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

The harmonic oscillator is the model system of model systems. We study it here to characterize differences in the dynamical behavior predicted by classical and quantum mechanics, stressing concepts and results. More details and mathematical formalism can be found in textbooks [1,2]. Our model system is a single particle moving in the x dimension connected by a spring to a fixed point. Its potential energy is $V(x) = k x^2 / 2$, where k is the spring constant. Stiff springs are described by large k's.

Classically, this oscillator undergoes sinusoidal oscillation of amplitude $A = \sqrt{2 E / k}$ and frequency $f = (k/m)^{1/2}/(2 \pi)$, 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 $k_BT/2$; 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 $E_n$ and eigenfunctions $\Psi_n(x)$. For the one dimensional harmonic oscillator, the energies are found to be $E_n = (n+1/2) hf$, where $h$ is Planck's constant, f is the classical frequency of motion (above), and n may take on integer values from 0 to infinity. The $\Psi_n(x)$ turn out to be real functions involving the Hermite polynomials. From equation 1, only the ground state ($n = 0$) is populated as the temperature $T \rightarrow 0$. The energy does not go to zero but to $hf/2$. The corresponding zero-point motion is a quantum mechanical phenomenon. Classically, there is no motion as $T \rightarrow 0$. Thus, we expect that quantum mechanics predicts more motion than classical mechanics, especially at low temperature.