections to heat capacity and thermal conduction.
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Looking at the green curve, for n=3 the wave function changes sign in three places. Is that a coincidence?
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No coincidence — the wave function with quantum number n always has exactly n nodes (zero-crossings). That is because the Hermite polynomial $H_n$ has exactly n real roots. n=0 has no nodes (just a Gaussian), n=1 has one node at the center, and n=3 has three. The "number of nodes" card in the simulator tracks this. It is a textbook example of the general rule "more nodes means higher energy".
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There is also a little bit of wave function leaking past the dashed lines at ±x_c, isn't there?
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Good catch. Classically the particle can never go beyond ±x_c — the kinetic energy would be negative. But quantum mechanically it bleeds exponentially into that forbidden region. This is the prototype of tunneling. Quantum-well lasers, the scanning tunneling microscope (STM), and even α-decay all depend on this same "leakage into a forbidden region". As you change n in the simulator, you can see the width of that leakage change too.
Frequently Asked Questions
Because the potential V(x) = ½mω²x² is a parabola, a very symmetric shape. Algebraically, the creation and annihilation operators a^†, a have the property that a^† "raises" the state by one level, and each step adds exactly ℏω. By contrast a 1/r potential, like the hydrogen atom, has spacings that shrink toward zero.
Yes — this is the "correspondence principle". For large n the envelope of the probability density |ψ_n(x)|² approaches the classical residence-time distribution (highest near the turning points and lowest at the center). Set n=10 in the simulator: the red |ψ|² curve oscillates rapidly, but its envelope traces out a U-shape that matches the classical probability distribution.
In quantum mechanics the wave function can penetrate classically forbidden regions (where E < V), decaying exponentially. This follows from the Schrödinger equation requiring the wave function to be continuous as long as the potential is not infinite. This "leakage" underlies tunneling, the operating principle of α-decay and the scanning tunneling microscope (STM).
Yes. Discretize the Schrödinger equation with finite differences or the Numerov method, and the energy levels and wave functions come out as an eigenvalue problem of a matrix. For the harmonic oscillator the analytic Hermite solution is a convenient benchmark, but for more general potentials (Morse, anharmonic, well-plus-barrier) numerical methods become essential.
Real-World Applications
Molecular vibrational spectroscopy: The stretching vibrations of diatomic molecules (H₂, CO, HCl, ...) sit in a potential well that is approximately parabolic near equilibrium, so the harmonic-oscillator model applies directly. Infrared (IR) and Raman spectra read off ℏω from the line positions and provide a direct measure of bond strength. A Morse potential and anharmonicity corrections refine the picture for excited states.
Phonons in solid-state physics: Lattice vibrations of atoms in a crystal can be viewed as a system of coupled springs. Diagonalizing into normal modes gives independent harmonic oscillators, and the quantized energy quanta are "phonons". The Einstein and Debye heat-capacity models, thermal conductivity, and the electron-phonon coupling of superconductivity all rest on this picture.
Quantum optics and lasers: Each mode of the electromagnetic field is itself a harmonic oscillator, and each energy quantum is a "photon". Coherent states |α⟩ describe laser light, while squeezed states reduce quantum noise in precision measurements such as gravitational-wave detectors. Stepping through n in this simulator is the same as stepping through the Fock states |n⟩ of a single mode of light.
Quantum computing and quantum information: In trapped-ion quantum computers the shared harmonic mode of the trapped ions serves as the data bus that mediates entanglement. Superconducting qubits rest on the quantization of an LC circuit, again a harmonic oscillator at its core, so understanding this model is the bridge to understanding quantum hardware.
Common Misconceptions and Cautions
The most common misconception is to think that the wave function ψ itself is a directly observable quantity. The actual observable is the probability density |ψ|², and the phase or sign of ψ has no standalone meaning (only relative phases become observable when interference takes place). In the simulator the green ψ swings between positive and negative, but the red |ψ|² is always non-negative — that is what represents "the probability of finding the particle there". Multiplying the wave function by −1 gives exactly the same physical state.
The next most common mistake is to picture the classical turning point x_c as a hard wall the particle can never cross. That is true classically, but quantum mechanically there is always exponential leakage. Look outside the dashed lines in the simulator: neither the green ψ nor the red |ψ|² is zero there — both decay rapidly but keep a tail. This leakage is the prototype of quantum tunneling that underlies α-decay, the scanning tunneling microscope, and many semiconductor devices. "Forbidden region" is a classical concept; in quantum mechanics the probability is small but never zero.
Finally, do not overuse the harmonic-oscillator model as an always-valid approximation. It is a second-order Taylor expansion around equilibrium, so when amplitudes grow the quartic and higher corrections (anharmonicity) become important. In real molecules the level spacing shrinks at high quantum number and the molecule eventually dissociates — captured by a Morse potential. The simulator is the pure harmonic limit; remember to add anharmonic corrections when modeling real systems.