Wave Basics: Wavelength and Period

Before quantum mechanics there were waves — and quantum mechanics turns out to be wave mechanics. Getting the vocabulary right now will pay off every time you encounter the Schrödinger equation, de Broglie wavelengths, or interference.

What a wave is

A travelling wave is a disturbance that moves through space at a definite speed without permanently transporting matter. The snapshot of the wave at any instant is periodic in space; the motion at any fixed point is periodic in time.

Two lengths fully characterise that periodicity:

| Symbol | Name | Definition | |--------|------|------------| | $\lambda$ | wavelength | spatial distance between successive identical phases (e.g. crest to crest), in metres | | $T$ | period | time for one complete oscillation at a fixed point, in seconds |

The corresponding rates are the wave number $k = 2\pi/\lambda$ (radians per metre) and the frequency $f = 1/T$ (cycles per second, or hertz).

The fundamental wave relation

A crest travels one wavelength $\lambda$ in exactly one period $T$ . Speed is distance divided by time, so

v = \frac{\lambda}{T} = \lambda f.

This is the single most useful equation in wave physics. For light in vacuum the speed is the universal constant $c = 2.998 \times 10^8\,\text{m/s}$ , giving

c = \lambda f \quad \Longrightarrow \quad \lambda = \frac{c}{f}.

Angular frequency and wave number

Physicists often prefer the angular frequency $\omega = 2\pi f$ (radians per second) and the wave number $k = 2\pi/\lambda$ (radians per metre). In these variables the wave relation becomes simply $\omega = k v$ , and a sinusoidal wave travelling in the $+x$ direction is written

y(x, t) = A\cos(kx - \omega t + \phi_0),

where $A$ is the amplitude and $\phi_0$ is an initial phase.

Why this matters for quantum mechanics

De Broglie (1924) proposed that every particle with momentum $p$ has an associated wavelength

\lambda = \frac{h}{p},

where $h = 6.626 \times 10^{-34}\,\text{J s}$ is Planck's constant. This one equation bridges the classical wave picture developed here and the quantum mechanical wave function developed in later modules. The smaller the wavelength (higher $k$ ), the higher the momentum — a direct consequence of $v = \lambda f$ applied to matter waves.

Try it

This is a numerical exercise — your code should return a number, not a circuit. Use the wave relation $\lambda = c/f$ to find the wavelength of a 100 MHz radio wave in metres. Take $c = 3.0 \times 10^8\,\text{m/s}$ .

Run your code to see the quantum state.