What is a wave function and what does its probability density tell us about a particle?

In one line

The wave function \(\psi\) is a mathematical function that describes the quantum state of a particle, and \(|\psi|^2\) gives the probability per unit length (or volume) of finding the particle at a location.

Explained simply

A weather forecast does not tell you exactly where the rain will fall; it gives a "chance of rain" map. The wave function works like that for an electron. You cannot say exactly where the electron is, but \(|\psi|^2\) tells you how likely you are to find it at each place. Where \(|\psi|^2\) is large, the electron is often found.

How it works

1. For a particle, a wave function \(\Psi(x, t)\) is a solution of Schrödinger's equation. It is generally complex, so it cannot be measured directly.
2. Max Born's interpretation: \(P(x) = |\Psi|^2 = \Psi^*\Psi\) is the probability density (\(\Psi^*\) is the complex conjugate).
3. Probability of finding the particle between \(x\) and \(x + dx\) is \(|\Psi|^2dx\); in 3D, between volume \(dV\): \(|\Psi|^2dV\).
4. Probability over a region \(a\) to \(b\): \(\int_a^b|\Psi|^2dx\).
5. Since the particle must be somewhere, the total probability is 1: \(\int_{-\infty}^{\infty}|\Psi|^2dx = 1\). This is the normalisation condition.
6. If a found \(\psi\) gives a total of \(N\) instead of 1, divide it by \(\sqrt{N}\) to normalise it.

Key formula

\(P = |\Psi|^2 = \Psi^*\Psi\); \(\int_{-\infty}^{\infty}|\Psi|^2dx = 1\). In one dimension \(|\Psi|^2\) has unit m⁻¹; in 3D, m⁻³; the probability itself is dimensionless.

Quick example

For a particle in a box of length \(L\), \(\psi = A\sin(\pi x/L)\). Normalising: \(A^2\int_0^L\sin^2(\pi x/L)dx = A^2L/2 = 1\), so \(A = \sqrt{2/L}\). The probability density \(\frac{2}{L}\sin^2(\pi x/L)\) is maximum at \(x = L/2\), so the particle is most likely found at the centre and never at the walls.

Common mistakes

• Saying \(\psi\) itself is the probability; it is \(|\psi|^2\).
• Forgetting the complex conjugate when \(\psi\) is complex.
• Not normalising before calculating probabilities.

How MU asks it

Asked as "What is a wave function? State its physical significance" (3 marks) or "Explain probability density and normalisation" (3 to 5 marks). Define \(\psi\), give Born's interpretation \(|\psi|^2\), write the normalisation integral, and note that \(\psi\) itself has no direct physical meaning.