Classical vs. Quantum Mechanics: The Harmonic Oscillator in One Dimension

**Classically**, this oscillator undergoes sinusoidal oscillation of amplitude
and frequency
, where *E* is the total energy, potential plus kinetic.
In equilibrium at temperture *T*, its average potential energy and kinetic energy are both equal to ;
they depend only on temperature, not on the motion's frequency.

**Quantum mechanically**, the *probability* of finding the particle at a given place is obtained from the solution of
Shrödinger's equation, yielding eigenvalues and eigenfunctions . For the one dimensional harmonic oscillator,
the energies are found to be
, where is Planck's constant, *f* is the classical frequency of motion (above), and *n* may take on integer values from
0 to infinity. The turn out to be real functions involving the *Hermite polynomials*. From equation 1, only the *ground state* () is populated as the temperature
. The energy does not go to zero but to .
The corresponding *zero-point motion* is a quantum mechanical phenomenon. Classically, there is no motion as
.
Thus, we expect that quantum mechanics predicts more motion than classical mechanics, especially at low temperature.

- Probability(x): Where is the oscillating particle?
- Average Energy
*U*and Heat Capacity - Mean-Square Fluctuation
- Overall Comparison - What does all this mean for simulation?