I recently wrote a Python script for plotting wave functions for the Quantum Harmonic Oscillator. This is one of the few problems in quantum mechanics that has a “nice” solution which can be obtained exactly. In simple terms the quantum harmonic oscillator refers to a quantum particle that is confined by a potential whose strength is proportional to the square of the distance away from the equilibrium position (i.e. the position where there is no force from the potential acting on the particle). A concrete physical situation where this applies might be an electron that is in a uniform magnetic field. There are many other physical situations that can either be informed by or very well approximated by the harmonic oscillator model.
Below are some plots of the wavefunction for a particle in a harmonic oscillator potential. Here n denotes the “energy level” where n = 0 corresponds to the ground state of the system. The energy level takes on discrete (or “quantized” hence the name “quantum” mechanics) values. The energy of a particle at the n-th energy level is
where ħ stands for Planck’s constant and ω stands for the natural angular frequency of the oscillator (this is related to the proportionality constant for the potential described in the first paragraph). Interestingly the lowest possible energy for a particle confined to this potential (set n = 0 in this equation) is not zero! This means it possesses so-called “zero point” energy which is something pretty weird that differs from potentials in the macroscopic world.
A wavefunction is a special mathematical object that encodes information about the dynamics of a quantum system. In general wavefunctions are not directly measurable, and many view them as a mathematical tool we use to represent quantum systems. The wavefunction can be used to compute the probability that, for example, the position of the particle will be within a particular range of values. So without any further adieu, here are the plots:
This is the “ground state” wave function for the quantum harmonic oscillator. The shape of this plot is called a Gaussian. Note that these plots are done in so-called “natural units” (where c = ħ = 1) that simplify the calculations a bit.
This is the first excited state wave function, note that it has two “bumps.”
Just for fun, here is the 134th excited state. It’s quite pretty. Notice the larger bumps on either end. As n gets large, we expect the oscillations in the middle to sort of “wash out.” These two large bumps on the ends correspond to the classical turning points for the harmonic oscillator. From Wikipedia:
As the energy increases, the probability density becomes concentrated at the classical “turning points”, where the state’s energy coincides with the potential energy. This is consistent with the classical harmonic oscillator, in which the particle spends most of its time (and is therefore most likely to be found) at the turning points, where it is the slowest. The correspondence principle is thus satisfied.