The quantum harmonic oscillator is one of the most important exactly solvable models in quantum mechanics. It describes a particle moving in the quadratic potential
\[
V(x)=\frac12 m\omega^2 x^2.
\]
Unlike the classical oscillator, which can have any energy, the quantum version has discrete allowed energies:
\[
E_n=\hbar\omega\left(n+\frac12\right),
\qquad n=0,1,2,\dots
\]
This means the system is quantized. Even the lowest state, the ground state \(n=0\), has nonzero energy:
\[
E_0=\frac12\hbar\omega.
\]
This is called the zero-point energy.
The corresponding stationary-state wavefunctions are
\[
\psi_n(x)=\frac{1}{\sqrt{2^n n!}}
\left(\frac{\alpha}{\pi}\right)^{1/4}
H_n(\sqrt{\alpha}\,x)e^{-\alpha x^2/2},
\qquad \alpha=\frac{m\omega}{\hbar},
\]
where \(H_n\) is the \(n\)-th Hermite polynomial. The Gaussian factor
\[
e^{-\alpha x^2/2}
\]
keeps the wavefunction localized, while the Hermite polynomial determines the oscillatory shape and the number of nodes.
For the sample input \(n=0\), \(\omega=1\), \(m=1\), and \(\hbar=1\), we have
\[
\alpha=\frac{m\omega}{\hbar}=1.
\]
Then the ground-state energy is
\[
E_0=\hbar\omega\left(0+\frac12\right)=\frac12.
\]
Since \(H_0=1\), the ground-state wavefunction simplifies to
\[
\psi_0(x)=\left(\frac{1}{\pi}\right)^{1/4}e^{-x^2/2}.
\]
This is a Gaussian centered at the origin. Its probability density is
\[
|\psi_0(x)|^2=\left(\frac{1}{\pi}\right)^{1/2}e^{-x^2},
\]
which shows that the particle is most likely to be found near \(x=0\).
As \(n\) increases, the energy increases linearly with \(n\), and the wavefunction develops more nodes. In fact, the \(n\)-th state has \(n\) nodes. These states alternate in parity: even \(n\) gives an even wavefunction, while odd \(n\) gives an odd wavefunction.
The quantity \(|\psi_n(x)|^2\) is the probability density. It is always nonnegative and tells how likely the particle is to be found near position \(x\). The wavefunction \(\psi_n(x)\) itself can be positive or negative, but the sign does not directly represent probability. Instead, it affects interference and nodal structure.
The quantum harmonic oscillator appears throughout physics: molecular vibrations, phonons in solids, quantum fields, and approximate motion near stable equilibrium points all use this model. It is central because it combines exact solvability with deep physical meaning.
This preview tool focuses on stationary states, their energy levels, Hermite-polynomial structure, and probability plots. It is meant to visualize the key ideas clearly, not to replace a full time-dependent quantum solver.
