Probability Density |ψ|²

From wavefunction to probability

In quantum mechanics a particle in one dimension is described by its wavefunction $\psi(x, t)$ — a complex-valued function of position and time. The wavefunction itself is not directly measurable, but its squared modulus

is the probability density: the probability of finding the particle in a small interval $[x,\, x + dx]$ at time $t$ is

This is Max Born's 1926 interpretation of the wavefunction, now called the Born rule. Because $|\psi|^2$ is a probability density, integrating over all space must give 1:

A wavefunction that satisfies this condition is said to be normalized.

Computing $|\psi|^2$ for a complex amplitude

For a general complex number $z = a + bi$,

where $z^*$ is the complex conjugate. When $\psi$ is real-valued (as in many textbook examples) this simplifies to $|\psi|^2 = \psi^2$. The squared modulus is always real and non-negative, which is necessary for it to be a probability density.

Example: Gaussian wavefunction

One of the most important wavefunctions is the Gaussian (also called a coherent state in the position basis):

where $\sigma$ controls the spatial spread and $A$ is determined by normalization. Inserting into the normalization condition and using the Gaussian integral $\int_{-\infty}^{\infty} e^{-u^2/\sigma^2}\, du = \sigma\sqrt{\pi}$:

For the standard choice $\sigma = 1$ this gives $|A|^2 = 1/\sqrt{\pi}$. It is conventional to write the normalization constant as

because then $|A|^2 = 1/\sqrt{\pi}$ and one can verify directly: $\int_{-\infty}^{\infty} \pi^{-1/2} e^{-x^2}\, dx = \pi^{-1/2} \cdot \sqrt{\pi} = 1$.

The probability density is

which is a normalized Gaussian with peak $1/\sqrt{\pi} \approx 0.564$ at $x = 0$.

Key points

• The wavefunction $\psi(x)$ can be complex; only $|\psi|^2$ has direct physical meaning.
• Probability density $\rho = |\psi|^2$ is always real and non-negative.
• To find the probability in a finite region $[a, b]$, integrate the density: $P = \int_a^b |\psi|^2\, dx$.
• The Born rule is a postulate — it connects the mathematical formalism to measured outcomes.

Try it

This is a numerical exercise — your code should return a number.

For the Gaussian wavefunction with $\sigma = 1$ and normalization constant $A = \pi^{-1/4}$, compute the probability density at $x = 0$, that is $|\psi(0)|^2$.

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