Oscillations and Frequency

Quantum mechanics was born from a puzzle about oscillating systems — the spectrum of light emitted by a hot blackbody. Before tackling that puzzle, it pays to have a firm grip on what an oscillation is and how to measure it.

Simple harmonic motion

A mass $m$ on a spring with spring constant $k$ obeys Hooke's law: the restoring force is $F = -kx$ , where $x$ is the displacement from equilibrium. Newton's second law then gives

$m\ddot{x} = -kx,$

whose general solution is

$x(t) = A\cos(\omega t + \varphi).$

Here $A$ is the amplitude (maximum displacement), $\varphi$ is the phase (determined by initial conditions), and $\omega$ is the angular frequency. Substituting back confirms that the equation is satisfied when

$\omega = \sqrt{\frac{k}{m}}.$

This quantity $\omega$ has units of radians per second. A larger spring constant (stiffer spring) or a smaller mass produces a higher angular frequency — the system oscillates faster.

Frequency and period

Two related quantities are more directly observable. The period $T$ is the time for one complete oscillation, and the frequency $f$ is the number of complete oscillations per second:

$f = \frac{1}{T}, \qquad \omega = 2\pi f.$

Frequency is measured in hertz (Hz): $1\,\text{Hz} = 1\,\text{s}^{-1}$ . The factor of $2\pi$ converts from full cycles to radians. For the spring–mass system,

$f = \frac{1}{2\pi}\sqrt{\frac{k}{m}},$

a result that makes physical sense in both limits: a stiffer spring drives the mass back to equilibrium faster, raising $f$ ; a heavier mass has more inertia and responds more slowly, lowering $f$ .

Energy in an oscillator

The total mechanical energy of a simple harmonic oscillator is conserved and equals the maximum potential energy stored in the spring at maximum displacement:

$E = \tfrac{1}{2}kA^2.$

This energy sloshes back and forth between kinetic energy ( $\tfrac{1}{2}mv^2$ ) and potential energy ( $\tfrac{1}{2}kx^2$ ) but their sum stays constant. The frequency $\omega$ does not appear in $E$ — the energy depends on the amplitude, not on how fast the system oscillates.

Why this matters for quantum mechanics

The harmonic oscillator is not just a classical curiosity. When Max Planck resolved the blackbody problem in 1900, he modelled the atoms in a hot object as a collection of oscillators, each with a definite frequency $f$ . He was forced to assume that the energy of each oscillator could only take discrete values $E_n = nhf$ (with $n = 0, 1, 2, \ldots$ ) rather than varying continuously. This quantisation hypothesis — with its explicit reference to frequency — was the first foothold of quantum mechanics.

Later, Einstein showed that the electromagnetic field itself can be treated as a collection of oscillators. Each normal mode of frequency $f$ carries energy in packets of size $hf$ , which we now call photons. The concept of frequency therefore bridges the classical world of springs and pendulums to the quantum world of photons and energy levels; it will reappear in every topic in this module.