The Wavefunction ψ(x)

In classical mechanics a particle is fully described at any instant by its position $x$ and momentum $p$ . Quantum mechanics replaces this sharp picture with something fundamentally different: a wavefunction $\psi(x, t)$ .

What the wavefunction is

The position-space wavefunction $\psi(x, t)$ is a complex-valued function defined over all of space at each instant of time. For a single particle moving in one dimension it maps every position $x \in \mathbb{R}$ and every time $t$ to a complex number:

$\psi : \mathbb{R} \times \mathbb{R} \;\longrightarrow\; \mathbb{C}.$

Writing the complex value out explicitly,

$\psi(x, t) = A(x, t)\,e^{i\phi(x, t)},$

where $A(x, t) \ge 0$ is the modulus and $\phi(x, t)$ is the phase. Both pieces matter: the modulus governs measurement probabilities, while the phase governs interference.

What the wavefunction is not

$\psi(x, t)$ is not a classical wave in physical space. It is not a field of matter density or a vibrating medium. It is a mathematical object that encodes the complete quantum state of the particle. There is no classical analogue for it.

The normalisation condition

For the probability interpretation to be consistent, the wavefunction must be square-integrable and we require it to be normalised:

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

This condition says that the total probability of finding the particle somewhere is exactly one. A wavefunction that satisfies this at $t = 0$ continues to satisfy it for all $t$ provided the particle evolves under the Schrödinger equation — a fact that follows from probability current conservation.

Dimension and units

Because $|\psi(x)|^2$ must have units of probability per unit length (so that integrating over a length gives a dimensionless probability), $\psi(x)$ itself carries units of $\text{length}^{-1/2}$ . In three dimensions the same argument gives units of $\text{length}^{-3/2}$ .

The state as a vector

There is a deeper way to read the wavefunction: it is a vector in a Hilbert space. The set of all square-integrable functions on $\mathbb{R}$ forms an infinite-dimensional complex vector space $L^2(\mathbb{R})$ , equipped with the inner product

$\langle \phi | \psi \rangle = \int_{-\infty}^{\infty} \phi^*(x)\,\psi(x)\, dx.$

Normalisation then reads $\langle \psi | \psi \rangle = 1$ , exactly the same condition as for a unit vector. This vector-space perspective is what makes Dirac notation useful and what connects wave mechanics to the broader framework you will explore through this module.

Summary

| Symbol | Meaning | |--------|---------| | $\psi(x, t)$ | Complex-valued wavefunction (position-space representation) | | $|\psi(x,t)|^2$ | Probability density — to be explored in the next lesson | | $\displaystyle\int_{-\infty}^{\infty}|\psi|^2\,dx = 1$ | Normalisation condition | | $L^2(\mathbb{R})$ | Hilbert space of square-integrable functions |

In the next lesson you will use $|\psi|^2$ to compute the probability of finding a particle inside a specific region.