From Duality to the Wavefunction

The experiments of this module — electron diffraction, single-particle double-slit build-up, wave-packet dispersion — each forced the same conclusion: a particle cannot be described by a definite trajectory. Something else is needed, something that carries interference information, something whose squared magnitude gives a probability. That something is the wavefunction.

Why a trajectory is not enough

In classical mechanics, knowing a particle's position $x_0$ and momentum $p_0$ at one instant determines its position at every later instant. The trajectory $x(t)$ is the complete description.

But the double-slit experiment demolishes this picture. Suppose an electron passes through the apparatus while both slits are open. There is no detector at the slits, so we cannot say which one it went through. Yet when we collect many such electrons the interference pattern appears — bright and dark fringes whose spacing is set by the de Broglie wavelength $\lambda = h/p$ . If the electron had a definite trajectory it would have passed through one slit or the other, and the pattern on the screen would be the incoherent sum of two single-slit patterns: no fringes. Fringes exist, so no definite trajectory exists.

What interference demands

The double-slit analysis showed that arrival probabilities involve a cross-term. Labelling the contribution from slit 1 as $\psi_1$ and from slit 2 as $\psi_2$ , the probability density at a screen position $x$ is

$P(x) = |\psi_1(x) + \psi_2(x)|^2 = |\psi_1|^2 + |\psi_2|^2 + 2\operatorname{Re}\!\left(\psi_1^* \psi_2\right).$

The cross-term $2\operatorname{Re}(\psi_1^* \psi_2)$ is the interference term. For it to be nonzero and spatially oscillating, $\psi_1$ and $\psi_2$ must be complex-valued functions, not real probabilities. Real non-negative probabilities cannot produce cancellation at dark fringes; complex amplitudes can.

This is the key insight: probabilities cannot be the fundamental description. We need a complex-valued object — a probability amplitude — whose squared modulus gives the probability.

Introducing the wavefunction

The wavefunction $\psi(x, t)$ is a complex-valued function of position $x$ and time $t$ that encodes the quantum state of a particle moving in one dimension. Its physical content is given by the Born rule: the probability of finding the particle in the interval $[a, b]$ at time $t$ is

$P(a \le x \le b,\, t) = \int_a^b |\psi(x, t)|^2 \, dx.$

For this to be a valid probability distribution, the wavefunction must be normalised:

$\int_{-\infty}^{\infty} |\psi(x, t)|^2 \, dx = 1.$

The function $|\psi(x,t)|^2$ is called the probability density. It is real and non-negative, as a probability density must be. The wavefunction $\psi$ itself is complex, and it is $\psi$ — not $|\psi|^2$ — that obeys a wave equation and that adds when paths are superposed.

What the wavefunction carries

A free particle with a definite momentum $p$ and energy $E = p^2 / (2m)$ is described by a plane-wave wavefunction

$\psi(x, t) = A\, e^{i(kx - \omega t)},$

where $k = p/\hbar$ is the wave number, $\omega = E/\hbar$ is the angular frequency, and $\hbar = h/(2\pi)$ is the reduced Planck constant.

Two things are worth noting. First, the de Broglie relation $\lambda = h/p$ is already embedded here: the spatial period of the plane wave is $2\pi/k = 2\pi\hbar/p = h/p = \lambda$ . The wavefunction formalism does not contradict de Broglie — it contains it.

Second, the plane wave has $|\psi|^2 = |A|^2$ , a constant everywhere. A particle described by a plane wave has a perfectly definite momentum but is completely delocalised in space. To build a localised particle — a wave packet — we must superpose many plane waves with a spread of momenta. This trade-off between spatial localisation and momentum spread is the origin of the Heisenberg uncertainty principle, which the next module treats quantitatively.

Three open questions the wavefunction raises

Accepting the wavefunction as the fundamental description immediately raises three questions that the rest of this course addresses.

1. What equation governs the wavefunction? The wavefunction must evolve in a way consistent with energy conservation and the de Broglie relation. Schrodinger derived the equation that does this in 1926; it plays the role in quantum mechanics that Newton's second law plays in classical mechanics.

2. What happens when we measure? Before measurement, the particle has a spread of possible positions (wherever $|\psi|^2$ is nonzero). After measurement, we find it at one definite location. How does the spread of the wavefunction become a single outcome? This is the measurement problem, and it connects directly to the Born rule.

3. How does this extend to more particles? A single particle in one dimension has a wavefunction $\psi(x,t)$ . Two particles require $\psi(x_1, x_2, t)$ — a function on a higher-dimensional space. The wavefunction does not live in ordinary three-dimensional space; it lives in configuration space. For $N$ particles in three dimensions that is a $3N$ -dimensional space, and it is there that entanglement — quantum correlations with no classical analog — lives.

The thread connecting this module to the next

This module started with the experimental facts: electrons diffract, single particles build up interference patterns, wave packets spread. These facts demanded a mathematical description that carries phase information, adds coherently, and yields probabilities when squared. That description is the wavefunction.

The next module takes the wavefunction as given and develops its properties systematically: normalisation, expectation values, the Born rule in detail, and its time evolution under the Schrodinger equation. The experiments you have studied are not left behind — they become the interpretive foundation for every calculation that follows.