Superposition of Waves

Classical wave superposition is the direct physical ancestor of quantum superposition. Before exploring quantum states it is worth understanding precisely how waves add.

What is a wave?

A transverse sinusoidal wave propagating in the $+x$ direction can be written

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

where $A$ is the amplitude (peak displacement), $k = 2\pi/\lambda$ is the wave number, $\omega = 2\pi f$ is the angular frequency, and $\phi$ is the initial phase.

The principle of superposition

When two or more waves occupy the same region of space, the principle of superposition states that the net displacement at any point is the algebraic sum of the individual displacements:

$y_{\text{net}}(x, t) = y_1(x, t) + y_2(x, t).$

This is an exact statement for linear media (including electromagnetic fields in vacuum). The waves do not alter one another — they pass through each other and the string (or field) responds to their combined effect.

Constructive and destructive interference

Consider two waves with equal frequency and wave number but possibly different amplitudes and phases:

$y_1 = A_1 \cos(kx - \omega t), \qquad y_2 = A_2 \cos(kx - \omega t + \delta),$

where $\delta$ is the phase difference. Adding them gives a single wave of the same frequency with a resultant amplitude $A_R$ that depends on $\delta$ .

Using the cosine addition formula one can show:

$A_R = \sqrt{A_1^2 + A_2^2 + 2A_1 A_2 \cos\delta}.$

Two special cases follow immediately:

Constructive interference ( $\delta = 0, \pm 2\pi, \pm 4\pi, \ldots$ ): The two crests (and troughs) align perfectly. $\cos\delta = 1$ , so

$A_R = \sqrt{(A_1 + A_2)^2} = A_1 + A_2.$

The resultant amplitude is the sum of the individual amplitudes — the waves reinforce each other.

Destructive interference ( $\delta = \pm\pi, \pm 3\pi, \ldots$ ): Crest meets trough. $\cos\delta = -1$ , so

$A_R = \sqrt{(A_1 - A_2)^2} = |A_1 - A_2|.$

When $A_1 = A_2$ the result is complete cancellation: $A_R = 0$ .

Why this matters for quantum mechanics

In quantum mechanics the wavefunction $\psi(x, t)$ obeys a linear equation (the Schrodinger equation), so the superposition principle applies there too. Quantum interference is the direct consequence of this linearity: probability amplitudes, not probabilities, add. The resulting constructive and destructive interference patterns are responsible for effects like the double-slit diffraction of electrons and the operation of quantum algorithms.

Try it

This is a numerical exercise — your code should return a number. Two waves with amplitudes $A_1 = 3.0\,\text{m}$ and $A_2 = 4.0\,\text{m}$ travel in phase ( $\delta = 0$ ). What is the resultant amplitude?

Run your code to see the quantum state.